CA Geometry: Triangles and parallelograms

WEBVTT
Kind: captions
Language: en

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We're on problem 21.

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In the figure below, n
is a whole number.

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All right.

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What is the smallest possible
value for n?

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OK, both of these sides are n.

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And so this is something that's
actually really good to

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get this intuition.

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Because this shows up on all
sorts of standardized tests.

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And so let's think about how
small can we make n.

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So the lower the top of
this pyramid becomes,

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the smaller n becomes.

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If we pushed this top of the
pyramid really high then n

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would have to be really big.

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For example, if we made the
triangle into that.

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Then clearly, this length is
shorter than that length.

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We want to keep lowering it to
get as small a possible n.

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But what happens if we lowered
it all the way?

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If we flattened this triangle.

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If we just flattened it
all the way down.

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So essentially this would be the
top of it and this would

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be n and this would be n.

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I hope you're visualizing
that properly.

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I've flattened the triangle.

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So these two sides would just
go flat with the base.

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And so this top if I
were to do it in

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magenta, is right here.

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So this is as small
as n could get.

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One could argue whether this is
a triangle at all anymore.

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It's really a line now, because
I've squished out all

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of the area in there.

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But even in this case, n would
have to be, in the smallest

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case, it would be 7.5.

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Each n would be half
of this 15.

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So, as we push this base down,
that's kind of the limit that

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n approaches.

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n cannot be any smaller
than 7.5.

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And they tell us that
n is a whole number.

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So n has to be greater
than 7.5 in order for

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this to be a triangle.

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And n is a whole number,
so n is equal to 8.

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That's choice C.

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And that's an important
intuition to have in general.

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That the third side of a
triangle can never be bigger

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than two of the other
sides combined.

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Then you're dealing with
something else.

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You're not dealing
with a triangle.

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Even if the third side is equal
to the other two sides,

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then you're actually dealing
with a line.

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Because you'd have to squish
out all of the area of the

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triangle in order
to get there.

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Anyway, I like that problem.

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Next question.

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I think, just eyeballing
it, they want us

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to do the same thing.

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Same type of intuition.

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And I had my rant in the last
video about how they weren't

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doing problems that give
you intuition or

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that test your intuition.

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But I'll take that back, because
I think that's what

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they are testing now.

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Which of the following sets of
numbers could represent the

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lengths of the sides
of a triangle.

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2, and 2, and 5.

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So this is the same
thing again.

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How can I have two sides of
a triangle combined being

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shorter than the third side.

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If I had a side of length 2, and
then I had another side of

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length 2, there's no way that
this last side could be 5.

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Even if I completely flattened
this triangle, 2 and 2, the

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longest that this last
side, this third

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side, could be is 4.

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So that can't be a triangle.

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Same here, let's look
at the other ones.

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3, 3, and 5.

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That's no reason why that
can't be a triangle.

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3 and 3 is 6.

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So even if I flattened it out
a lot, then this side can be

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as long as 6.

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And obviously I could squeeze
them together like this.

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3 and 3, and then this little
third side could be something

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really small.

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Because it's anything
in between 0 and 6.

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So obviously it can be 5.

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So that's going to
be the answer.

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4, 4 and 8.

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Same problem.

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I would have to squish
the triangle so much.

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If those are two of the sides,
4 and 4, this last side is

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still going to be less than 8.

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The only way to get it to 8 is
if I push the top of this

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triangle all the way down and
essentially make it a line.

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Once again, one side of a
triangle can't be bigger than

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two of the other
sides combined.

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In fact, it can't be
equal to two of

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the other sides combined.

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Then we're dealing
with a line.

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So they're really testing
that same intuition.

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And the choice is B.

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23.

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Copy and paste this.

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OK.

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I try to avoid talking when
I'm copying and pasting

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because I think it slows
down my computer.

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In the accompanying diagram,
parallel lines L and M are cut

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by transversal T.

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So it's a classic parallel
line with

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a transversal problem.

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And they're parallel.

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That's why I did those arrows.

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Which statements about angle
1 and 2 must be true?

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I don't know if you've seen the
Khan Academy videos of the

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angle game, but that's what
we're going to play here, the

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angle game.

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So angle 1, if you want to
look at its corresponding

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angle, its corresponding angle
on the other parallel line, or

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with the transversal and
the other parallel

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line, is right there.

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And they're going
to be congruent.

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Those two angles are going
to be congruent.

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So you could say this is equal
to the measure of angle 1.

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I've picked up their terminology
well, I think.

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So this is equal to the
measure of angle 1.

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And this is obviously angle 2.

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You see immediately that they
have to be supplementary.

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Because when you add them
together you get 180 degrees.

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Together they go all the
way around and they

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kind of form a line.

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So you know that if this angle
and this angle are

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supplementary.

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And this angle is congruent to
angle 1, then angle 1 and

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angle 2 must be supplementary.

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So what do they say?

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Angle 1 is definitely
not necessarily

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congruent to angle 2.

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It's congruent to
this angle here.

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Angle 1 is a complement
of angle 2.

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Complement means you
add up to 90.

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No, we're talking about
supplement.

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So it's not that.

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Angle 1 is a supplement
of angle 2.

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There you go.

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And there's nothing that says
that they're right angles,

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that's silly.

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All right, next problem.

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Let me copy and paste it.

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OK.

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And paste it.

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Ready to go.

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What values-- let me pick a good
color-- what values of A

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and B make the quadrilateral
MNOP a parallelogram.

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For this to be a parallelogram,
the opposite

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sides have to be equal.

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And I challenge you to
experiment to draw a

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parallelogram where opposite
sides are parallel where the

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opposite sides are
also not equal.

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If you make two of the sides not
equal, then the other two

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lines aren't going to
be parallel anymore.

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And you can play with
that if you like.

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But if opposite sides are going
to be equal that means

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4A plus B is equal to 21.

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Because they're opposite sides,
so they should be equal

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to each other.

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Similarly, 3A minus 2B should be
equal to 13 because they're

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opposite sides.

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So 3A minus 2B is equal to 13.

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And now we have two linear
equations with two unknowns.

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So this is really an Algebra
1 problem in disguise.

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So let's see, they want
us to solve for both.

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Let's see if we can
cancel out B.

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So let's multiply this
top equation by 2.

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And I'm doing that to
cancel out the B's.

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So you get 8A plus 2B
is equal to 42.

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And I did that so that the 2B
and the minus 2B cancel out.

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So let's add these two
equations to each.

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The left-hand side,
3A plus 8A is 11A.

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The B's cancel out; minus 2B,
plus 2B, that's no B's.

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Is equal to 42 plus 13 is 55.

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That worked out well.

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Divide both sides by 11, you
get A is equal to 5.

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Now if A is equal
to 5, what's B?

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Let's substitute back into
that first equation.

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Because you picked either.

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So 4 times 5 plus B
is equal to 21.

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20 plus B is equal to 21.

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So subtract 20 from both
sides, B is equal to 1.

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So A is 5, B is 1.

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That is choice B.

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Problem 25.

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I think I have to
clear this now.

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Quadrilateral ABCD is
a parallelogram.

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If adjacent angles are
congruent, which statement

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must be true?

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All right.

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So let me draw a
parallelogram.

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Well, I'll do it quick
and dirty.

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So a parallelogram says that
opposite sides are parallel.

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That's parallel to that and
that's parallel to that.

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But they give us another
statement.

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They say if the adjacent
sides are congruent.

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So they're saying that this
is congruent to that.

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They are saying if adjacent
angles are congruent.

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So I don't know if they're
saying it's just one of them

00:09:06.200 --> 00:09:06.660
or all of them.

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But it's the same
thing actually.

00:09:07.590 --> 00:09:09.970
If that angle is congruent
to that angle, something

00:09:09.970 --> 00:09:11.840
interesting has to happen.

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They both have to
be 90 degrees.

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And I want you to think about
that a little bit.

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So let me just draw the kind
of bottom part of that

00:09:17.350 --> 00:09:18.600
parellelogram.

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And I drew it more so if you say
that this line right here

00:09:30.450 --> 00:09:31.700
is a transversal.

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And that since it's a
parallelogram, we know that

00:09:37.370 --> 00:09:39.155
this line is parallel
to that line.

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So we have a transversal
between parallel lines.

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This angle is congruent
to that angle, they're

00:09:50.310 --> 00:09:51.810
corresponding angles.

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And this angle is a supplement
to this angle.

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They have to add up to 180.

00:09:57.620 --> 00:10:00.910
So this angle, this red angle
and this brown angle have to

00:10:00.910 --> 00:10:01.770
be supplements.

00:10:01.770 --> 00:10:04.760
Or if we said angle
1 and angle 2.

00:10:04.760 --> 00:10:07.200
They have to add up to 180.

00:10:07.200 --> 00:10:11.480
Measure of angle 1 plus measure
of angle 2 have to be

00:10:11.480 --> 00:10:13.060
equal to 180.

00:10:13.060 --> 00:10:14.020
But then they tell
us even more.

00:10:14.020 --> 00:10:15.880
They tell you that they're
congruent.

00:10:15.880 --> 00:10:17.470
These are adjacent
angles, right?

00:10:17.470 --> 00:10:18.950
That's angle 1 and
that's angle 2.

00:10:18.950 --> 00:10:19.910
They're adjacent.

00:10:19.910 --> 00:10:23.320
So if both of these are
congruent, if their measures

00:10:23.320 --> 00:10:28.310
equal each other, they both
have to be equal to 90.

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So if adjacent angles
are congruent in the

00:10:31.290 --> 00:10:33.430
parallelogram, then these
angles are 90.

00:10:33.430 --> 00:10:35.760
And if these angles are 90,
those are going to be 90 by

00:10:35.760 --> 00:10:36.990
the same argument.

00:10:36.990 --> 00:10:39.430
So now we're not dealing with
just a parallelogram.

00:10:39.430 --> 00:10:41.815
Well, I did want to draw
it all filled in.

00:10:41.815 --> 00:10:44.170
It looks tacky.

00:10:44.170 --> 00:10:46.330
It's a rectangle.

00:10:46.330 --> 00:10:48.070
They say quadrilateral
ABCD is a square.

00:10:48.070 --> 00:10:50.550
It could be a square, but in
order for it to be a square

00:10:50.550 --> 00:10:52.000
they'd have to tell us that
all the sides are equal to

00:10:52.000 --> 00:10:53.040
each other.

00:10:53.040 --> 00:10:55.510
So it's not A.

00:10:55.510 --> 00:10:57.350
That could happen, it
doesn't have to be.

00:10:57.350 --> 00:10:59.230
ABCD is a rhombus.

00:10:59.230 --> 00:11:01.050
No, in a rhombus all the
sides have to be

00:11:01.050 --> 00:11:02.330
equal to each other.

00:11:02.330 --> 00:11:03.610
They didn't tell us that.

00:11:03.610 --> 00:11:04.780
ABCD is a rectangle.

00:11:04.780 --> 00:11:05.330
Sure.

00:11:05.330 --> 00:11:09.280
Because we know all the angles
now are 90 degrees.

00:11:09.280 --> 00:11:11.180
And that is choice C.

00:11:11.180 --> 00:11:12.430
See you in the next video.


CA Geometry: Area, pythagorean theorem

WEBVTT
Kind: captions
Language: en

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All right.

00:00:01.270 --> 00:00:02.860
We're on problem 26.

00:00:02.860 --> 00:00:06.250
For the quadrilateral shown
below, a quadrilateral has

00:00:06.250 --> 00:00:10.120
four sides, measure of angle A
plus the measure of angle C is

00:00:10.120 --> 00:00:11.310
equal to what?

00:00:11.310 --> 00:00:14.770
And here, you should know that
the sum of all the angles in a

00:00:14.770 --> 00:00:17.890
quadrilateral are equal
to 360 degrees.

00:00:17.890 --> 00:00:22.410
And you might say, OK, I'll add
that to my memory bank of

00:00:22.410 --> 00:00:23.290
things to memorize.

00:00:23.290 --> 00:00:25.340
Like the angles in a triangle
are equal to 180.

00:00:25.340 --> 00:00:27.110
And I'll show you no, you don't
have to memorize that.

00:00:27.110 --> 00:00:30.010
Because if you imagine any
quadrilateral, let me draw a

00:00:30.010 --> 00:00:31.960
quadrilateral for you.

00:00:31.960 --> 00:00:35.970
And this is true
of any polygon.

00:00:35.970 --> 00:00:38.770
So let's say this is
some quadrilateral.

00:00:38.770 --> 00:00:41.660
You don't have to memorize that
the sum of the angles is

00:00:41.660 --> 00:00:42.270
equal to 360.

00:00:42.270 --> 00:00:43.880
Although it might be useful
for a quadrilateral.

00:00:43.880 --> 00:00:45.040
But I'll show you how to always

00:00:45.040 --> 00:00:46.630
prove it for any polygon.

00:00:46.630 --> 00:00:49.160
You just break it up
into triangles.

00:00:49.160 --> 00:00:50.770
Then you only have to
memorize one thing.

00:00:50.770 --> 00:00:57.660
If you break it up into
triangles, this angle plus

00:00:57.660 --> 00:01:02.080
that angle plus that angle
has to be equal to 180.

00:01:02.080 --> 00:01:06.480
And this angle plus that angle
plus that angle have to be

00:01:06.480 --> 00:01:08.710
equal to 180.

00:01:08.710 --> 00:01:11.630
So the angles in the
quadrilateral itself are this

00:01:11.630 --> 00:01:13.970
angle and this angle.

00:01:13.970 --> 00:01:17.550
And then this angle
and this angle.

00:01:17.550 --> 00:01:19.680
Well this one is just the sum
of those two, and this one's

00:01:19.680 --> 00:01:21.180
just the sum of those two.

00:01:21.180 --> 00:01:23.340
So if these three
added up to 180.

00:01:23.340 --> 00:01:25.150
And these three added
up to 180.

00:01:25.150 --> 00:01:29.870
This plus this plus this, plus
this will add up to 360.

00:01:29.870 --> 00:01:32.150
And you can do that with an
arbitrarily shaped polygon.

00:01:32.150 --> 00:01:34.760
Let's do five sides, let's
do a pentagon.

00:01:34.760 --> 00:01:40.290
So one, two, three,
four, five sides.

00:01:40.290 --> 00:01:42.230
Wow, how many angles are
there in a pentagon.

00:01:42.230 --> 00:01:44.180
Just break it up
into triangles.

00:01:44.180 --> 00:01:45.760
How many triangles can
you fit in it?

00:01:45.760 --> 00:01:46.000
Let's see.

00:01:46.000 --> 00:01:49.060
One, two.

00:01:49.060 --> 00:01:52.670
Each of these triangles, their
angles, they add up to 180.

00:01:52.670 --> 00:01:59.970
So if you want to know that,
that, that, plus that, that,

00:01:59.970 --> 00:02:02.560
that, plus that,
that, and that.

00:02:02.560 --> 00:02:06.490
That would just be 180 times
3, which is 540.

00:02:06.490 --> 00:02:09.080
And that also would be the angle
measures of the polygon.

00:02:09.080 --> 00:02:11.440
Because these three angles
add up to that angle.

00:02:11.440 --> 00:02:12.150
That's that.

00:02:12.150 --> 00:02:13.940
Those angles add
up to that one.

00:02:13.940 --> 00:02:17.300
Those angles add up to that one,
and those angles add up

00:02:17.300 --> 00:02:17.960
to that one.

00:02:17.960 --> 00:02:21.670
So now hopefully, if I gave you
a 20 sided polygon, you

00:02:21.670 --> 00:02:24.030
can figure out how many times
can I fit triangles into it.

00:02:24.030 --> 00:02:25.670
And you'll know how many
angles there are.

00:02:25.670 --> 00:02:26.910
And the sum of all of them.

00:02:26.910 --> 00:02:29.980
But anyway, back to
the quadrilateral.

00:02:29.980 --> 00:02:31.610
A quadrilateral, the sum
of the angles are

00:02:31.610 --> 00:02:33.530
going to be 360 degrees.

00:02:33.530 --> 00:02:38.540
So, if we say, measure of angle
A, plus measure of angle

00:02:38.540 --> 00:02:42.530
C, plus these two angles.

00:02:42.530 --> 00:02:43.130
Let me write it down.

00:02:43.130 --> 00:02:48.940
Plus 95 plus 32 is going
to be equal to 360.

00:02:48.940 --> 00:02:52.620
So I'll just write A plus C,
just a quick notation.

00:02:52.620 --> 00:02:56.060
Let's see, 95 plus 32 is 127.

00:02:56.060 --> 00:03:00.020
Plus 127 is equal to 360.

00:03:00.020 --> 00:03:05.062
A plus C is equal to
360 minus 127.

00:03:05.062 --> 00:03:06.070
And what is that?

00:03:06.070 --> 00:03:08.680
That is 233.

00:03:08.680 --> 00:03:11.190
Right, and that's the choice.

00:03:11.190 --> 00:03:12.520
Fair enough.

00:03:12.520 --> 00:03:14.370
Question 27.

00:03:14.370 --> 00:03:18.330
If ABCD is a parallelogram,
that's the sides are parallel,

00:03:18.330 --> 00:03:21.810
what is the length
of segment BD?

00:03:21.810 --> 00:03:23.960
So they want from
here to here.

00:03:23.960 --> 00:03:25.730
And this is just another
interesting thing, I'm not

00:03:25.730 --> 00:03:28.320
going to prove it right now,
but this is a good thing to

00:03:28.320 --> 00:03:30.350
know, especially if you
become a mathlete.

00:03:30.350 --> 00:03:32.010
Because it shows up in
math competitions

00:03:32.010 --> 00:03:33.340
every now and then.

00:03:33.340 --> 00:03:36.020
If you have a parallelogram,
the opposite sides are

00:03:36.020 --> 00:03:39.070
parallel, then their diagonals
are actually

00:03:39.070 --> 00:03:40.640
bisecting each other.

00:03:40.640 --> 00:03:42.950
Which means that they split
the other diagonal in two.

00:03:42.950 --> 00:03:45.160
So this diagonal splits
this diagonal in two.

00:03:45.160 --> 00:03:48.180
So if this is 6, this is
also going to be 6.

00:03:48.180 --> 00:03:50.570
And this diagonal splits
BD in two.

00:03:50.570 --> 00:03:54.010
So if this is 5, then
this is also 5.

00:03:54.010 --> 00:03:58.390
So BE is 5, ED is 5, then
BD has to be 10.

00:03:58.390 --> 00:03:59.640
Choice A.

00:04:02.040 --> 00:04:15.950
Let me copy and paste
28 in here.

00:04:15.950 --> 00:04:19.110
A right circular cone
has radius 5 inches

00:04:19.110 --> 00:04:21.089
and height 8 inches.

00:04:21.089 --> 00:04:23.620
Fair enough, they've
drawn it for us.

00:04:23.620 --> 00:04:27.160
What is the lateral
area of the cone?

00:04:27.160 --> 00:04:28.370
Good, they gave us
a definition.

00:04:28.370 --> 00:04:34.720
Lateral area of a cone is equal
to pi times r times l,

00:04:34.720 --> 00:04:36.730
where l is the slant height.

00:04:36.730 --> 00:04:41.500
So we know what r is, they
give us r. r is 5.

00:04:41.500 --> 00:04:43.780
So we just have to figure
out what the slant

00:04:43.780 --> 00:04:45.330
height is, this l.

00:04:45.330 --> 00:04:48.430
Well this looks like a
Pythagorean theorem problem.

00:04:48.430 --> 00:04:50.280
This is a right angle, I know
it's all weird because it's

00:04:50.280 --> 00:04:51.040
three dimensions.

00:04:51.040 --> 00:04:54.170
But this forms a
right triangle.

00:04:54.170 --> 00:04:56.360
We're just kind of picking one
slice of the cone that

00:04:56.360 --> 00:04:59.630
includes the pointy part.

00:04:59.630 --> 00:05:09.500
We say 5 squared, plus 8 squared
is equal to l squared.

00:05:09.500 --> 00:05:11.960
This is a right angle,
l is the hypotenuse.

00:05:11.960 --> 00:05:17.610
So we get 25 plus 64 is
equal to l squared.

00:05:17.610 --> 00:05:26.820
So that's 89 is equal
to l squared.

00:05:26.820 --> 00:05:30.760
And so, l is equal to the
square root of 89.

00:05:30.760 --> 00:05:33.140
Unless, I've made a
mistake someplace.

00:05:33.140 --> 00:05:36.160
Square root of 89.

00:05:36.160 --> 00:05:38.170
Oh good, I see a square root
of 89 there already.

00:05:38.170 --> 00:05:40.130
So we probably are on
the right track.

00:05:40.130 --> 00:05:41.430
So l is equal to the
square root of 89.

00:05:41.430 --> 00:05:44.650
And they give us the formula
for the lateral area of a

00:05:44.650 --> 00:05:46.530
cone as pi r l.

00:05:46.530 --> 00:05:53.120
So pi r l is equal to pi times
r, the radius of the base,

00:05:53.120 --> 00:05:54.760
which is 5.

00:05:54.760 --> 00:06:01.255
Times this slant height, which
is the square root of 89.

00:06:01.255 --> 00:06:04.730
This equals 5 pi times the
square root of 89.

00:06:04.730 --> 00:06:08.690
Which is, I just peeked
and saw, choice D.

00:06:08.690 --> 00:06:11.510
Whenever you see a number like
89, you begin to get worried.

00:06:11.510 --> 00:06:15.070
But it's good that that was
one of the choices.

00:06:15.070 --> 00:06:18.150
Problem 29.

00:06:18.150 --> 00:06:19.285
OK.

00:06:19.285 --> 00:06:20.870
Let me copy and paste it.

00:06:24.460 --> 00:06:25.710
Clear this image.

00:06:28.885 --> 00:06:30.760
It's early on a Saturday
morning, my

00:06:30.760 --> 00:06:33.310
wife is still sleeping.

00:06:33.310 --> 00:06:36.370
We're expecting our first
child in a month.

00:06:36.370 --> 00:06:39.560
So I figure the sleep
is good for her.

00:06:39.560 --> 00:06:42.140
Gives me more time to
record math videos.

00:06:42.140 --> 00:06:45.480
OK, I don't know why I go
onto those tangents.

00:06:45.480 --> 00:06:48.140
OK, figure ABCD is a kite.

00:06:48.140 --> 00:06:50.440
And it looks like a kite.

00:06:50.440 --> 00:06:56.350
What is the area of figure ABCD
in square centimeters.

00:06:56.350 --> 00:06:58.540
Well everything they're giving
us is in centimeters.

00:06:58.540 --> 00:07:00.030
So if we just stay
in centimeters we

00:07:00.030 --> 00:07:00.590
won't have a problem.

00:07:00.590 --> 00:07:02.520
So what's the area of this?

00:07:02.520 --> 00:07:06.730
So we just figure out the area
of each of these triangles.

00:07:06.730 --> 00:07:08.170
And what's the area
of a triangle?

00:07:08.170 --> 00:07:09.860
The area of a triangle
is equal to

00:07:09.860 --> 00:07:14.540
1/2 base times height.

00:07:14.540 --> 00:07:16.880
So what's the area
of this triangle?

00:07:16.880 --> 00:07:18.380
Well, actually, this
is symmetric.

00:07:18.380 --> 00:07:20.310
If we know the area of this
triangle, we know the area of

00:07:20.310 --> 00:07:20.970
this triangle.

00:07:20.970 --> 00:07:23.550
Because this is 6 and
8, this is 6 and 8.

00:07:23.550 --> 00:07:26.570
So the area of this one
is 6 times 8 is 48.

00:07:26.570 --> 00:07:32.450
48 times 1/2 is 24.

00:07:32.450 --> 00:07:34.950
This one is also going to be
24 by the same argument.

00:07:34.950 --> 00:07:37.580
So when you add them together,
you get 48.

00:07:37.580 --> 00:07:41.150
Those two combined are
going to be 48.

00:07:41.150 --> 00:07:45.330
Now, this triangle, 8
times 15 times 1/2.

00:07:45.330 --> 00:07:48.120
That's 4 times 15, which
is equal to 60.

00:07:48.120 --> 00:07:49.120
And this is going
to have the same

00:07:49.120 --> 00:07:50.960
area by the same argument.

00:07:50.960 --> 00:07:51.460
60.

00:07:51.460 --> 00:07:53.410
We don't even have to multiply
by 1/2, because we're going to

00:07:53.410 --> 00:07:55.070
multiply by 2 eventually.

00:07:55.070 --> 00:07:57.090
Or add it to each other again.

00:07:57.090 --> 00:08:00.840
So anyway, we have 60 plus 60
plus 24 plus 24, that's 120

00:08:00.840 --> 00:08:04.470
plus 48, so 168.

00:08:04.470 --> 00:08:07.660
Choice C.

00:08:07.660 --> 00:08:10.030
Next problem.

00:08:10.030 --> 00:08:11.750
Problem 31.

00:08:11.750 --> 00:08:15.500
I like these problems. Now that
we're out of the whole

00:08:15.500 --> 00:08:18.490
part that they were
getting into

00:08:18.490 --> 00:08:20.606
congruencies and similars.

00:08:20.606 --> 00:08:21.830
And I thought they
made a couple of

00:08:21.830 --> 00:08:22.920
mistakes on some of those.

00:08:22.920 --> 00:08:27.210
Anyway, if a cylindrical barrel
measures 22 inches in

00:08:27.210 --> 00:08:32.390
diameter, how many inches will
it roll in 8 revolutions along

00:08:32.390 --> 00:08:34.320
a smooth surface?

00:08:34.320 --> 00:08:37.679
So we could imagine a wheel.

00:08:37.679 --> 00:08:39.740
It's a tire of some kind.

00:08:39.740 --> 00:08:41.309
So let me draw a circle.

00:08:41.309 --> 00:08:43.440
So if we look at the cylindrical
barrel from the

00:08:43.440 --> 00:08:46.440
side, because I think that's
all we care about.

00:08:46.440 --> 00:08:48.390
That's its side.

00:08:48.390 --> 00:08:50.190
They say, a cylindrical
barrel measures

00:08:50.190 --> 00:08:51.700
22 inches in diameter.

00:08:51.700 --> 00:08:54.090
So this distance right here.

00:08:57.650 --> 00:09:02.260
That distance right
there is 22.

00:09:02.260 --> 00:09:05.450
And what they say is, this
thing is going to roll 8

00:09:05.450 --> 00:09:06.670
revolutions on a
smooth surface.

00:09:06.670 --> 00:09:07.980
It's going to go
around 8 times.

00:09:07.980 --> 00:09:11.220
It's going to roll and
move to the right.

00:09:11.220 --> 00:09:13.090
So how long will it roll?

00:09:13.090 --> 00:09:15.070
So if you think about it,
it's going to cover its

00:09:15.070 --> 00:09:18.080
circumference 8 times.

00:09:18.080 --> 00:09:22.460
If this point is starting
touching the ground, after it

00:09:22.460 --> 00:09:24.610
moves a circumference of
distance, that point will be

00:09:24.610 --> 00:09:27.040
touching the ground again.

00:09:27.040 --> 00:09:29.590
An easy way to think about it
is, as this thing moves to the

00:09:29.590 --> 00:09:35.240
right, as it rolls, when it
moves 1 foot, 1 foot along of

00:09:35.240 --> 00:09:37.880
circumference will then be
touching the ground.

00:09:37.880 --> 00:09:41.770
Or 1 cm, or 2 inches
or whatever.

00:09:41.770 --> 00:09:45.080
Then 2 inches along its
circumference will be touching

00:09:45.080 --> 00:09:46.470
the ground.

00:09:46.470 --> 00:09:48.860
So it's going to go 8
circumferences in 8

00:09:48.860 --> 00:09:50.070
revolutions.

00:09:50.070 --> 00:09:51.370
So what's the circumference
of this?

00:09:51.370 --> 00:09:55.700
Circumference is equal to
pi times the diameter.

00:09:55.700 --> 00:09:57.410
The diameter they already
gave us is 22.

00:09:57.410 --> 00:10:00.360
So the circumference
is equal to 22 pi.

00:10:00.360 --> 00:10:02.630
So it's going to move 8
circumferences in 8

00:10:02.630 --> 00:10:03.830
revolutions.

00:10:03.830 --> 00:10:12.970
So 22 pi times 8 is 176 pi.

00:10:12.970 --> 00:10:15.570
And that's choice C.

00:10:15.570 --> 00:10:17.450
See you in the next video.


CA Geometry: More on congruent and similar triangles

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:01.200
All right.

00:00:01.200 --> 00:00:02.890
We're on problem 17.

00:00:02.890 --> 00:00:05.450
It says, which of the following
best describes the

00:00:05.450 --> 00:00:07.505
triangles shown below?

00:00:10.810 --> 00:00:11.955
OK, they want to know,
are the similar?

00:00:11.955 --> 00:00:12.550
Are they congruent?

00:00:12.550 --> 00:00:13.800
Et cetera.

00:00:15.740 --> 00:00:17.590
They tell us that this
is a 60 degree angle.

00:00:17.590 --> 00:00:20.190
This is a 90 degree angle, they
do this little square

00:00:20.190 --> 00:00:22.390
thing there.

00:00:22.390 --> 00:00:25.390
The angles in a triangle
have to add up to 180.

00:00:25.390 --> 00:00:29.190
So if this is 90, this is
60, that adds up to 150.

00:00:29.190 --> 00:00:31.420
So this has to be
180 minus 150.

00:00:31.420 --> 00:00:34.080
So this has to be a
30 degree angle.

00:00:34.080 --> 00:00:37.460
So that has to be 30 degrees
right there.

00:00:37.460 --> 00:00:38.200
Fair enough.

00:00:38.200 --> 00:00:39.460
Now let's get this one.

00:00:39.460 --> 00:00:41.280
That's 30, that's 90.

00:00:41.280 --> 00:00:43.450
Well, by the same argument,
this has got to be 60.

00:00:43.450 --> 00:00:45.960
Because they all have
to add up to 180.

00:00:45.960 --> 00:00:46.430
All right.

00:00:46.430 --> 00:00:49.770
So just there we know that all
of the angles in both of the

00:00:49.770 --> 00:00:52.400
triangles are congruent.

00:00:52.400 --> 00:00:54.320
Or that their measures
are equal.

00:00:54.320 --> 00:00:56.230
So we know already that
these are definitely

00:00:56.230 --> 00:00:57.480
both similar triangles.

00:01:01.970 --> 00:01:06.300
Now, a similar triangle also
tells us that the ratio of all

00:01:06.300 --> 00:01:07.600
of the sides are equal.

00:01:07.600 --> 00:01:08.450
So.

00:01:08.450 --> 00:01:10.530
if you just sort of eyeball it,
if you said, OK, the side

00:01:10.530 --> 00:01:13.100
opposite the 90 degree, these
are the corresponding sides,

00:01:13.100 --> 00:01:14.170
the ratios are equal.

00:01:14.170 --> 00:01:16.080
But we see that they give
us the actual lengths.

00:01:16.080 --> 00:01:19.290
So the hypotenuse of both
of these triangles is 8.

00:01:19.290 --> 00:01:21.350
So the ratio is actually 1:1.

00:01:21.350 --> 00:01:23.740
And when the ratio of the sides
is 1:1, when the sides

00:01:23.740 --> 00:01:26.400
are actually congruent, and if
you're just given one side,

00:01:26.400 --> 00:01:27.040
that's enough.

00:01:27.040 --> 00:01:29.770
Then you could actually figure
out the rest of them using a

00:01:29.770 --> 00:01:31.430
little trigonometry or
something like that.

00:01:31.430 --> 00:01:33.120
We're not going to go
there just yet.

00:01:33.120 --> 00:01:35.630
But in geometry class, you
learned that if something is

00:01:35.630 --> 00:01:40.300
similar and and at least one of
the corresponding sides is

00:01:40.300 --> 00:01:41.370
congruent, then the whole thing

00:01:41.370 --> 00:01:42.250
is going to be congruent.

00:01:42.250 --> 00:01:46.495
So these are both similar
and congruent triangles.

00:01:50.340 --> 00:01:54.410
Both similar and congruent,
that's A.

00:01:54.410 --> 00:01:59.580
Problem 18.

00:01:59.580 --> 00:02:01.140
OK, let me cut and paste it.

00:02:07.530 --> 00:02:10.500
Which of the following
statements must be true if

00:02:10.500 --> 00:02:13.070
triangle GHI is similar.

00:02:15.690 --> 00:02:17.570
So if they write this that
means congruent.

00:02:17.570 --> 00:02:20.020
If they just write that,
that means similar.

00:02:20.020 --> 00:02:22.130
Which of the following
statements must be true if

00:02:22.130 --> 00:02:26.540
triangle GHI is similar
to triangle JKL?

00:02:26.540 --> 00:02:28.410
So even before looking at the
choices, that means that the

00:02:28.410 --> 00:02:31.200
ratio of all the sides are
the same, or all of the

00:02:31.200 --> 00:02:32.620
angles are the same.

00:02:32.620 --> 00:02:35.140
Let's see what they give us.

00:02:35.140 --> 00:02:37.660
The two triangles
must be scalene.

00:02:37.660 --> 00:02:39.990
Now, you have similar triangles
that are isosceles

00:02:39.990 --> 00:02:40.670
or equilateral.

00:02:40.670 --> 00:02:42.440
That's not right.

00:02:42.440 --> 00:02:46.320
The two triangles must have
exactly one acute angle.

00:02:54.740 --> 00:02:56.210
No, they could have
two acute angles.

00:02:56.210 --> 00:02:57.570
They could have three
acute angles.

00:02:57.570 --> 00:02:59.370
The way that they've drawn
it here, actually

00:02:59.370 --> 00:03:00.440
all of them are acute.

00:03:00.440 --> 00:03:02.120
None of these angles are greater
than 90 degrees just

00:03:02.120 --> 00:03:02.840
the way they've drawn.

00:03:02.840 --> 00:03:04.530
So that's not right.

00:03:04.530 --> 00:03:07.620
Some of these statements
are so crazy that

00:03:07.620 --> 00:03:08.590
they're hard to process.

00:03:08.590 --> 00:03:13.050
Anyway, C, at least one of the
sides of the two triangles

00:03:13.050 --> 00:03:16.530
must be parallel.

00:03:16.530 --> 00:03:17.780
I don't care how they're
oriented.

00:03:22.430 --> 00:03:26.630
You don't care about the
orientation of the triangles.

00:03:26.630 --> 00:03:29.540
The corresponding sides of
the triangles must be

00:03:29.540 --> 00:03:30.300
proportional.

00:03:30.300 --> 00:03:33.040
Yeah, that's one of the ways
that you know that something

00:03:33.040 --> 00:03:33.500
is similar.

00:03:33.500 --> 00:03:36.290
That the corresponding sides
are proportional.

00:03:36.290 --> 00:03:37.790
So that is, D.

00:03:37.790 --> 00:03:39.600
So this is almost, do you
know the definition

00:03:39.600 --> 00:03:41.380
of a similar triangle?

00:03:41.380 --> 00:03:48.860
Question 19.

00:03:48.860 --> 00:03:52.010
Let me erase this.

00:03:52.010 --> 00:03:54.480
OK.

00:03:54.480 --> 00:03:57.120
I have copied it.

00:03:57.120 --> 00:04:00.760
Now I am pasting it.

00:04:00.760 --> 00:04:06.490
In the figure below, AC
is congruent to DF.

00:04:06.490 --> 00:04:08.240
OK, so they're equal
to each other.

00:04:08.240 --> 00:04:11.300
AC and DF are congruent.

00:04:11.300 --> 00:04:14.185
And angle A is congruent
to angle D.

00:04:20.970 --> 00:04:21.480
Fair enough.

00:04:21.480 --> 00:04:22.900
That's angle A, that's
angle D.

00:04:22.900 --> 00:04:25.000
That's what they tell us.

00:04:25.000 --> 00:04:28.120
Which additional information
would be enough to prove that

00:04:28.120 --> 00:04:34.070
triangle ABC is congruent
to DEF?

00:04:34.070 --> 00:04:36.520
So they just gave us one
side and one angle.

00:04:36.520 --> 00:04:39.000
If they gave us another side,
if they said that DE is

00:04:39.000 --> 00:04:42.130
congruent to AB, that'd
be pretty cool.

00:04:42.130 --> 00:04:47.410
If they gave us this angle, if
they said angle F is congruent

00:04:47.410 --> 00:04:48.720
to angle C, that'd be good.

00:04:48.720 --> 00:04:50.360
Let's see what they give us.

00:04:50.360 --> 00:04:52.610
AB is congruent to DE.

00:04:52.610 --> 00:04:53.940
Yeah, sure.

00:04:53.940 --> 00:04:58.420
If AB is congruent to DE
then we definitely

00:04:58.420 --> 00:05:01.260
have congruent triangles.

00:05:01.260 --> 00:05:05.090
And you know the theorem that
you would have to say in your

00:05:05.090 --> 00:05:08.270
geometry class is, I have a
side, an angle, and a side.

00:05:08.270 --> 00:05:14.180
So you would say by SAS, by
side, angle, side, I know that

00:05:14.180 --> 00:05:17.390
these two triangles
are congruent.

00:05:17.390 --> 00:05:18.520
So AB is congruent to DE.

00:05:18.520 --> 00:05:19.410
Let's look at the other
ones to make sure

00:05:19.410 --> 00:05:20.390
we didn't miss anything.

00:05:20.390 --> 00:05:21.640
AB is congruent to BC.

00:05:26.680 --> 00:05:27.560
Well, that's fine.

00:05:27.560 --> 00:05:30.180
But that doesn't tell us
how AB relates to DE.

00:05:30.180 --> 00:05:32.730
So that's a useless statement.

00:05:32.730 --> 00:05:34.420
BC is congruent to EF.

00:05:40.440 --> 00:05:43.130
Well, see, this is another
time that I have a slight

00:05:43.130 --> 00:05:44.870
problem with the way they're
going with this.

00:05:44.870 --> 00:05:47.580
Because if BC were
congruent to EF.

00:05:55.300 --> 00:05:56.360
Let me think about that.

00:05:56.360 --> 00:05:59.680
Could I draw this triangle
in a way where

00:05:59.680 --> 00:06:01.530
they're still not congruent.

00:06:01.530 --> 00:06:03.950
Because I have this angle
here constraining it.

00:06:03.950 --> 00:06:05.090
They told us that.

00:06:05.090 --> 00:06:07.100
So it's not like I can
draw this line,

00:06:07.100 --> 00:06:11.920
FE, coming out here.

00:06:11.920 --> 00:06:13.920
Because if it came out here,
then DE would have

00:06:13.920 --> 00:06:15.680
to come like that.

00:06:15.680 --> 00:06:19.990
And then this angle couldn't
be what they said it was.

00:06:19.990 --> 00:06:22.590
So I'm just trying to think, I
actually think that would be

00:06:22.590 --> 00:06:23.840
sufficient.

00:06:28.280 --> 00:06:31.970
If you're given that this side
is congruent to that side.

00:06:31.970 --> 00:06:34.420
I think you could make a
trigonometric argument very

00:06:34.420 --> 00:06:37.440
easily to show that these two
triangles have equal sides.

00:06:37.440 --> 00:06:39.960
But anyway, I'm not going
to bother with that.

00:06:39.960 --> 00:06:40.360
Let's see.

00:06:40.360 --> 00:06:41.490
Let's look at choice D.

00:06:41.490 --> 00:06:43.000
BC is congruent to DE.

00:06:47.340 --> 00:06:49.070
Well, these aren't even
corresponding sides.

00:06:49.070 --> 00:06:51.890
So that's clearly useless.

00:06:51.890 --> 00:06:53.620
I have a suspicion that
this would have also

00:06:53.620 --> 00:06:54.800
been enough to prove.

00:06:54.800 --> 00:07:02.500
But anyway, I don't want
to insult anyone in the

00:07:02.500 --> 00:07:04.770
California Department of
Education, but I'm slightly

00:07:04.770 --> 00:07:06.250
disappointed by some
of these questions.

00:07:06.250 --> 00:07:09.270
Because I feel like they
really aren't testing

00:07:09.270 --> 00:07:12.060
intuition, they're just testing
to see whether you

00:07:12.060 --> 00:07:14.910
know the definitions of some of
these geometric terms. And

00:07:14.910 --> 00:07:17.780
whether you can spout out, side,
angle, side, and angle,

00:07:17.780 --> 00:07:18.370
side, angle.

00:07:18.370 --> 00:07:19.450
And things like that.

00:07:19.450 --> 00:07:23.230
And you're going to forget those
about three hours after

00:07:23.230 --> 00:07:25.910
you take the test. That's
pretty useless.

00:07:25.910 --> 00:07:28.490
What's useful is if you know
something that gives you an

00:07:28.490 --> 00:07:29.740
intuition about triangles.

00:07:29.740 --> 00:07:31.830
That's going to be useful for
you on the SAT, that's going

00:07:31.830 --> 00:07:34.690
to be useful for you when
you take trigonometry.

00:07:34.690 --> 00:07:38.500
And I'll tell you
a dirty secret.

00:07:38.500 --> 00:07:44.620
You will never use ASA theorem
or SAS theorem or anything

00:07:44.620 --> 00:07:47.560
like that again in your
mathematical careers.

00:07:47.560 --> 00:07:51.350
Your 9th or 10th grade geometry
class is the first

00:07:51.350 --> 00:07:53.180
and the last time that
you'll ever see them.

00:07:53.180 --> 00:07:56.530
So I have a slight problem
where they want you to

00:07:56.530 --> 00:07:58.980
memorize these theorems
and all of that.

00:07:58.980 --> 00:08:02.770
And even some of this notation
never shows up again in your

00:08:02.770 --> 00:08:03.980
mathematical careers.

00:08:03.980 --> 00:08:06.120
Even if you do a PhD
in mathematics.

00:08:06.120 --> 00:08:07.990
The only time you'll probably
see it again is if you become

00:08:07.990 --> 00:08:09.320
a geometry math teacher.

00:08:09.320 --> 00:08:10.980
Anyway, but it's good.

00:08:10.980 --> 00:08:13.710
I mean you should know how to do
this stuff at minimum just

00:08:13.710 --> 00:08:15.740
to jump through that
loop that society

00:08:15.740 --> 00:08:17.150
makes us all jump through.

00:08:17.150 --> 00:08:20.560
So problem 20.

00:08:20.560 --> 00:08:22.700
You don't want someone else to
get paid more just because

00:08:22.700 --> 00:08:25.600
they were willing
to say SAS, ASA.

00:08:25.600 --> 00:08:27.660
Anyway, all right, problem 20.

00:08:27.660 --> 00:08:31.290
Given AB and CD intersect
at point E.

00:08:34.450 --> 00:08:37.240
And just another aside, I think
you can even tell from

00:08:37.240 --> 00:08:39.669
my tone that I enjoy the SAT
problems a lot more.

00:08:39.669 --> 00:08:43.510
Because in some ways, in fact,
in every way, the SAT problems

00:08:43.510 --> 00:08:47.070
really test your understanding
of geometry, but never do they

00:08:47.070 --> 00:08:52.180
mention the words similar,
congruent, SAS, ASA.

00:08:52.180 --> 00:08:54.220
They never mention all these
things that you essentially

00:08:54.220 --> 00:08:55.960
memorize in your
geometry class.

00:08:55.960 --> 00:08:58.860
And I know tons of people who
get A's in geometry and then

00:08:58.860 --> 00:09:00.310
they don't do well on the SAT.

00:09:00.310 --> 00:09:02.050
And I know people who
do the other way.

00:09:02.050 --> 00:09:03.700
And frankly, I'd rather
hire the person who

00:09:03.700 --> 00:09:04.590
does well on the SAT.

00:09:04.590 --> 00:09:07.710
Because that's the person who
I think has the intuition.

00:09:07.710 --> 00:09:08.950
But anyway, we have
to do this.

00:09:08.950 --> 00:09:12.490
And I probably shouldn't
rant like that.

00:09:12.490 --> 00:09:15.666
Given AB and CD intersect
at point E.

00:09:20.500 --> 00:09:21.570
Fair enough.

00:09:21.570 --> 00:09:27.100
And they tell us that angle
1 is congruent to angle 2.

00:09:27.100 --> 00:09:30.330
So that and that are equal.

00:09:30.330 --> 00:09:31.990
All right, so already
those look like

00:09:31.990 --> 00:09:33.940
alternate interior angles.

00:09:33.940 --> 00:09:36.010
If this line were parallel.

00:09:36.010 --> 00:09:37.940
In fact, I think that's enough
to show that this line is

00:09:37.940 --> 00:09:39.190
parallel to this line.

00:09:42.120 --> 00:09:45.620
Because, if you view this as a
transversal, if you view DC as

00:09:45.620 --> 00:09:49.310
a transversal, then you see
that's a transversal between

00:09:49.310 --> 00:09:50.530
these two lines.

00:09:50.530 --> 00:09:53.600
And because the alternate
interior angles are the same,

00:09:53.600 --> 00:09:55.540
or they're congruent, you know
that those are going to be

00:09:55.540 --> 00:09:56.650
parallel lines.

00:09:56.650 --> 00:09:59.210
But anyway, I don't know if
that's at all useful.

00:09:59.210 --> 00:10:00.350
What are they going to ask us?

00:10:00.350 --> 00:10:09.190
Which theorem or postulate can
be used to prove that AED is

00:10:09.190 --> 00:10:12.480
similar to BEC.

00:10:12.480 --> 00:10:13.000
OK.

00:10:13.000 --> 00:10:15.470
So let's see, so I didn't
have to even say those

00:10:15.470 --> 00:10:18.495
are parallel lines.

00:10:18.495 --> 00:10:22.390
So what do they tell us?

00:10:22.390 --> 00:10:27.040
First of all, we know that 3
and 4 are congruent angles,

00:10:27.040 --> 00:10:28.990
because they're opposite
angles.

00:10:28.990 --> 00:10:31.120
Once again, I don't like the
word vertical angles, because

00:10:31.120 --> 00:10:33.060
these angles are clearly
not vertical.

00:10:33.060 --> 00:10:34.270
They're more side by side.

00:10:34.270 --> 00:10:35.470
But they're definitely
opposite.

00:10:35.470 --> 00:10:36.890
So those two angles
are the same.

00:10:40.750 --> 00:10:44.560
1 and 2 are the same, 3
and 4 are the same.

00:10:44.560 --> 00:10:46.250
If you know two of the
angles in a triangle,

00:10:46.250 --> 00:10:46.910
you know the third.

00:10:46.910 --> 00:10:49.670
So this angle and that angle
have to be the same.

00:10:49.670 --> 00:10:52.070
But in general, if you know that
two angles of a triangle

00:10:52.070 --> 00:10:53.890
are the same, the third
has to be the same.

00:10:53.890 --> 00:10:57.130
So, that tells you that it's
a similar triangle.

00:10:57.130 --> 00:10:59.350
So we could use angle, angle.

00:10:59.350 --> 00:11:02.140
We know that two angles are the
same as two other angles.

00:11:02.140 --> 00:11:05.430
So we know we're dealing
with similar triangles.

00:11:05.430 --> 00:11:08.520
Anyway, all out of time
because of my rant.

00:11:08.520 --> 00:11:10.500
See you in the next video.


CA Geometry: Similar triangles 1

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.730
Actually, just right after
stopping that video, I

00:00:02.730 --> 00:00:06.770
realized a very simple way
of showing you that RP is

00:00:06.770 --> 00:00:09.930
congruent to TA, a little bit
more of a rigorous definition.

00:00:09.930 --> 00:00:23.150
If we can show that this
triangle right there, that one

00:00:23.150 --> 00:00:25.990
I drew in purple, and this
triangle right here are

00:00:25.990 --> 00:00:33.140
congruent, then we could make
a fairly reasonable argument

00:00:33.140 --> 00:00:35.920
that RP is going to be congruent
to TA, because

00:00:35.920 --> 00:00:38.190
they're essentially the
corresponding sides of the two

00:00:38.190 --> 00:00:39.130
congruent triangles.

00:00:39.130 --> 00:00:41.000
This congruent triangles
would be kind of

00:00:41.000 --> 00:00:42.730
flipping each other.

00:00:42.730 --> 00:00:44.430
So how can we make
that argument?

00:00:44.430 --> 00:00:48.650
Well, on the purple triangle,
this angle is going to be

00:00:48.650 --> 00:00:55.850
equal to this angle on
the yellow triangle.

00:00:55.850 --> 00:00:58.640
Actually, we got that from the
fact that this is an isosceles

00:00:58.640 --> 00:01:01.380
trapezoid, so the base angles
are going to be the same.

00:01:01.380 --> 00:01:03.730
They told us this is an
isosceles trapezoid, so we

00:01:03.730 --> 00:01:07.480
know that this side, right
there, is going to be

00:01:07.480 --> 00:01:11.110
congruent to this side.

00:01:11.110 --> 00:01:13.880
And then finally, they both
share this side right here.

00:01:16.460 --> 00:01:19.750
So we could use the argument--
once again, side, angle,

00:01:19.750 --> 00:01:22.770
side-- that the side, angle, and
side are congruent to this

00:01:22.770 --> 00:01:24.160
side, angle, and side.

00:01:24.160 --> 00:01:37.930
So you could say by SAS,
triangle TRP is congruent to

00:01:37.930 --> 00:01:40.230
triangle TAP.

00:01:42.760 --> 00:01:44.870
And if they're congruent, then
all of the corresponding sides

00:01:44.870 --> 00:01:49.790
are equal, so then TA
is congruent to RP.

00:01:49.790 --> 00:01:51.140
Once again, you didn't
have to do all that.

00:01:51.140 --> 00:01:52.830
It's a multiple choice test. But
I wanted to give you that.

00:01:52.830 --> 00:01:57.490
I felt bad I wasn't giving you
a more rigorous definition.

00:01:57.490 --> 00:02:00.040
A more rigorous proof.

00:02:00.040 --> 00:02:14.360
So anyway, problem number 11.

00:02:14.360 --> 00:02:16.970
A conditional statement
is shown below.

00:02:16.970 --> 00:02:20.530
If a quadrilateral has
perpendicular diagonals, then

00:02:20.530 --> 00:02:22.010
it is a rhombus.

00:02:22.010 --> 00:02:23.140
Fair enough.

00:02:23.140 --> 00:02:26.100
Which of the following is
a counterexample to the

00:02:26.100 --> 00:02:27.460
statement above?

00:02:27.460 --> 00:02:30.120
So they're saying, if it's
perpendicular diagonals, then

00:02:30.120 --> 00:02:31.040
it's a rhombus.

00:02:31.040 --> 00:02:33.260
So if we could find something
that has perpendicular

00:02:33.260 --> 00:02:36.190
diagonals that is not a rhombus,
then we have a

00:02:36.190 --> 00:02:37.870
counterexample.

00:02:37.870 --> 00:02:39.900
Then this would not be true.

00:02:39.900 --> 00:02:43.600
So let's find something with
perpendicular diagonals that

00:02:43.600 --> 00:02:46.670
is not a rhombus.

00:02:46.670 --> 00:02:49.310
Well, this one has perpendicular
diagonals.

00:02:49.310 --> 00:02:51.180
The diagonals are perpendicular
to each other,

00:02:51.180 --> 00:02:52.300
all 90-degree angles.

00:02:52.300 --> 00:02:54.000
And this is clearly
not a rhombus.

00:02:54.000 --> 00:02:55.650
This is like a kite.

00:02:55.650 --> 00:02:58.260
This is not parallel to this and
that is not parallel, so

00:02:58.260 --> 00:02:59.140
this is not a rhombus.

00:02:59.140 --> 00:03:02.570
So this is definitely
a counterexample.

00:03:02.570 --> 00:03:04.520
This one does have perpendicular
diagonals, but

00:03:04.520 --> 00:03:05.430
it's also a rhombus.

00:03:05.430 --> 00:03:06.530
So it's not a counterexample.

00:03:06.530 --> 00:03:08.850
It's just an example of what
they're trying to say.

00:03:08.850 --> 00:03:10.860
This has perpendicular
diagonals, it's a square, but

00:03:10.860 --> 00:03:12.900
a square is a subset
of rhombuses.

00:03:12.900 --> 00:03:15.060
So this is another example.

00:03:15.060 --> 00:03:16.240
And, of course, this
one does not have

00:03:16.240 --> 00:03:17.630
perpendicular diagonals.

00:03:17.630 --> 00:03:19.900
This is not a right angle.

00:03:19.900 --> 00:03:23.250
So A is the counterexample.

00:03:23.250 --> 00:03:24.500
Next question.

00:03:27.110 --> 00:03:36.040
Problem 12.

00:03:36.040 --> 00:03:38.035
Which triangles must
be similar?

00:03:40.970 --> 00:03:43.430
Two obtuse triangles.

00:03:43.430 --> 00:03:48.340
Well, obtuse just means that
they have two angles, one

00:03:48.340 --> 00:03:51.710
obtuse angle might look like
that where that is greater

00:03:51.710 --> 00:03:53.990
than 90 degrees there,
and the other obtuse

00:03:53.990 --> 00:03:54.880
might be super obtuse.

00:03:54.880 --> 00:03:58.120
It might be like that.

00:03:58.120 --> 00:03:59.230
And clearly these
aren't similar.

00:03:59.230 --> 00:04:02.270
Well, this angle is obviously
larger than that one.

00:04:02.270 --> 00:04:04.390
OK, so this is not similar.

00:04:04.390 --> 00:04:06.080
Similar means all the
angles are the same.

00:04:06.080 --> 00:04:09.020
So it's like congruent, but you
can scale them in size.

00:04:09.020 --> 00:04:10.650
That's how I think of them.

00:04:10.650 --> 00:04:12.560
Like this triangle.

00:04:16.100 --> 00:04:18.300
I'm trying to draw it so it
looks exactly the same.

00:04:18.300 --> 00:04:19.600
That triangle.

00:04:19.600 --> 00:04:22.270
Well no, but you can imagine
if I cut and

00:04:22.270 --> 00:04:23.790
pasted that, right?

00:04:23.790 --> 00:04:26.020
It would only be similar to
this triangle if I drew

00:04:26.020 --> 00:04:27.820
everything to scale.

00:04:27.820 --> 00:04:30.040
Because the sides are different
sizes, but all the

00:04:30.040 --> 00:04:31.680
angles are the same.

00:04:31.680 --> 00:04:35.000
That's what similar means.

00:04:35.000 --> 00:04:35.630
So let's see.

00:04:35.630 --> 00:04:38.440
Two scalene triangles with
congruent bases.

00:04:41.600 --> 00:04:43.810
Well no, that's not true.

00:04:43.810 --> 00:04:46.330
That's not similar.

00:04:46.330 --> 00:04:48.710
Let's say they share
the same base.

00:04:48.710 --> 00:04:50.745
One scalene triangle might
look like this.

00:04:50.745 --> 00:04:56.320
It might come out a little bit
and then go down like that.

00:04:56.320 --> 00:05:00.580
And the other scalene triangle
has the same base right there.

00:05:00.580 --> 00:05:02.850
The sides might be a little
closer to each other.

00:05:02.850 --> 00:05:04.650
So clearly, these two things
aren't similar.

00:05:04.650 --> 00:05:06.110
This angle is different
than that angle.

00:05:06.110 --> 00:05:07.160
All the angles are
different, so

00:05:07.160 --> 00:05:08.870
they're not similar triangles.

00:05:08.870 --> 00:05:10.870
So that's not right.

00:05:10.870 --> 00:05:13.040
Two right triangles.

00:05:13.040 --> 00:05:14.090
Do those have to be similar?

00:05:14.090 --> 00:05:14.650
Well, no.

00:05:14.650 --> 00:05:17.440
You could have a right triangle
that looks like this,

00:05:17.440 --> 00:05:22.320
where maybe the two sides
are equal, right?

00:05:22.320 --> 00:05:25.020
That's a 45-45-90 triangle.

00:05:25.020 --> 00:05:31.370
Or you could have something
like this where you have a

00:05:31.370 --> 00:05:35.100
30-60-90 triangle.

00:05:35.100 --> 00:05:36.440
These clearly are not similar.

00:05:36.440 --> 00:05:37.660
All the angles are
not the same.

00:05:37.660 --> 00:05:39.980
They both have a 90-degree
angle.

00:05:39.980 --> 00:05:43.250
So I'm already guessing that D
is our right answer, but let's

00:05:43.250 --> 00:05:44.100
see how it works out.

00:05:44.100 --> 00:05:48.925
Two isosceles triangles with
congruent vertex angles.

00:05:53.960 --> 00:05:58.610
So I'm assuming when they say
congruent vertex angles, I'm

00:05:58.610 --> 00:06:00.906
assuming they mean all of the
angles are congruent.

00:06:03.910 --> 00:06:05.295
So two isosceles triangles.

00:06:08.750 --> 00:06:11.690
Let me think about
it a little bit.

00:06:11.690 --> 00:06:13.990
Oh, actually, I think what
they mean is angle in the

00:06:13.990 --> 00:06:15.460
middle when they say
vertex angle.

00:06:19.300 --> 00:06:22.760
If that's one of my isosceles
triangles.

00:06:26.560 --> 00:06:29.390
And isosceles triangle means
that that side is equal to

00:06:29.390 --> 00:06:31.980
that side and that angle
is equal to that angle.

00:06:31.980 --> 00:06:34.380
The vertex angle, I'm guessing,
they mean is this

00:06:34.380 --> 00:06:35.720
angle right there.

00:06:35.720 --> 00:06:37.760
So if I had another isosceles
triangle.

00:06:37.760 --> 00:06:41.140
Let's say maybe it's a
little bit smaller.

00:06:41.140 --> 00:06:43.800
It looks something like that.

00:06:43.800 --> 00:06:46.690
And their vertex angles
are the same.

00:06:46.690 --> 00:06:50.410
That angle is equal
to this angle.

00:06:50.410 --> 00:06:52.770
Well, if that angle is equal to
this angle and we know it's

00:06:52.770 --> 00:06:56.167
isosceles, so if we know it's
isosceles, that is equal to

00:06:56.167 --> 00:06:58.930
that, then that has to be equal
to that, we know that

00:06:58.930 --> 00:07:00.230
all the angles are the same.

00:07:00.230 --> 00:07:02.380
How do we know that this angle
is equal to this angle?

00:07:02.380 --> 00:07:03.580
Well, think about it.

00:07:03.580 --> 00:07:07.680
Whatever angle this us,
let's call this x.

00:07:07.680 --> 00:07:10.650
Let's call this angle
y and this angle y.

00:07:10.650 --> 00:07:19.340
We know that x plus 2y is equal
to 180, or that 2y is

00:07:19.340 --> 00:07:27.170
equal to 180 minus x, Or y is
equal to 90 minus x over 2.

00:07:27.170 --> 00:07:33.100
Now if this is x, And let's call
these z and z, So we know

00:07:33.100 --> 00:07:37.190
that x plus 2z is
equal to 180.

00:07:37.190 --> 00:07:40.160
All the angles in a triangle
have to add up to 180.

00:07:40.160 --> 00:07:42.970
Subtract x from both sides,
you get 2z is equal

00:07:42.970 --> 00:07:45.320
to 180 minus x.

00:07:45.320 --> 00:07:49.890
Divide by 2, you get z is equal
to 90 minus x over 2.

00:07:49.890 --> 00:07:51.960
So z and y are going to
be the same angles.

00:07:51.960 --> 00:07:54.230
So all the angles are the same,
so we're dealing with

00:07:54.230 --> 00:07:55.170
similar triangles.

00:07:55.170 --> 00:08:00.230
So choice D was definitely
correct.

00:08:00.230 --> 00:08:11.340
13.

00:08:11.340 --> 00:08:12.930
OK.

00:08:12.930 --> 00:08:16.130
Which of the following facts
would be sufficient to prove

00:08:16.130 --> 00:08:21.630
that triangles ABC, that's the
big triangle, and triangle

00:08:21.630 --> 00:08:24.270
DBE, so that's a small
one, are similar?

00:08:24.270 --> 00:08:26.600
So we have to prove that all of
their angles are similar.

00:08:26.600 --> 00:08:29.300
I cannot even look at the
choices and I can guess where

00:08:29.300 --> 00:08:29.840
this is going.

00:08:29.840 --> 00:08:30.740
So we want to prove that
those are similar.

00:08:30.740 --> 00:08:32.730
So first of all, they share
the same angle.

00:08:32.730 --> 00:08:38.340
Angle ABC, this angle, is
the same as angle DBE.

00:08:38.340 --> 00:08:39.750
So they share that same angle.

00:08:39.750 --> 00:08:42.549
So we got one angle down.

00:08:42.549 --> 00:08:43.620
Now let's think about it.

00:08:43.620 --> 00:08:46.750
If we knew that this angle is
equal to that angle and that

00:08:46.750 --> 00:08:49.220
angle is equal to that
angle, we'd be done.

00:08:49.220 --> 00:08:52.240
And the best way to come to
that conclusion is if they

00:08:52.240 --> 00:08:55.340
told us that this and
this are parallel.

00:08:55.340 --> 00:08:56.670
I'm guessing that's where
they're going.

00:08:56.670 --> 00:08:58.340
Now I might have gone on a
completely wrong tangent.

00:08:58.340 --> 00:09:01.440
Because it those two are
parallel, then these two lines

00:09:01.440 --> 00:09:03.450
are transversals of
the parallel.

00:09:03.450 --> 00:09:10.310
So that angle and that angle
would be corresponding angles,

00:09:10.310 --> 00:09:12.750
so they would be congruent, and
then that angle and that

00:09:12.750 --> 00:09:15.240
angle would be corresponding
angles so

00:09:15.240 --> 00:09:16.080
they'd also be congruent.

00:09:16.080 --> 00:09:18.240
So if they told us that these
are parallel, we're done.

00:09:18.240 --> 00:09:20.860
These are definitely
similar triangles.

00:09:20.860 --> 00:09:27.190
And sure enough, choice C, they
tell us that AC and DE

00:09:27.190 --> 00:09:27.860
are parallel.

00:09:27.860 --> 00:09:30.440
These are parallel, that's
a transversal, this is a

00:09:30.440 --> 00:09:33.110
corresponding angle of that,
so they're congruent.

00:09:33.110 --> 00:09:36.370
This is a corresponding angle to
this, congruent, so all of

00:09:36.370 --> 00:09:37.360
the angles are congruent.

00:09:37.360 --> 00:09:39.185
So we have a similar triangle.

00:09:44.030 --> 00:09:53.190
Problem 14.

00:09:53.190 --> 00:09:54.440
OK.

00:10:01.080 --> 00:10:03.940
Parallelogram ABCD
is shown below.

00:10:03.940 --> 00:10:04.740
Fair enough.

00:10:04.740 --> 00:10:05.860
Parallelogram: that
tells us that the

00:10:05.860 --> 00:10:06.870
opposites sides are parallel.

00:10:06.870 --> 00:10:12.650
That's parallel to that, and
then this is parallel to that.

00:10:12.650 --> 00:10:14.680
And all of the choices got
clipped at the bottom, but

00:10:14.680 --> 00:10:17.455
I'll copy them over.

00:10:20.500 --> 00:10:24.200
Maybe I'll copy them
above the question.

00:10:24.200 --> 00:10:25.570
Well, let me see
what I can do.

00:10:29.300 --> 00:10:31.276
I think that's good enough.

00:10:31.276 --> 00:10:33.070
A little unconventional.

00:10:33.070 --> 00:10:34.080
OK.

00:10:34.080 --> 00:10:35.100
Parallelogram is shown below.

00:10:35.100 --> 00:10:37.280
They say which pair of triangles
can be established

00:10:37.280 --> 00:10:47.160
to be congruent to prove
that angle DAB is

00:10:47.160 --> 00:10:48.410
congruent to angle BCD?

00:10:53.400 --> 00:10:55.020
So DAB is this.

00:10:55.020 --> 00:10:56.380
Let me do it in another color.

00:10:56.380 --> 00:11:02.060
DAB is that angle, is
congruent to BCD.

00:11:02.060 --> 00:11:04.470
They want us to show
that those have

00:11:04.470 --> 00:11:06.640
the same angle measure.

00:11:06.640 --> 00:11:08.310
OK, and what do we
have to show?

00:11:08.310 --> 00:11:10.120
They say what pair of triangles
can be established

00:11:10.120 --> 00:11:12.250
to be congruent to prove that.

00:11:12.250 --> 00:11:16.830
OK, if these are both part of
two different congruent

00:11:16.830 --> 00:11:19.440
triangles and they are the
corresponding angles, then we

00:11:19.440 --> 00:11:21.090
know that they're congruent
and we'd be done.

00:11:21.090 --> 00:11:22.330
So let's see what they say.

00:11:22.330 --> 00:11:23.980
Triangle ADC and BCD.

00:11:31.870 --> 00:11:34.650
BCD has this angle in it.

00:11:34.650 --> 00:11:37.640
BCD does help us because it
has this angle in it, but

00:11:37.640 --> 00:11:41.780
triangle ADC does not have
this angle in it, right?

00:11:41.780 --> 00:11:45.010
Triangle ADC has this the
smaller angle in it.

00:11:45.010 --> 00:11:47.560
ADC doesn't involve this whole
thing, so that's not

00:11:47.560 --> 00:11:49.490
going to help us.

00:11:49.490 --> 00:11:54.610
Triangle AED, once again, does
not involve this larger angle,

00:11:54.610 --> 00:11:56.820
does not involve
the angle DAB.

00:11:56.820 --> 00:11:58.850
It only involves the little
smaller angle, so that's not

00:11:58.850 --> 00:12:00.250
going to help us.

00:12:00.250 --> 00:12:02.060
Triangle DAB.

00:12:02.060 --> 00:12:02.690
That looks good.

00:12:02.690 --> 00:12:04.260
That has this whole
angle in it.

00:12:04.260 --> 00:12:05.700
DAB.

00:12:05.700 --> 00:12:08.190
And then BCD.

00:12:08.190 --> 00:12:15.780
If we showed that that triangle
is congruent to this

00:12:15.780 --> 00:12:20.780
triangle right here,
I think we're done.

00:12:20.780 --> 00:12:25.150
That would be enough to show
that this angle is congruent

00:12:25.150 --> 00:12:27.160
to that angle, because they
would be the corresponding

00:12:27.160 --> 00:12:29.210
angles of a congruent
triangle.

00:12:29.210 --> 00:12:31.010
So I think C is where
we're going to go.

00:12:31.010 --> 00:12:32.750
Let's just look at choice D.

00:12:32.750 --> 00:12:34.670
DEC.

00:12:34.670 --> 00:12:36.420
Once again, triangle DEC.

00:12:36.420 --> 00:12:38.050
Let me make this point clear.

00:12:38.050 --> 00:12:43.420
Triangle DEC does not involve
either of the angles we care.

00:12:43.420 --> 00:12:45.610
It clearly does not involve
this angle, and it only

00:12:45.610 --> 00:12:47.570
involves part of this angle,
only this part.

00:12:47.570 --> 00:12:50.120
It doesn't involve this whole
angle, so that's not going to

00:12:50.120 --> 00:12:51.060
help us either.

00:12:51.060 --> 00:12:52.470
So the answer is C.

00:12:52.470 --> 00:12:54.680
Anyway, see you in
the next video.


CA Geometry: Proof by contradiction

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.620
We're on problem number four,
and they give us a theorem.

00:00:03.620 --> 00:00:08.240
It says a triangle has, at
most, one obtuse angle.

00:00:08.240 --> 00:00:09.240
Fair enough.

00:00:09.240 --> 00:00:11.850
Eduardo is proving the theorem
above by contradiction.

00:00:14.630 --> 00:00:16.750
So the way you prove by a
contradiction, you're like,

00:00:16.750 --> 00:00:18.330
well what if this
weren't true.

00:00:18.330 --> 00:00:20.230
Let me prove that that
can't happen.

00:00:20.230 --> 00:00:21.930
Well let's see what
he did anyway.

00:00:21.930 --> 00:00:26.480
He began by assuming, that in
triangle ABC, angle A and B

00:00:26.480 --> 00:00:28.030
are both obtuse.

00:00:28.030 --> 00:00:32.520
Which theorem will Eduardo use
to reach a contradiction?

00:00:32.520 --> 00:00:38.680
OK, let me draw this, what
Eduardo is trying to do.

00:00:38.680 --> 00:00:40.780
The way I'm drawing it is
actually very hard.

00:00:40.780 --> 00:00:42.610
So this is actually not
drawn at all to scale.

00:00:42.610 --> 00:00:46.010
So he's saying that angle A and
angle B are both obtuse.

00:00:46.010 --> 00:00:48.570
So this means that this angle
is greater than 90.

00:00:48.570 --> 00:00:50.050
Let's say that's angle A.

00:00:50.050 --> 00:00:51.290
And this is angle B.

00:00:51.290 --> 00:00:52.460
And it's also greater than 90.

00:00:52.460 --> 00:00:55.070
That's what obtuse means.

00:00:55.070 --> 00:00:58.510
Which theorem will Eduardo use
to reach a contradiction?

00:00:58.510 --> 00:01:01.420
Well, before even reading the
choices, think about it.

00:01:01.420 --> 00:01:02.970
What do we know about
triangles?

00:01:02.970 --> 00:01:07.580
That all of the angles add
up to 180 degrees, right?

00:01:07.580 --> 00:01:10.000
So if this is angle A, this is
angle B, and then let's call

00:01:10.000 --> 00:01:12.460
this angle C.

00:01:12.460 --> 00:01:18.480
We know A plus B plus C
have to be equal to

00:01:18.480 --> 00:01:20.910
180 degrees, right?

00:01:20.910 --> 00:01:29.240
Or another way to view it
is, C is equal to 180

00:01:29.240 --> 00:01:32.100
minus A minus B.

00:01:32.100 --> 00:01:33.750
Or another way you can think of
it, I'm just writing it a

00:01:33.750 --> 00:01:34.470
bunch of different ways.

00:01:34.470 --> 00:01:40.300
C is equal to 180 minus
A plus B, right?

00:01:40.300 --> 00:01:42.110
Now, let me ask you
a question.

00:01:42.110 --> 00:01:46.400
If we assume from the get-go,
as Eduardo did, if we assume

00:01:46.400 --> 00:01:50.920
that both A and B are greater
than 90 degrees, what's A plus

00:01:50.920 --> 00:01:53.600
B going to be at least
greater than?

00:01:53.600 --> 00:01:56.160
If this is greater than 90 and
that's greater than 90, then A

00:01:56.160 --> 00:01:59.010
plus B is going to be greater
than 90 plus 90.

00:01:59.010 --> 00:02:01.660
So this has to be greater
than 180.

00:02:01.660 --> 00:02:04.240
So if this is greater than 180,
and we're subtracting it

00:02:04.240 --> 00:02:10.690
from 180, so this essentially
says if angle A is greater

00:02:10.690 --> 00:02:15.340
than 90, and angle B is greater
than 90, than what we

00:02:15.340 --> 00:02:17.770
can deduce is, from this
statement right here.

00:02:17.770 --> 00:02:20.310
From this equation right here.

00:02:20.310 --> 00:02:22.370
If this and this is greater than
90 then this whole term

00:02:22.370 --> 00:02:23.560
is greater than 180.

00:02:23.560 --> 00:02:26.760
So then the deduction would be
that C has to be less than

00:02:26.760 --> 00:02:29.480
zero, and we can't have
negative angles.

00:02:29.480 --> 00:02:34.910
So right there, that is
the contradiction.

00:02:34.910 --> 00:02:37.850
And then you would say, OK,
therefore you cannot have two

00:02:37.850 --> 00:02:40.490
angles that are more than
90 degrees or two

00:02:40.490 --> 00:02:42.490
angles that are obtuse.

00:02:42.490 --> 00:02:46.090
And that would be your proof
by contradiction.

00:02:46.090 --> 00:02:49.400
Let's see if what we did
can be phrased in

00:02:49.400 --> 00:02:50.170
one of these choices.

00:02:50.170 --> 00:02:53.010
If two angles of a triangle are
equal, the sides opposite

00:02:53.010 --> 00:02:53.820
the angles are equal.

00:02:53.820 --> 00:02:54.380
No.

00:02:54.380 --> 00:03:01.410
If two supplementary angles
are equal, the angles each

00:03:01.410 --> 00:03:02.410
measure 90.

00:03:02.410 --> 00:03:05.290
Well, we didn't use that.

00:03:05.290 --> 00:03:07.020
The largest angle of a triangle
is opposite the

00:03:07.020 --> 00:03:07.580
longest side.

00:03:07.580 --> 00:03:07.860
No.

00:03:07.860 --> 00:03:10.710
The sum of the measures of the
angles of a triangle is 180.

00:03:10.710 --> 00:03:13.370
That's the first thing we
wrote down right there.

00:03:13.370 --> 00:03:15.720
So it's choice D.

00:03:15.720 --> 00:03:20.160
That's the theorem Eduardo used
to reach a contradiction.

00:03:20.160 --> 00:03:23.160
Next question.

00:03:23.160 --> 00:03:24.410
Problem five.

00:03:27.320 --> 00:03:28.570
OK, this one.

00:03:31.860 --> 00:03:36.210
OK, it's a big question.

00:03:36.210 --> 00:03:39.830
Let me see if I can copy and
paste the whole thing.

00:03:39.830 --> 00:03:41.080
I've copied it.

00:03:44.270 --> 00:03:45.520
All right.

00:03:50.110 --> 00:03:52.040
I think it all fits
in the window.

00:03:52.040 --> 00:03:54.960
Let's see, it says use
the proof to answer

00:03:54.960 --> 00:03:56.110
the question below.

00:03:56.110 --> 00:04:02.860
So given that side AB is
congruent to side BC.

00:04:02.860 --> 00:04:05.810
So we could say that side
is equal to that side.

00:04:05.810 --> 00:04:06.820
That's given.

00:04:06.820 --> 00:04:08.440
D is the midpoint of AC.

00:04:08.440 --> 00:04:10.220
So that means D is equidistant
between AC.

00:04:10.220 --> 00:04:13.470
So that means that AD and
DC are equal length.

00:04:13.470 --> 00:04:14.720
Let me write that.

00:04:19.910 --> 00:04:27.225
Prove that triangle ABD is
congruent to to triangle CBD.

00:04:30.440 --> 00:04:32.980
All right, and just so you know,
congruent triangles are

00:04:32.980 --> 00:04:35.920
triangles that are the same in
every way, except they might

00:04:35.920 --> 00:04:38.770
have been rotated.

00:04:38.770 --> 00:04:41.040
They could have been rotated
in some way.

00:04:41.040 --> 00:04:43.480
If you had similar triangles,
then you could also have

00:04:43.480 --> 00:04:44.520
different side measures.

00:04:44.520 --> 00:04:46.250
They're just kind of the same
shape, but they could be

00:04:46.250 --> 00:04:50.120
expanded or contracted
in some way.

00:04:50.120 --> 00:04:53.390
If you're congruent, you have
similar triangles but they

00:04:53.390 --> 00:04:54.980
also have the same
side lengths.

00:04:54.980 --> 00:04:57.060
But even though they have the
same side lengths, they could

00:04:57.060 --> 00:04:57.900
be flipped over.

00:04:57.900 --> 00:04:59.440
Like, you can just
look at this one.

00:04:59.440 --> 00:05:05.420
ABD looks like it's a
mirror image of DBC.

00:05:05.420 --> 00:05:07.610
So, just eyeballing it, it
already feels like they're

00:05:07.610 --> 00:05:09.230
congruent triangles.

00:05:09.230 --> 00:05:11.920
Let's see how they go
about proving it.

00:05:11.920 --> 00:05:16.720
So statement one, AB
is congruent to BC,

00:05:16.720 --> 00:05:18.240
they give us that.

00:05:18.240 --> 00:05:19.740
D is the midpoint of AC.

00:05:19.740 --> 00:05:22.070
That was given, fair enough.

00:05:22.070 --> 00:05:23.870
AD is congruent to CD.

00:05:26.750 --> 00:05:29.640
That's because D is the
midpoint of AC.

00:05:29.640 --> 00:05:32.060
We did that part right there,
definition of midpoint.

00:05:32.060 --> 00:05:33.420
Fair enough.

00:05:33.420 --> 00:05:37.790
BD is congruent to
BD, of course.

00:05:37.790 --> 00:05:40.300
Anything is congruent
to itself.

00:05:40.300 --> 00:05:43.990
So that just says the BD for
that triangle is the same

00:05:43.990 --> 00:05:45.450
length as the BD for
this triangle.

00:05:45.450 --> 00:05:47.270
Fair enough, reflexive
property.

00:05:47.270 --> 00:05:50.300
Fancy word for a very
simple idea.

00:05:50.300 --> 00:05:54.870
And then finally, they
say triangle ABD is

00:05:54.870 --> 00:05:57.270
congruent to CBD.

00:05:57.270 --> 00:06:01.180
OK, well from the get-go, using
these statments, we've

00:06:01.180 --> 00:06:02.640
already shown that they
have the same

00:06:02.640 --> 00:06:05.050
exact three side measures.

00:06:05.050 --> 00:06:07.470
Both triangles have a
side of length BD.

00:06:07.470 --> 00:06:10.960
Both triangles have a side
of length AD or DC.

00:06:10.960 --> 00:06:13.930
And both triangles have
a side of length BA.

00:06:13.930 --> 00:06:16.570
So all of their sides
are the same length.

00:06:16.570 --> 00:06:21.100
That's what we know after
the first three steps.

00:06:21.100 --> 00:06:24.670
So what reason can be used
to prove that the

00:06:24.670 --> 00:06:25.420
triangles are congruent?

00:06:25.420 --> 00:06:28.230
Well we just said, these three
steps showed that all the

00:06:28.230 --> 00:06:29.650
sides are the same.

00:06:29.650 --> 00:06:32.060
So this SSS that you see.

00:06:32.060 --> 00:06:32.660
What reason?

00:06:32.660 --> 00:06:34.463
That means side, side, side.

00:06:37.440 --> 00:06:42.020
And that's just the argument
that you use in your geometry

00:06:42.020 --> 00:06:45.500
class to say that all
three sides of both

00:06:45.500 --> 00:06:47.910
triangles are congruent.

00:06:47.910 --> 00:06:50.340
This means that you have an
angle, an angle, and a side.

00:06:50.340 --> 00:06:52.570
This means that you have an
angle, and then the side

00:06:52.570 --> 00:06:54.010
between the two angles.

00:06:54.010 --> 00:06:56.360
And then the next angle that
all of those are congruent.

00:06:56.360 --> 00:06:59.130
And this says that one of the
sides and the angle, and the

00:06:59.130 --> 00:07:00.340
other side, that those
are congruent.

00:07:00.340 --> 00:07:01.530
We'll probably run
into those in the

00:07:01.530 --> 00:07:02.330
next couple of questions.

00:07:02.330 --> 00:07:05.520
But anyway, this shows that
all three sides of both

00:07:05.520 --> 00:07:06.690
triangles are equal.

00:07:06.690 --> 00:07:12.630
And then so, we could say
by the side, side, side

00:07:12.630 --> 00:07:16.730
reasoning, I'm not that
good with terminology.

00:07:16.730 --> 00:07:19.230
By the side, side, side
reasoning, these are both

00:07:19.230 --> 00:07:20.500
congruent triangles.

00:07:20.500 --> 00:07:23.820
And I said, that's one of the
ways of thinking about a

00:07:23.820 --> 00:07:25.540
congruent triangle, is that all
the sides are going to be

00:07:25.540 --> 00:07:28.150
the same length.

00:07:28.150 --> 00:07:29.400
Next question.

00:07:33.260 --> 00:07:34.510
All right.

00:07:42.420 --> 00:07:47.590
In the figure below, AB
is greater than BC.

00:07:47.590 --> 00:07:49.740
OK, so this side is greater
than that side.

00:07:49.740 --> 00:07:52.830
Although the way they drew it,
they all look the same.

00:07:52.830 --> 00:07:54.760
So let's see what we can do.

00:07:54.760 --> 00:07:59.250
If we assume that measure of
angle A is equal to measure of

00:07:59.250 --> 00:08:19.030
angle C, it follows that
AB is equal to BC.

00:08:19.030 --> 00:08:22.510
AB is equal to BC.

00:08:22.510 --> 00:08:25.150
And I don't know if you've run
into this already, but you

00:08:25.150 --> 00:08:28.910
learned that if you have two
angles that are congruent, or

00:08:28.910 --> 00:08:30.030
if the measures are the same.

00:08:30.030 --> 00:08:31.480
This is essentially saying
that angle A is

00:08:31.480 --> 00:08:32.950
congruent to angle C.

00:08:32.950 --> 00:08:35.230
They instead just wrote it as
that the measures of the

00:08:35.230 --> 00:08:36.690
angles are equal.

00:08:36.690 --> 00:08:38.570
That's what the definition of
congruence is, is that the

00:08:38.570 --> 00:08:40.539
measures of the angles
are equal.

00:08:40.539 --> 00:08:45.940
You could have written angle
A is congruent to angle C.

00:08:45.940 --> 00:08:50.440
But anyway, if you have two
angles that are equal, then

00:08:50.440 --> 00:08:52.670
the sides that are opposite
those angles are

00:08:52.670 --> 00:08:53.960
also going to be equal.

00:08:53.960 --> 00:08:55.750
So this side right here
is going to be

00:08:55.750 --> 00:08:57.010
equal to that side.

00:08:57.010 --> 00:08:57.980
And that's what they
wrote here.

00:08:57.980 --> 00:09:01.830
It follows that AB
is equal to BC.

00:09:01.830 --> 00:09:03.160
Fair enough.

00:09:03.160 --> 00:09:07.180
Then they say this contradicts
the given statement that AB is

00:09:07.180 --> 00:09:09.760
greater than BC.

00:09:09.760 --> 00:09:12.880
Right, it says, it follows that
AB is equal to BC and it

00:09:12.880 --> 00:09:14.680
contradicts this statement.

00:09:14.680 --> 00:09:16.140
Where are they going
with this?

00:09:16.140 --> 00:09:19.705
What conclusion can be drawn
from this contradiction?

00:09:23.190 --> 00:09:25.100
Let's see, measure of
angle A is equal to

00:09:25.100 --> 00:09:26.350
measure of angle B.

00:09:31.970 --> 00:09:33.830
No, that's not the case.

00:09:33.830 --> 00:09:34.900
I can think of an example.

00:09:34.900 --> 00:09:37.450
These can both be 30
degree angles.

00:09:37.450 --> 00:09:41.430
If these are both 30 degree
angles, add up to 60, then

00:09:41.430 --> 00:09:43.840
this would have to be 120 for
them to all add up to 180.

00:09:43.840 --> 00:09:45.550
And it would completely
gel with

00:09:45.550 --> 00:09:46.490
everything else we've learned.

00:09:46.490 --> 00:09:48.280
So, A is definitely not right.

00:09:48.280 --> 00:09:51.560
That A does not have
to be equal to B.

00:09:51.560 --> 00:09:54.680
Measure of A does not equal
the measure of angle B.

00:09:58.800 --> 00:10:00.670
Well, they could, right?

00:10:00.670 --> 00:10:03.770
All of these angles could
be 60 degrees.

00:10:03.770 --> 00:10:06.380
We haven't said that B
definitely does not equal A.

00:10:06.380 --> 00:10:07.970
This could be 60, that
could be 60, and so

00:10:07.970 --> 00:10:08.790
could this be 60.

00:10:08.790 --> 00:10:12.530
And we'd be dealing with an
equilateral triangle.

00:10:12.530 --> 00:10:15.100
So I don't think that's
right either.

00:10:15.100 --> 00:10:19.070
Measure of angle A is equal
to measure of angle C.

00:10:24.780 --> 00:10:27.560
I see what they're
saying here.

00:10:27.560 --> 00:10:29.330
Sorry, and this is my bad.

00:10:29.330 --> 00:10:32.350
They're saying, AB is definitely
greater than BC.

00:10:36.920 --> 00:10:41.800
Now, they said if we assume
that measure of angle A is

00:10:41.800 --> 00:10:43.630
equal to measure of angle
C, it follows that

00:10:43.630 --> 00:10:45.260
AB is equal to BC.

00:10:45.260 --> 00:10:47.550
They didn't say that this
is definitely true.

00:10:47.550 --> 00:10:51.520
They just said that if we assume
that this is true.

00:10:51.520 --> 00:10:53.990
But they didn't say this
is a definite case.

00:10:53.990 --> 00:10:55.710
And that's where the
contradiction came.

00:10:55.710 --> 00:10:58.690
Because if we assumed it,
then AB could not be

00:10:58.690 --> 00:11:00.540
greater than BC.

00:11:00.540 --> 00:11:03.700
Because then AB would
equal BC.

00:11:03.700 --> 00:11:05.590
So now I see what
they're asking.

00:11:05.590 --> 00:11:07.040
So this is an assumption.

00:11:07.040 --> 00:11:09.070
This isn't actually
proven to be true.

00:11:09.070 --> 00:11:11.920
So this contradicts the given
statement that AB is

00:11:11.920 --> 00:11:13.130
greater than BC.

00:11:13.130 --> 00:11:14.410
Right, that's true.

00:11:14.410 --> 00:11:17.640
What conclusion can be drawn
from this contradiction?

00:11:17.640 --> 00:11:21.000
So we made the assumption that
the measure of angle A is

00:11:21.000 --> 00:11:22.290
equal to the measure
of angle C.

00:11:22.290 --> 00:11:24.740
That follows that these two
sides are equal, which

00:11:24.740 --> 00:11:26.840
contradicted the given
statement.

00:11:26.840 --> 00:11:29.790
Therefore, we know that the
measures of these two angles

00:11:29.790 --> 00:11:31.710
cannot be equal to each other.

00:11:31.710 --> 00:11:35.220
Because if they were, then we
would contradict the given

00:11:35.220 --> 00:11:36.310
assumption.

00:11:36.310 --> 00:11:39.440
So, we know from the
contradiction that the measure

00:11:39.440 --> 00:11:44.580
of angle A cannot equal the
measure of angle C.

00:11:44.580 --> 00:11:46.490
And we can't make that
assumption because it leads to

00:11:46.490 --> 00:11:48.390
a contradiction.

00:11:48.390 --> 00:11:52.620
So the correct answer is D.

00:11:52.620 --> 00:11:55.900
All right, I'll see you
in the next video.


CA Geometry: Deductive reasoning

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.830
All right, we're doing the
California Standards released

00:00:03.830 --> 00:00:05.530
questions in geometry now.

00:00:05.530 --> 00:00:06.990
And here's the first question.

00:00:06.990 --> 00:00:10.120
It says, which of the following
best describes

00:00:10.120 --> 00:00:12.330
deductive reasoning?

00:00:12.330 --> 00:00:15.420
And I'm not a huge fan of when
they ask essentially

00:00:15.420 --> 00:00:17.890
definitional questions
in math class.

00:00:17.890 --> 00:00:19.950
But we'll do it and hopefully
it will help you understand

00:00:19.950 --> 00:00:21.070
what deductive reasoning is.

00:00:21.070 --> 00:00:25.380
Although, I do think that
deductive reasoning itself is

00:00:25.380 --> 00:00:28.380
probably more natural than the
definition they'll give here.

00:00:28.380 --> 00:00:30.360
Well, actually before I even
look at the definitions, let

00:00:30.360 --> 00:00:31.300
me tell you what it is.

00:00:31.300 --> 00:00:33.670
And then we can see which of
these definitions matches it.

00:00:33.670 --> 00:00:36.650
Deductive reasoning is, if I
give you a bunch of statements

00:00:36.650 --> 00:00:40.140
and then from those statements
you deduce, or you come to

00:00:40.140 --> 00:00:43.380
some conclusion that you
know must be true.

00:00:43.380 --> 00:00:54.070
Like if, I said that
all boys are tall.

00:00:54.070 --> 00:00:56.000
If I told you all
boys are tall.

00:00:56.000 --> 00:00:58.780
And if I told you that
Bill is a boy.

00:00:58.780 --> 00:01:01.790
Bill is a boy.

00:01:01.790 --> 00:01:04.430
That's two separate
words, a and boy.

00:01:04.430 --> 00:01:08.580
So if you say, OK, if these two
statements are true, what

00:01:08.580 --> 00:01:09.420
can you deduce?

00:01:09.420 --> 00:01:12.220
Well I say well, Bill is a boy
and all boys are tall.

00:01:12.220 --> 00:01:15.160
Then Bill must be tall.

00:01:15.160 --> 00:01:17.820
You deduced this last statement
from these two other

00:01:17.820 --> 00:01:18.880
statements that you
knew were true.

00:01:18.880 --> 00:01:22.090
And this one has to be true
if those two are true.

00:01:22.090 --> 00:01:24.630
So Bill must be tall.

00:01:24.630 --> 00:01:27.540
That is deductive reasoning.

00:01:27.540 --> 00:01:30.170
Deductive reasoning.

00:01:30.170 --> 00:01:33.790
Bill must be tall, not Bill
must be deductive.

00:01:33.790 --> 00:01:36.250
So anyway, you have some
statements and you deduce

00:01:36.250 --> 00:01:39.920
other statements that must
be true given those.

00:01:39.920 --> 00:01:43.580
And you often hear
another type of

00:01:43.580 --> 00:01:45.850
reasoning, inductive reasoning.

00:01:45.850 --> 00:01:48.330
And that's when you're given a
couple of examples and you

00:01:48.330 --> 00:01:49.580
generalize.

00:01:53.080 --> 00:01:55.305
Well, I don't want to get too
complicated here, because this

00:01:55.305 --> 00:01:56.810
is a question on deductive
reasoning.

00:01:56.810 --> 00:01:59.830
But generalizations often
aren't a good thing.

00:01:59.830 --> 00:02:03.080
But if you see a couple of
examples and you see a pattern

00:02:03.080 --> 00:02:05.540
there you can often extrapolate
and get to a kind

00:02:05.540 --> 00:02:07.070
of broader generalization.

00:02:07.070 --> 00:02:08.080
That's inductive reasoning.

00:02:08.080 --> 00:02:09.979
But that's not what they're
asking us about this.

00:02:09.979 --> 00:02:12.310
Let's see if we can find the
definition of deductive

00:02:12.310 --> 00:02:18.000
reasoning in the California
Standards language.

00:02:18.000 --> 00:02:21.510
Use logic to draw conclusions
based on accepted statements.

00:02:21.510 --> 00:02:23.520
Yeah, well actually that
sounds about right.

00:02:23.520 --> 00:02:24.130
That's what we did here.

00:02:24.130 --> 00:02:26.760
We used logic to draw
conclusions based on accepted

00:02:26.760 --> 00:02:29.320
statements, which
were those two.

00:02:29.320 --> 00:02:31.800
So I'm going to go
with A, so far.

00:02:31.800 --> 00:02:36.260
Accepting the meaning of a
term without definition.

00:02:36.260 --> 00:02:38.630
Well, I don't even know
how one can do that.

00:02:38.630 --> 00:02:41.200
How do you accept the meaning of
something without it having

00:02:41.200 --> 00:02:43.210
being defined?

00:02:43.210 --> 00:02:44.650
Let's see, so it's not B.

00:02:44.650 --> 00:02:46.990
I don't think anything
is really B.

00:02:46.990 --> 00:02:49.820
C, defining mathematical
terms to correspond

00:02:49.820 --> 00:02:51.420
with physical objects.

00:02:51.420 --> 00:02:54.980
No, that's not really
anything related to

00:02:54.980 --> 00:02:56.810
deductive reasoning either.

00:02:56.810 --> 00:03:00.060
D, inferring a general truth
by examining a number of

00:03:00.060 --> 00:03:01.040
specific examples.

00:03:01.040 --> 00:03:03.290
Well, this is more of what
I had just talked about,

00:03:03.290 --> 00:03:04.870
inductive reasoning.

00:03:04.870 --> 00:03:07.270
So, they want to know what
deductive reasoning is, so I'm

00:03:07.270 --> 00:03:10.780
going to go with A.

00:03:10.780 --> 00:03:13.770
Use logic to draw conclusions
based on accepted statements.

00:03:19.080 --> 00:03:20.330
Next problem.

00:03:22.430 --> 00:03:24.290
OK, let me copy and paste
the whole thing.

00:03:27.470 --> 00:03:29.996
Copy and paste is essential with
these geometry problems,

00:03:29.996 --> 00:03:32.750
we won't have to redraw
everything.

00:03:32.750 --> 00:03:38.230
OK, in the diagram below, angle
1, and this right here,

00:03:38.230 --> 00:03:39.510
you should learn,
means congruent.

00:03:42.510 --> 00:03:45.110
When you say two angles are
congruent, so in this case

00:03:45.110 --> 00:03:48.200
they're saying angle 1 is
congruent to angle 4, that

00:03:48.200 --> 00:03:49.990
means that they have the
same angle measure.

00:03:49.990 --> 00:03:53.240
And the only difference between
congruent and being

00:03:53.240 --> 00:03:57.210
equal is that congruent says
they can have the same angle

00:03:57.210 --> 00:03:59.040
measure, but they could be
in different directions.

00:03:59.040 --> 00:04:02.440
And the rays that come out
from them could be of

00:04:02.440 --> 00:04:03.380
different lengths.

00:04:03.380 --> 00:04:05.370
Although I'd often say
that's equal as well.

00:04:05.370 --> 00:04:07.160
But if we're dealing
with congruency,

00:04:07.160 --> 00:04:07.800
that's what it means.

00:04:07.800 --> 00:04:10.110
It essentially just means the
angle measures are equal.

00:04:10.110 --> 00:04:11.560
So we could draw that here.

00:04:11.560 --> 00:04:15.030
Angle 1 is congruent
to angle 4.

00:04:15.030 --> 00:04:16.740
It just mean that these angle
measures are the same.

00:04:16.740 --> 00:04:20.019
Whether we're measuring them
in degrees or radians.

00:04:20.019 --> 00:04:22.230
All right, now what do they
want us to come to a

00:04:22.230 --> 00:04:22.760
conclusion?

00:04:22.760 --> 00:04:24.400
Which of the following
conclusions does

00:04:24.400 --> 00:04:26.550
not have to be true?

00:04:26.550 --> 00:04:28.480
Does not have to be true.

00:04:28.480 --> 00:04:29.430
Not.

00:04:29.430 --> 00:04:31.640
Angles 3 and 4 are supplementary
angles.

00:04:31.640 --> 00:04:32.890
What does supplementary mean?

00:04:32.890 --> 00:04:37.620
That means that angle
3 plus angle 4 have

00:04:37.620 --> 00:04:38.740
to be equal to 180.

00:04:38.740 --> 00:04:41.660
This is the definition of
a supplementary angle.

00:04:46.000 --> 00:04:47.250
Supplementary.

00:04:48.965 --> 00:04:51.320
So angles 3 and 4, they're
actually opposite angles.

00:04:51.320 --> 00:04:52.150
And you can play with these.

00:04:52.150 --> 00:04:55.195
If you had these two lines and
you kind of changed the angle

00:04:55.195 --> 00:04:57.870
at which they instersect, you
would see that angles 3 and 4

00:04:57.870 --> 00:04:59.680
are actually going to
be congruent angles.

00:04:59.680 --> 00:05:00.880
They're always going
to be equal to

00:05:00.880 --> 00:05:02.930
each other in measure.

00:05:02.930 --> 00:05:04.610
So they are equal
to each other.

00:05:04.610 --> 00:05:08.620
And if angle 4 is, we don't
know what it is.

00:05:08.620 --> 00:05:11.450
If angle 4 is 95 degrees,
then angle 3 is also

00:05:11.450 --> 00:05:13.270
going to be 95 degrees.

00:05:13.270 --> 00:05:15.790
If angle 4 is 30 degrees,
angle 3 is also

00:05:15.790 --> 00:05:17.120
going to be 30 degrees.

00:05:17.120 --> 00:05:20.890
So I can think of a bunch of
cases where this will not be

00:05:20.890 --> 00:05:22.260
equal to 180.

00:05:22.260 --> 00:05:24.820
The only way that this would be
equal to 180, angle 3 plus

00:05:24.820 --> 00:05:28.470
angle 4, is if angle 4 were a
90 degree angle, and angle 3

00:05:28.470 --> 00:05:29.570
were also a 90 degree angle.

00:05:29.570 --> 00:05:31.020
But they don't tell us that.

00:05:31.020 --> 00:05:34.240
All they tell us is that angle 4
and angle 1 are the same, at

00:05:34.240 --> 00:05:37.020
least when you measure
the angles.

00:05:37.020 --> 00:05:38.710
So I would already
go with choice A.

00:05:38.710 --> 00:05:40.000
That does not have to be true.

00:05:40.000 --> 00:05:42.050
That'll only be true
if both of those

00:05:42.050 --> 00:05:44.000
angles were 90 degrees.

00:05:44.000 --> 00:05:47.790
Let's see, line L is
parallel to line M.

00:05:47.790 --> 00:05:49.700
That's true.

00:05:49.700 --> 00:05:54.160
If that angle is equal to this
angle, the best way to think

00:05:54.160 --> 00:05:57.490
about it is, this angle is also
equal to, and you could

00:05:57.490 --> 00:06:00.910
watch the videos on the
angle game that I do.

00:06:00.910 --> 00:06:02.620
We do this quite a bit.

00:06:02.620 --> 00:06:03.850
But opposite angles are equal.

00:06:03.850 --> 00:06:05.710
And that should be intuitive
to you at this point.

00:06:05.710 --> 00:06:10.380
Because you can imagine that if
these two lines, if I were

00:06:10.380 --> 00:06:12.620
to change the angle at which
they cross, no matter what

00:06:12.620 --> 00:06:14.670
angle I do it at, that's
always going

00:06:14.670 --> 00:06:15.970
to be equal to this.

00:06:15.970 --> 00:06:18.640
So angle 1 is going to
congruent to angle 2.

00:06:18.640 --> 00:06:21.970
And if these two lines are
parallel, if L and M are

00:06:21.970 --> 00:06:24.970
parallel, then 2 and 4 are
going to be the same.

00:06:24.970 --> 00:06:26.770
Or you can think of
it the other way.

00:06:26.770 --> 00:06:31.080
If 4 and 1 are the same, and 1
is the same as 2, then that

00:06:31.080 --> 00:06:32.330
means 4 is the same as 2.

00:06:32.330 --> 00:06:35.020
And if 4 and 2 are the same,
then that means that these two

00:06:35.020 --> 00:06:36.350
lines are parallel.

00:06:39.080 --> 00:06:41.540
So this is definitely true.

00:06:41.540 --> 00:06:45.400
Angle 1 is congruent
to angle 3.

00:06:45.400 --> 00:06:50.170
Once again, if angle 1 is
congruent to angle 4, so those

00:06:50.170 --> 00:06:53.220
two are congruent, and angle
3 is congruent to angle 4,

00:06:53.220 --> 00:06:55.880
because they're opposite
angles.

00:06:55.880 --> 00:06:59.780
Or instead of saying congruent,
I could say equal.

00:06:59.780 --> 00:07:02.080
If this is equal to this, and
this is equal to this, then

00:07:02.080 --> 00:07:04.120
this is equal to that.

00:07:04.120 --> 00:07:06.350
All right.

00:07:06.350 --> 00:07:10.420
And then, the last one,
2 is congruent to 3.

00:07:10.420 --> 00:07:12.070
2 is congruent to 3.

00:07:12.070 --> 00:07:21.070
Well, by the same logic, if 1 is
congruent to 4, and since 1

00:07:21.070 --> 00:07:24.440
and 2 are opposite, it's
also the same as 2.

00:07:24.440 --> 00:07:28.470
And 4, because it's opposite of
3, it's congruent to that,

00:07:28.470 --> 00:07:30.180
all these angles have to
be the same thing.

00:07:30.180 --> 00:07:33.640
So 2 and 3 would also
be congruent angles.

00:07:33.640 --> 00:07:36.170
So all of the other ones must
be true, B, C and D.

00:07:36.170 --> 00:07:37.635
So A is definitely our choice.

00:07:40.280 --> 00:07:41.530
Next problem.

00:07:44.916 --> 00:07:48.630
Let me copy and paste it.

00:07:48.630 --> 00:07:49.880
OK.

00:07:54.010 --> 00:07:56.200
Consider the arguments below.

00:07:56.200 --> 00:08:01.310
Every multiple of 4 is even,
376 is a multiple of 4.

00:08:01.310 --> 00:08:03.800
Therefore, 376 is even.

00:08:03.800 --> 00:08:05.030
Fair enough.

00:08:05.030 --> 00:08:06.990
A number can be written
as a repeating

00:08:06.990 --> 00:08:09.060
decimal if it is rational.

00:08:09.060 --> 00:08:12.730
Pi cannot be written as
a repeating decimal.

00:08:12.730 --> 00:08:15.120
Therefore, pi is not rational.

00:08:15.120 --> 00:08:18.920
Which ones, if any, use
deductive reasoning?

00:08:18.920 --> 00:08:22.300
OK, so statement one, every
multiple of 4 is even.

00:08:22.300 --> 00:08:25.250
376 is a multiple of 4.

00:08:25.250 --> 00:08:27.270
So this is deductive
reasoning.

00:08:27.270 --> 00:08:30.680
Because you know that every
multiple of 4 is even.

00:08:30.680 --> 00:08:33.059
So you pick any multiple of
4, it's going to be even.

00:08:33.059 --> 00:08:34.530
376 is a multiple of 4.

00:08:34.530 --> 00:08:35.530
Therefore, it has to be even.

00:08:35.530 --> 00:08:37.250
So this is correct logic.

00:08:37.250 --> 00:08:40.620
So statement one is definitely
deductive reasoning.

00:08:40.620 --> 00:08:41.590
Let's see, statement
number two.

00:08:41.590 --> 00:08:43.429
A number can be written
as a repeating

00:08:43.429 --> 00:08:45.800
decimal if it is rational.

00:08:45.800 --> 00:08:54.000
So if you're rational, that
means that you can write it as

00:08:54.000 --> 00:08:55.250
a repeating decimal.

00:08:57.600 --> 00:08:59.330
That's like 0.33333.

00:08:59.330 --> 00:09:00.650
That's 1/3.

00:09:00.650 --> 00:09:02.175
That's all they mean by
a repeating decimal.

00:09:05.960 --> 00:09:08.010
But notice, this statement right
here, a number can be

00:09:08.010 --> 00:09:11.510
written as a repeating decimal
if it is rational.

00:09:11.510 --> 00:09:17.050
That doesn't say that a
repeating decimal means that

00:09:17.050 --> 00:09:17.610
it's rational.

00:09:17.610 --> 00:09:19.720
It just means that a rational
number can be written as a

00:09:19.720 --> 00:09:20.740
repeating decimal.

00:09:20.740 --> 00:09:22.390
This statement doesn't let
us go the other way.

00:09:22.390 --> 00:09:24.330
It doesn't say that a repeating
decimal can

00:09:24.330 --> 00:09:26.500
definitely be written as
a rational number.

00:09:26.500 --> 00:09:30.840
It just says that if it is
rational, a number can be

00:09:30.840 --> 00:09:32.330
written as a repeating
decimal.

00:09:32.330 --> 00:09:33.510
Fair enough.

00:09:33.510 --> 00:09:36.110
And then it says pi
cannot be written

00:09:36.110 --> 00:09:38.000
as a repeating decimal.

00:09:38.000 --> 00:09:41.590
Pi cannot be written as
a repeating decimal.

00:09:41.590 --> 00:09:44.890
So if pi cannot be written
as a repeating

00:09:44.890 --> 00:09:47.180
decimal, pi be rational?

00:09:47.180 --> 00:09:52.450
Well if pi was rational, if pi
was in this set, if pi were

00:09:52.450 --> 00:09:56.730
rational, then you could write
it as a repeating decimal.

00:09:56.730 --> 00:09:57.954
But it said that you
cannot write it

00:09:57.954 --> 00:09:58.840
as a repeating decimal.

00:09:58.840 --> 00:10:01.460
So therefore, pi cannot
be rational.

00:10:01.460 --> 00:10:03.520
Cannot be in the set
of rationals.

00:10:03.520 --> 00:10:05.760
So therefore this is also sound
deductive reasoning.

00:10:05.760 --> 00:10:10.950
So both one and two use
deductive reasoning.

00:10:10.950 --> 00:10:14.060
As far as I know.

00:10:14.060 --> 00:10:14.710
I'm out of time.

00:10:14.710 --> 00:10:16.490
See you in the next video.