Other triangle congruence postulates

WEBVTT
Kind: captions
Language: en

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We now know that if
we have two triangles

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and all of their
corresponding sides

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are the same, so by
side, side, side-- so

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if the corresponding
sides, all three

00:00:09.670 --> 00:00:12.150
of the corresponding sides,
have the same length,

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we know that those
triangles are congruent.

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What I want to do
in this video is

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explore if there
are other properties

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that we can find between
the triangles that

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can help us feel pretty good
that those two triangles would

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be congruent.

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So side, side, side works.

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What about angle, angle, angle?

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So let me do that over here.

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What about angle angle angle?

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So what I'm saying
is, is if-- let's say

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I have a triangle
like this, like I

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have a triangle like that, and
I have a triangle like this.

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And if we know that this angle
is congruent to that angle,

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if this angle is congruent
to that angle, which

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means that their measures are
equal, or-- and-- I should say

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and-- and that angle is
congruent to that angle,

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can we say that these are
two congruent triangles?

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And at first case, it
looks like maybe it

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is, at least the
way I drew it here.

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But when you think
about it, you can

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have the exact same
corresponding angles,

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having the same measure
or being congruent,

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but you could actually scale one
of these triangles up and down

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and still have that property.

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For example, if I had this
triangle right over here,

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it looks similar--
and I'm using that

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in just the everyday
language sense--

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it has the same shape as these
triangles right over here.

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And it has the same angles.

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That angle is congruent
to that angle,

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this angle down here is
congruent to this angle

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over here, and this
angle over here

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is congruent to this
angle over here.

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So all of the angles in all
three of these triangles

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are the same.

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The corresponding angles
have the same measure.

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But clearly, clearly this
triangle right over here

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is not the same.

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It is not congruent
to the other two.

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The sides have a very
different length.

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This side is much shorter than
this side right over here.

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This side is much shorter
than that side over there.

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And this side is much
shorter over here.

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So with just angle,
angle, angle,

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you cannot say that a triangle
has the same size and shape.

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It does have the same shape
but not the same size.

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So this does not
imply congruency.

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So angle, angle, angle
does not imply congruency.

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What it does imply, and we
haven't talked about this yet,

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is that these are
similar triangles.

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So angle, angle,
angle implies similar.

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So let me write it over here.

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It implies similar triangles.

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And similar-- you
probably are use

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to the word in just everyday
language-- but similar

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has a very specific
meaning in geometry.

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And similar things have
the same shape but not

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necessarily the same size.

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So anything that is
congruent, because it

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has the same size and
shape, is also similar.

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But not everything that is
similar is also congruent.

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So for example, this
triangle is similar-- all

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of these triangles are
similar to each other,

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but they aren't all congruent.

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These two are congruent
if their sides

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are the same-- I didn't
make that assumption.

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But if we know that
their sides are the same,

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then we can say that
they're congruent.

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But neither of these are
congruent to this one right

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over here, because this
is clearly much larger.

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It has the same shape
but a different size.

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So we can't have an AAA
postulate or an AAA axiom

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to get to congruency.

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What about side, angle, side?

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So let's try this out,
side, angle, side.

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So let's start off with one
triangle right over here.

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So let's start off with a
triangle that looks like this.

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I have my blue side,
I have my pink side,

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and I have my magenta side.

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And let's say that I have
another triangle that

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has this blue side.

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It has the same side, same
length as that blue side.

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So let me draw it like that.

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It has the same length
as that blue side.

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So that length and that length
are going to be the same.

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It has a congruent
angle right after that.

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So this angle and the next
angle for this triangle

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are going to have
the same measure,

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or they're going
to be congruent.

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And then the next
side is going to have

00:04:07.207 --> 00:04:08.790
the same length as
this one over here.

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So that's going to be the
same length as this over here.

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So it's going to
be the same length.

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And because we only know that
two of the corresponding sides

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have the same length, and
the angle between them--

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and this is important--
the angle between the two

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corresponding sides also
have the same measure,

00:04:28.630 --> 00:04:32.230
we can do anything we want with
this last side on this one.

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We can essentially--
it's going to have

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to start right over here.

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You could start from this point.

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And we can pivot it to
form any triangle we want.

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But we can see, the only
way we can form a triangle

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is if we bring this side
all the way over here

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and close this right over there.

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And so we can see just
logically for two triangles,

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they have one side that
has the length the same,

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the next side has
a length the same,

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and the angle in between
them-- so this angle--

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let me do that in the
same color-- this angle in

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between them, this is the angle.

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This A is this angle
and that angle.

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It's the angle in between them.

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This first side is in blue.

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And this second side right,
over here, is in pink.

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Well, it's already
written in pink.

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So we can see that if two
sides are the same, have

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the same length-- two
corresponding sides have

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the same length, and
the corresponding angle

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between them, they
have to be congruent.

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There's no other one place
to put this third side.

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So SAS-- and sometimes,
it's once again

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called a postulate, an axiom,
or if it's kind of proven,

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sometimes is called
a theorem-- this

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does imply that the two
triangles are congruent.

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So we will give ourselves
this tool in our tool kit.

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We had the SSS postulate.

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Now we have the SAS postulate.

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Two sides are equal and
the angle in between them,

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for two triangles,
corresponding sides and angles,

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then we can say that
it is definitely--

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these are congruent triangles.

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Now what about--
and I'm just going

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to try to go through all the
different combinations here--

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what if I have
angle, side, angle?

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So let me try that.

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So what happens if I
have angle, side, angle?

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So let's go back to this
one right over here.

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So actually, let me
just redraw a new one

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for each of these cases.

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So angle, side, angle, so
I'll draw a triangle here.

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So I have this triangle.

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So this would be maybe the side.

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That would be the side.

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So let me draw the whole
triangle, actually, first.

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So I have this triangle.

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Let me draw one side over here.

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And then let me draw
one side over there.

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And this angle right
over here, I'll

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call it-- I'll do it in orange.

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And this angle over here,
I will do it in yellow.

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So if I have another
triangle that

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has one side having
equal measure--

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so I'll use it as this
blue side right over here.

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So it has one side
that has equal measure.

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And the two angles on
either side of that side,

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or at either end of
that side, are the same,

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will this triangle
necessarily be congruent?

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And we're just going to
try to reason it out.

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These aren't formal proofs.

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We're really just
trying to set up

00:07:05.986 --> 00:07:07.680
what are reasonable
postulates, or what

00:07:07.680 --> 00:07:11.400
are reasonable assumptions
we can have in our tool kit

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as we try to prove other things.

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So that angle, let's call it
that angle, right over there,

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they're going to have the
same measure in this triangle.

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And this angle right
over here in yellow

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is going to have the same
measure on this triangle

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right over here.

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So regardless,
I'm not in any way

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constraining the
sides over here.

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So this side right over
here could have any length.

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It could have any length, but it
has to form this angle with it.

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So it could have any length.

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And it can just go as
far as it wants to go.

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In no way have we constrained
what the length of that is.

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And actually, let me
mark this off, too.

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So this is the same as this.

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So that side can be anything.

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We haven't constrained it all.

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And once again, this
side could be anything.

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We haven't
constrained it at all.

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But we know it has
to go at this angle.

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So it has to go at that angle.

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Well, once again,
there's only one triangle

00:08:00.424 --> 00:08:01.590
that can be formed this way.

00:08:01.590 --> 00:08:03.298
We can say all day
that this length could

00:08:03.298 --> 00:08:05.230
be as long as we want
or as short as we want.

00:08:05.230 --> 00:08:06.900
And this one could be as
long as we want and as short

00:08:06.900 --> 00:08:07.750
as we want.

00:08:07.750 --> 00:08:10.083
But the only way that they
can actually touch each other

00:08:10.083 --> 00:08:12.760
and form a triangle and
have these two angles,

00:08:12.760 --> 00:08:18.390
is if they are the exact same
length as these two sides

00:08:18.390 --> 00:08:19.140
right over here.

00:08:19.140 --> 00:08:22.761
So this side will actually have
to be the same as that side.

00:08:22.761 --> 00:08:24.760
And this would have to
be the same as that side.

00:08:24.760 --> 00:08:26.790
Once again, this isn't a proof.

00:08:26.790 --> 00:08:30.080
I'd call it more of a reasoning
through it or an investigation,

00:08:30.080 --> 00:08:32.919
really just to establish
what reasonable baselines,

00:08:32.919 --> 00:08:36.120
or axioms, or assumptions, or
postulates that we could have.

00:08:36.120 --> 00:08:40.070
So for my purposes, I
think ASA does show us

00:08:40.070 --> 00:08:44.727
that two triangles
are congruent.

00:08:44.727 --> 00:08:45.810
Now let's try another one.

00:08:45.810 --> 00:08:49.010
Let's try angle, angle, side.

00:08:49.010 --> 00:08:50.510
Let's try angle, angle, side.

00:08:50.510 --> 00:08:52.571
And in some geometry
classes, maybe

00:08:52.571 --> 00:08:54.320
if you have to go
through an exam quickly,

00:08:54.320 --> 00:08:57.086
you might memorize, OK, side,
side, side implies congruency.

00:08:57.086 --> 00:08:58.210
And that's kind of logical.

00:08:58.210 --> 00:08:59.751
Side, angle, side
implies congruency,

00:08:59.751 --> 00:09:01.660
and so on, and so forth.

00:09:01.660 --> 00:09:03.140
I'm not a fan of memorizing it.

00:09:03.140 --> 00:09:04.880
It might be good
for time pressure.

00:09:04.880 --> 00:09:07.470
It is good to, sometimes, even
just go through this logic.

00:09:07.470 --> 00:09:09.420
If you're like, wait, does
angle, angle, angle work?

00:09:09.420 --> 00:09:10.878
Well, no, I can
find this case that

00:09:10.878 --> 00:09:12.440
breaks down angle, angle, angle.

00:09:12.440 --> 00:09:14.540
If these work, just try
to verify for yourself

00:09:14.540 --> 00:09:17.880
that they make logical sense
why they would imply congruency.

00:09:17.880 --> 00:09:20.910
Now, let's try
angle, angle, side.

00:09:20.910 --> 00:09:24.620
Let's try angle, angle, side.

00:09:24.620 --> 00:09:27.670
So once again, let's have
a triangle over here.

00:09:27.670 --> 00:09:28.936
It has some side.

00:09:28.936 --> 00:09:31.310
So this one is going to be a
little bit more interesting.

00:09:31.310 --> 00:09:32.270
So it has some side.

00:09:32.270 --> 00:09:34.184
That's the side
right over there.

00:09:34.184 --> 00:09:35.350
And then, it has two angles.

00:09:35.350 --> 00:09:38.750
So let me draw the other
sides of this triangle.

00:09:38.750 --> 00:09:42.000
I'll draw one in magenta
and then one in green.

00:09:42.000 --> 00:09:45.720
And there's two angles
and then the side.

00:09:45.720 --> 00:09:48.200
So let's say you
have this angle-- you

00:09:48.200 --> 00:09:49.650
have that angle
right over there.

00:09:49.650 --> 00:09:50.720
Actually, I didn't
have to put a double,

00:09:50.720 --> 00:09:53.053
because that's the first angle
that I'm-- So I have that

00:09:53.053 --> 00:09:55.580
angle, which we'll refer
to as that first A.

00:09:55.580 --> 00:10:01.110
Then we have this angle,
which is that second A.

00:10:01.110 --> 00:10:05.570
So if I know that there's
another triangle that

00:10:05.570 --> 00:10:08.860
has one side having the
same length-- so let me draw

00:10:08.860 --> 00:10:11.680
it like that-- it has one
side having the same length.

00:10:11.680 --> 00:10:16.140
It has one angle on that side
that has the same measure.

00:10:16.140 --> 00:10:18.520
So it has a measure like that.

00:10:18.520 --> 00:10:23.950
And so this side right over
here could be of any length.

00:10:23.950 --> 00:10:26.590
We aren't constraining what
the length of that side is.

00:10:26.590 --> 00:10:28.980
But whatever the angle is on
the other side of that side

00:10:28.980 --> 00:10:32.350
is going to be the same as this
green angle right over here.

00:10:32.350 --> 00:10:34.327
So for example, it
could be like that.

00:10:34.327 --> 00:10:36.410
And then you could have a
green side go like that.

00:10:36.410 --> 00:10:39.650
It could be like that and have
the green side go like that.

00:10:39.650 --> 00:10:41.560
And if we have--
so the only thing

00:10:41.560 --> 00:10:46.490
we're assuming is that this
is the same length as this,

00:10:46.490 --> 00:10:49.360
and that this angle is the
same measure as that angle,

00:10:49.360 --> 00:10:51.710
and that this measure is the
same measure as that angle.

00:10:51.710 --> 00:10:54.390
And this magenta line
can be of any length,

00:10:54.390 --> 00:10:56.330
and this green line
can be of any length.

00:10:56.330 --> 00:10:58.440
We in no way have
constrained that.

00:10:58.440 --> 00:11:01.590
But can we form any triangle
that is not congruent to this?

00:11:01.590 --> 00:11:04.000
Because the bottom line
is, this green line

00:11:04.000 --> 00:11:06.330
is going to touch this
one right over there.

00:11:06.330 --> 00:11:10.040
And the only way it's going to
touch that one right over there

00:11:10.040 --> 00:11:13.370
is if it starts right
over here, because we're

00:11:13.370 --> 00:11:15.210
constraining this
angle right over here.

00:11:15.210 --> 00:11:16.970
We're constraining that angle.

00:11:16.970 --> 00:11:18.630
And so it looks
like angle, angle,

00:11:18.630 --> 00:11:23.220
side does indeed
imply congruency.

00:11:23.220 --> 00:11:26.110
So that does imply congruency.

00:11:26.110 --> 00:11:28.860
So let's just do one more
just to kind of try out

00:11:28.860 --> 00:11:30.806
all of the different situations.

00:11:30.806 --> 00:11:32.180
What if we have--
and I'm running

00:11:32.180 --> 00:11:33.240
out of a little
bit of real estate

00:11:33.240 --> 00:11:34.970
right over here at
the bottom-- what

00:11:34.970 --> 00:11:39.030
if we tried out
side, side, angle?

00:11:39.030 --> 00:11:42.000
So once again, draw a triangle.

00:11:42.000 --> 00:11:44.310
So it has one side there.

00:11:44.310 --> 00:11:46.710
It has another side there.

00:11:46.710 --> 00:11:49.750
And then-- I don't have to
do those hash marks just yet.

00:11:49.750 --> 00:11:54.497
So one side, then another
side, and then another side.

00:11:54.497 --> 00:11:57.080
And what happens if we know that
there's another triangle that

00:11:57.080 --> 00:12:00.470
has two of the sides the same
and then the angle after it?

00:12:00.470 --> 00:12:04.480
So for example, we would have
that side just like that,

00:12:04.480 --> 00:12:05.940
and then it has another side.

00:12:05.940 --> 00:12:09.540
But we're not
constraining the angle.

00:12:09.540 --> 00:12:11.910
We aren't constraining
this angle right over here,

00:12:11.910 --> 00:12:14.021
but we're constraining
the length of that side.

00:12:14.021 --> 00:12:15.020
So let me color code it.

00:12:15.020 --> 00:12:17.500
So that blue side
is that first side.

00:12:17.500 --> 00:12:21.150
Then we have this magenta
side right over there.

00:12:21.150 --> 00:12:23.835
So this is going to be the same
length as this right over here.

00:12:23.835 --> 00:12:25.460
But let me make it
at a different angle

00:12:25.460 --> 00:12:27.350
to see if I can disprove it.

00:12:27.350 --> 00:12:29.270
So let's say it looks like that.

00:12:29.270 --> 00:12:31.840
Or actually let me make
it even more interesting.

00:12:36.340 --> 00:12:39.600
Let me try to make it like that.

00:12:39.600 --> 00:12:41.090
So it's a very different angle.

00:12:41.090 --> 00:12:43.600
But now, it has to have
the same angle out here.

00:12:43.600 --> 00:12:45.700
It has to have that
same angle out here.

00:12:45.700 --> 00:12:47.640
So it has to be
roughly that angle.

00:12:47.640 --> 00:12:50.680
So it actually looks like we
can draw a triangle that is not

00:12:50.680 --> 00:12:54.429
congruent that has two
sides being the same length

00:12:54.429 --> 00:12:55.720
and then an angle is different.

00:12:55.720 --> 00:12:57.660
For example, this
is pretty much that.

00:12:57.660 --> 00:13:00.550
I made this angle
smaller than this angle.

00:13:00.550 --> 00:13:02.250
These two sides are the same.

00:13:02.250 --> 00:13:05.495
This angle is the same now, but
what the byproduct of that is,

00:13:05.495 --> 00:13:08.310
is that this green side is going
to be shorter on this triangle

00:13:08.310 --> 00:13:09.470
right over here.

00:13:09.470 --> 00:13:12.740
So you don't necessarily
have congruent triangles

00:13:12.740 --> 00:13:16.220
with side, side, angle.

00:13:16.220 --> 00:13:19.350
So this is not
necessarily congruent,

00:13:19.350 --> 00:13:23.520
not necessarily, or similar.

00:13:23.520 --> 00:13:27.650
It gives us neither
congruency nor similarity.


SSS to show a radius is perpendicular to a chord that it bisects

WEBVTT
Kind: captions
Language: en

00:00:00.384 --> 00:00:02.050
In the last video,
we learned that if we

00:00:02.050 --> 00:00:05.035
have two different
triangles and if all

00:00:05.035 --> 00:00:08.060
of the corresponding
sides of the two triangles

00:00:08.060 --> 00:00:11.770
have the same length,
then by side-side-side,

00:00:11.770 --> 00:00:14.640
we know that the two
triangles are congruent.

00:00:14.640 --> 00:00:16.580
And I also touched a
little bit on the idea

00:00:16.580 --> 00:00:18.030
of an axiom or postulate.

00:00:18.030 --> 00:00:19.230
But I want to be clear.

00:00:19.230 --> 00:00:21.070
Sometimes you will
hear this referred

00:00:21.070 --> 00:00:25.880
to as a side-side-side
theorem, and sometimes you'll

00:00:25.880 --> 00:00:30.900
hear it as a side-side-side
postulate or axiom.

00:00:30.900 --> 00:00:33.090
And I think it's
worth differentiating

00:00:33.090 --> 00:00:34.700
what these mean.

00:00:34.700 --> 00:00:36.200
A postulate or an
axiom is something

00:00:36.200 --> 00:00:38.430
that you just assume, you
assume from the get-go,

00:00:38.430 --> 00:00:42.480
while a theorem is something
you prove using more basic

00:00:42.480 --> 00:00:46.050
or using some
postulates or axioms.

00:00:46.050 --> 00:00:50.660
Really in all of mathematics,
you make some core assumptions.

00:00:50.660 --> 00:00:54.010
You call these the
axioms or the postulates.

00:00:59.480 --> 00:01:02.900
And then using those, you
try to prove theorems.

00:01:02.900 --> 00:01:07.030
So maybe using that one, I can
prove some theorem over here.

00:01:07.030 --> 00:01:09.610
And maybe using that
theorem, and then this axiom,

00:01:09.610 --> 00:01:11.880
I can prove another
theorem over here.

00:01:11.880 --> 00:01:13.900
And then using both
of those theorems,

00:01:13.900 --> 00:01:15.640
I can prove another
theorem over here.

00:01:15.640 --> 00:01:16.806
I think you get the picture.

00:01:16.806 --> 00:01:19.470
This axiom might lead
us to this theorem.

00:01:19.470 --> 00:01:23.290
These two might lead us to
this theorem right over here.

00:01:23.290 --> 00:01:26.110
And we essentially try
to build our knowledge

00:01:26.110 --> 00:01:30.110
or we build a mathematics
around these core assumptions.

00:01:30.110 --> 00:01:33.270
In an introductory
geometry class,

00:01:33.270 --> 00:01:38.967
we don't rigorously prove
the side-side-side theorem.

00:01:38.967 --> 00:01:40.800
And that's why in a lot
of geometry classes,

00:01:40.800 --> 00:01:42.520
you kind of just
take it as a given,

00:01:42.520 --> 00:01:44.136
as a postulate or an axiom.

00:01:44.136 --> 00:01:45.760
And the whole reason
why I'm doing this

00:01:45.760 --> 00:01:48.134
is one, just so you know the
difference between the words

00:01:48.134 --> 00:01:50.490
theorem and postulate or axiom.

00:01:50.490 --> 00:01:52.209
And also so that you
don't get confused.

00:01:52.209 --> 00:01:54.000
It is just a given,
but in a lot of books--

00:01:54.000 --> 00:01:55.416
and I've looked
at several books--

00:01:55.416 --> 00:01:57.629
they do refer to it as the
side-side-side theorem,

00:01:57.629 --> 00:01:59.420
even though they never
prove it rigorously.

00:01:59.420 --> 00:02:00.600
They do just assume it.

00:02:00.600 --> 00:02:03.690
So it really is more of
a postulate or an axiom.

00:02:03.690 --> 00:02:05.494
Now with that out
of the way, we're

00:02:05.494 --> 00:02:06.910
just going to
assume going forward

00:02:06.910 --> 00:02:08.410
that we just know
that this is true.

00:02:08.410 --> 00:02:09.820
We're going to
take it as a given.

00:02:09.820 --> 00:02:11.403
I want to show you
that we can already

00:02:11.403 --> 00:02:13.507
do something pretty
useful with it.

00:02:13.507 --> 00:02:14.965
So let's say that
we have a circle.

00:02:17.530 --> 00:02:20.710
And there's many useful things
that we can already do with it.

00:02:20.710 --> 00:02:24.720
And this circle has a
center, right here at A.

00:02:24.720 --> 00:02:28.870
And let's say that we have
a chord in the circle that

00:02:28.870 --> 00:02:30.440
is not a diameter.

00:02:30.440 --> 00:02:32.780
So let me draw a chord here.

00:02:32.780 --> 00:02:35.150
So let me draw a
chord in this circle.

00:02:35.150 --> 00:02:38.350
So it's kind of a
segment of a secant line.

00:02:38.350 --> 00:02:42.690
And let's say that
I have a line that

00:02:42.690 --> 00:02:47.473
bisects this chord
from the center.

00:02:47.473 --> 00:02:49.223
And I guess I call it
a radius because I'm

00:02:49.223 --> 00:02:53.600
going to go from the center
to the edge of the circle

00:02:53.600 --> 00:02:54.570
right over there.

00:02:54.570 --> 00:02:57.830
So I'm going to the center
to the circle itself.

00:02:57.830 --> 00:02:59.760
And when I say bisects
it-- so these are all--

00:02:59.760 --> 00:03:01.870
I'm just setting up
the problem right now.

00:03:01.870 --> 00:03:03.490
When I say bisecting
it, it means

00:03:03.490 --> 00:03:05.830
it splits that line
segment in half.

00:03:05.830 --> 00:03:08.560
So what it tells is, is that
the length of this segment

00:03:08.560 --> 00:03:11.400
right over here is going to
be equivalent to the length

00:03:11.400 --> 00:03:15.170
of this segment
right over there.

00:03:15.170 --> 00:03:15.795
I've set it up.

00:03:15.795 --> 00:03:16.910
I have a circle.

00:03:16.910 --> 00:03:20.380
This radius bisects this
chord right over here.

00:03:20.380 --> 00:03:27.720
And the goal here
is to prove that it

00:03:27.720 --> 00:03:29.909
bisects this chord
at a right angle.

00:03:29.909 --> 00:03:32.200
Or another way to say it--
let me add some points here.

00:03:32.200 --> 00:03:35.890
Let's call this B.
Let's call this C,

00:03:35.890 --> 00:03:38.940
And let's call this
D. I want to prove

00:03:38.940 --> 00:03:43.290
that segment AB
is perpendicular.

00:03:43.290 --> 00:03:45.030
It intersects it
at a right angle.

00:03:45.030 --> 00:03:48.170
It is perpendicular
to segment CD.

00:03:50.682 --> 00:03:52.140
And as you could
imagine, I'm going

00:03:52.140 --> 00:03:55.790
to prove it pretty much using
the side-side-side whatever you

00:03:55.790 --> 00:03:59.220
want to call it, side-side-side
theorem, postulate, or axiom.

00:03:59.220 --> 00:04:00.001
So let's do it.

00:04:00.001 --> 00:04:01.250
Let's think about it this way.

00:04:01.250 --> 00:04:02.610
So you can imagine if
I'm going to use this,

00:04:02.610 --> 00:04:03.859
I need to have some triangles.

00:04:03.859 --> 00:04:05.630
There's no triangles
here right now.

00:04:05.630 --> 00:04:08.130
But I can construct triangles,
and I can construct triangles

00:04:08.130 --> 00:04:09.640
based on things I know.

00:04:09.640 --> 00:04:14.105
For example, I can construct--
this has some radius.

00:04:16.740 --> 00:04:18.074
That's a radius right over here.

00:04:18.074 --> 00:04:20.739
The length of that is just going
to be the radius of the circle.

00:04:20.739 --> 00:04:22.390
But I can also do
it right over here.

00:04:22.390 --> 00:04:26.520
The length of AC is also going
to be the radius of the circle.

00:04:26.520 --> 00:04:29.450
So we know that these two lines
have the same length, which

00:04:29.450 --> 00:04:32.070
is the radius of the circle.

00:04:32.070 --> 00:04:34.500
Or we could say that
AD is congruent to AC,

00:04:34.500 --> 00:04:36.990
or they have the
exact same lengths.

00:04:36.990 --> 00:04:40.380
We know from the
set-up in the problem

00:04:40.380 --> 00:04:43.310
that this segment
is equal in length

00:04:43.310 --> 00:04:45.310
to this segment over here.

00:04:45.310 --> 00:04:47.500
Let me add a point here
so I can refer to it.

00:04:47.500 --> 00:04:49.310
So if I call that
point E, we know

00:04:49.310 --> 00:04:54.210
from the set-up in the problem,
that CE is congruent to ED,

00:04:54.210 --> 00:04:55.620
or they have the same lengths.

00:04:55.620 --> 00:04:58.260
CE has the same length as ED.

00:04:58.260 --> 00:05:01.700
And we also know that both of
these triangles, the one here

00:05:01.700 --> 00:05:06.900
on the left and the one here
on the right, they both share

00:05:06.900 --> 00:05:08.410
the side EA.

00:05:08.410 --> 00:05:11.540
So EA is clearly equal to EA.

00:05:11.540 --> 00:05:13.990
So this is clearly
equal to itself.

00:05:13.990 --> 00:05:14.790
It's the same side.

00:05:14.790 --> 00:05:16.820
The same side is being
used for both triangles.

00:05:16.820 --> 00:05:19.240
The triangles are
adjacent to each other.

00:05:19.240 --> 00:05:23.250
And so we see a situation where
we have two different triangles

00:05:23.250 --> 00:05:26.220
that have corresponding
sides being equal.

00:05:26.220 --> 00:05:29.230
This side is equivalent to
this side right over here.

00:05:29.230 --> 00:05:33.760
This side is equal in length
to that side over there.

00:05:33.760 --> 00:05:37.364
And then, obviously, AE
is equivalent to itself.

00:05:37.364 --> 00:05:38.530
It's a side on both of them.

00:05:38.530 --> 00:05:41.820
It's the corresponding side
on both of these triangles.

00:05:41.820 --> 00:06:05.440
And so by side-side-side, AEC.

00:06:05.440 --> 00:06:07.140
Let me write it over here.

00:06:07.140 --> 00:06:17.304
By side-side-side, we
know that triangle AEC

00:06:17.304 --> 00:06:19.940
is congruent to triangle AED.

00:06:24.080 --> 00:06:25.500
But how does that help us?

00:06:25.500 --> 00:06:27.490
How does that help
us knowing that we

00:06:27.490 --> 00:06:29.110
used our little theorem?

00:06:29.110 --> 00:06:31.860
But how does that
actually help us here?

00:06:31.860 --> 00:06:34.430
Well, what's cool is once we
know that two triangles are

00:06:34.430 --> 00:06:41.050
congruent-- so because they
are congruent, from that,

00:06:41.050 --> 00:06:43.470
we can deduce that all
the angles are the same.

00:06:43.470 --> 00:06:45.950
And in particular,
we can deduce that

00:06:45.950 --> 00:06:53.270
this angle right over here,
that the measure of angle CEA

00:06:53.270 --> 00:06:55.806
is equivalent to the
measure of angle DEA.

00:07:00.860 --> 00:07:03.900
And the reason why that's
useful is that we also

00:07:03.900 --> 00:07:05.680
see, just by looking
at this, that they're

00:07:05.680 --> 00:07:06.905
supplementary to each other.

00:07:06.905 --> 00:07:07.920
They're adjacent angles.

00:07:07.920 --> 00:07:10.830
Their outer sides
form a straight angle.

00:07:10.830 --> 00:07:16.830
So CEA is supplementary
and equivalent to DEA.

00:07:16.830 --> 00:07:18.790
So, they're are
also supplementary.

00:07:18.790 --> 00:07:24.480
So we also have the
measure of angle CEA

00:07:24.480 --> 00:07:30.030
plus the measure of angle
DEA is equal to 180 degrees.

00:07:30.030 --> 00:07:31.730
But they're equivalent
to each other.

00:07:31.730 --> 00:07:35.090
So I could replace the measure
of DEA with the measure of CEA.

00:07:37.690 --> 00:07:42.830
Or I could rewrite this as 2
times the measure of angle CEA

00:07:42.830 --> 00:07:45.110
is equal to 180 degrees.

00:07:45.110 --> 00:07:47.130
Or I could divide
both sides by 2.

00:07:47.130 --> 00:07:49.870
And I say the
measure of angle CEA

00:07:49.870 --> 00:07:52.180
is equal to 90
degrees, which is going

00:07:52.180 --> 00:07:54.710
to be the same as the
measure of angle DEA

00:07:54.710 --> 00:07:56.820
because they're equivalent.

00:07:56.820 --> 00:07:59.495
So we know that this angle
right over here is 90 degrees.

00:07:59.495 --> 00:08:01.600
So I can do it with
that little box.

00:08:01.600 --> 00:08:04.540
And this angle right
over here is 90 degrees.

00:08:04.540 --> 00:08:10.290
And because AB intersects
where it intersects CD,

00:08:10.290 --> 00:08:12.290
we have a 90 degree
angle here and there.

00:08:12.290 --> 00:08:14.900
And we could also prove
this over there as well.

00:08:14.900 --> 00:08:19.090
They are perpendicular
to each other.


Congruent triangles and SSS

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.880
Let's talk a little
bit about congruence.

00:00:04.880 --> 00:00:06.510
And one way to think
about congruence,

00:00:06.510 --> 00:00:09.540
it's really a kind of
equivalence for shape.

00:00:09.540 --> 00:00:12.120
So in algebra when something
is equal to another thing,

00:00:12.120 --> 00:00:13.960
it means that their
quantities are the same.

00:00:13.960 --> 00:00:16.209
But if we're now all of a
sudden talking about shapes,

00:00:16.209 --> 00:00:22.832
and we say that those shapes
are the same size and shape,

00:00:22.832 --> 00:00:24.290
then we say that
they're congruent.

00:00:24.290 --> 00:00:26.830
And just to see a
simple example here.

00:00:26.830 --> 00:00:30.060
I have this triangle
right over there.

00:00:30.060 --> 00:00:33.700
And let's say I have this
triangle right over here.

00:00:33.700 --> 00:00:38.520
And if you are able
to shift this triangle

00:00:38.520 --> 00:00:41.120
and rotate this triangle
and flip this triangle,

00:00:41.120 --> 00:00:43.500
you can make it look
exactly like this triangle.

00:00:43.500 --> 00:00:45.520
As long as you're not
changing the lengths

00:00:45.520 --> 00:00:47.220
of any of the sides
or the angles here.

00:00:47.220 --> 00:00:52.311
But you can flip it, you
can shift it, and rotate it.

00:00:52.311 --> 00:00:53.060
Let me write this.

00:00:53.060 --> 00:00:58.360
You can shift it, you can
flip it, and you can rotate.

00:00:58.360 --> 00:01:00.800
If you can do those
three procedures to make

00:01:00.800 --> 00:01:03.360
these the exact same triangle,
to make them look exactly

00:01:03.360 --> 00:01:06.170
the same, then
they are congruent.

00:01:06.170 --> 00:01:08.290
And if you say that a
triangle is congruent--

00:01:08.290 --> 00:01:09.240
let me label these.

00:01:09.240 --> 00:01:14.090
So let's call this
triangle A, B, and C.

00:01:14.090 --> 00:01:22.180
And let's call this
X, Y, and Z. So

00:01:22.180 --> 00:01:24.340
if we were to say,
if we make the claim,

00:01:24.340 --> 00:01:26.530
that both of these
triangles are congruent,

00:01:26.530 --> 00:01:30.809
so if we say triangle
ABC is congruent--

00:01:30.809 --> 00:01:32.600
and the way you specify
it, it looks almost

00:01:32.600 --> 00:01:34.260
like an equal sign,
but it's an equal sign

00:01:34.260 --> 00:01:35.760
with this little
curly thing on top.

00:01:35.760 --> 00:01:37.640
Let me write a
little bit neater.

00:01:37.640 --> 00:01:39.600
So we would write it like this.

00:01:39.600 --> 00:01:48.070
If we know that triangle ABC
is congruent to triangle XYZ,

00:01:48.070 --> 00:01:50.860
that means that their
corresponding sides have

00:01:50.860 --> 00:01:55.270
the same length, and
their corresponding angles

00:01:55.270 --> 00:01:56.760
have the same measure.

00:01:56.760 --> 00:01:59.960
So if we make this assumption,
or if someone tells us

00:01:59.960 --> 00:02:04.270
that this is true, then
we know, for example,

00:02:04.270 --> 00:02:06.540
that AB is going
to be equal to XY.

00:02:06.540 --> 00:02:08.820
The length of
segment AB is going

00:02:08.820 --> 00:02:12.010
to be equal to the
length of segment XY.

00:02:12.010 --> 00:02:14.030
And we could denote
it like this.

00:02:14.030 --> 00:02:17.140
And I'm assuming that these
are the corresponding sides.

00:02:17.140 --> 00:02:19.220
And you can see it,
actually, by the way

00:02:19.220 --> 00:02:20.800
we've defined these triangles.

00:02:20.800 --> 00:02:24.510
A corresponds to X,
B corresponds to Y,

00:02:24.510 --> 00:02:27.440
and then C corresponds
to Z right over there.

00:02:27.440 --> 00:02:31.564
So side AB is going to have
the same length as side XY.

00:02:31.564 --> 00:02:33.730
And you can sometimes, if
you don't have the colors,

00:02:33.730 --> 00:02:35.960
you would denote
it just like that.

00:02:35.960 --> 00:02:40.830
These two line segments
have the same length.

00:02:40.830 --> 00:02:43.310
And you could actually say
this, and you don't always

00:02:43.310 --> 00:02:47.230
see it written this way, you
could also make the statement

00:02:47.230 --> 00:02:53.940
that line segment AB is
congruent to line segment XY.

00:02:53.940 --> 00:02:56.670
But congruence of
line segments really

00:02:56.670 --> 00:02:59.650
just means that their
lengths are equivalent.

00:02:59.650 --> 00:03:01.957
So these two things
mean the same thing.

00:03:01.957 --> 00:03:04.540
If one line segment is congruent
to another line segment, that

00:03:04.540 --> 00:03:06.360
just means the measure
of one line segment

00:03:06.360 --> 00:03:09.104
is equal to the measure
of the other line segment.

00:03:09.104 --> 00:03:11.270
And so we can go through
all the corresponding sides

00:03:11.270 --> 00:03:13.430
if these two characters
are congruent.

00:03:13.430 --> 00:03:23.120
We also know the length of BC
is going to be the length of YZ,

00:03:23.120 --> 00:03:25.820
assuming that those are
the corresponding sides.

00:03:25.820 --> 00:03:28.180
And we could put these double
hash marks right over here

00:03:28.180 --> 00:03:31.760
to show that these two
lengths are the same.

00:03:31.760 --> 00:03:34.160
And then if we go
to the third side,

00:03:34.160 --> 00:03:37.230
we also know that these are
going to have the same length,

00:03:37.230 --> 00:03:40.730
or the line segments themselves
are going to be congruent.

00:03:40.730 --> 00:03:44.220
So we also know that
the length of AC

00:03:44.220 --> 00:03:47.220
is going to be equal
to the length of XZ.

00:03:50.770 --> 00:03:53.767
Not only do we know that all the
corresponding sides are going

00:03:53.767 --> 00:03:55.600
to have the same length,
if someone tells us

00:03:55.600 --> 00:03:57.400
that a triangle is
congruent, we also

00:03:57.400 --> 00:03:59.120
know that all the
corresponding angles

00:03:59.120 --> 00:04:01.070
are going to have
the same measure.

00:04:01.070 --> 00:04:06.130
So for example, we also know
that this angle's measure

00:04:06.130 --> 00:04:09.710
is going to be the same as the
corresponding angle's measure.

00:04:09.710 --> 00:04:12.060
The corresponding angle
is right over here.

00:04:12.060 --> 00:04:14.290
It's between this orange
side and this blue side.

00:04:14.290 --> 00:04:16.779
Or this orange side and this
purple side, I should say.

00:04:16.779 --> 00:04:19.390
In between this orange
side and this purple side.

00:04:19.390 --> 00:04:27.680
And so it also tells us that
the measure of angle BAC

00:04:27.680 --> 00:04:34.112
is equal to the
measure of angle YXZ.

00:04:34.112 --> 00:04:36.320
Let me write that angle
symbol a little less like a--

00:04:41.570 --> 00:04:45.910
We could also write
that, as angle BAC

00:04:45.910 --> 00:04:50.180
is congruent to angle YXZ.

00:04:50.180 --> 00:04:51.970
And once again,
like line segments,

00:04:51.970 --> 00:04:55.210
if one line segment is congruent
to another line segment,

00:04:55.210 --> 00:04:57.250
it just means that
their lengths are equal.

00:04:57.250 --> 00:04:59.990
And if one angle is
congruent to another angle,

00:04:59.990 --> 00:05:02.755
it just means that their
measures are equal.

00:05:02.755 --> 00:05:04.880
So we know that those two
corresponding angles have

00:05:04.880 --> 00:05:07.000
the same measure,
they're congruent.

00:05:07.000 --> 00:05:09.260
We also know that these two
corresponding angles have

00:05:09.260 --> 00:05:11.051
the same measure, and
I'll use a double arc

00:05:11.051 --> 00:05:13.790
to specify that this has
the same measure as that.

00:05:13.790 --> 00:05:21.970
So we also know that
the measure of angle ABC

00:05:21.970 --> 00:05:24.940
is equal to the
measure of angle XYZ.

00:05:27.970 --> 00:05:33.910
And then finally, we
know that this angle,

00:05:33.910 --> 00:05:36.036
if we know that these two
characters are congruent,

00:05:36.036 --> 00:05:38.701
that this angle is going to have
the same measure as this angle,

00:05:38.701 --> 00:05:40.110
as its corresponding angle.

00:05:40.110 --> 00:05:46.280
So we know that the
measure of angle ACB

00:05:46.280 --> 00:05:49.510
is going to be equal to
the measure of angle XZY.

00:05:53.067 --> 00:05:55.400
Now what we're going to concern
ourselves a lot with, is

00:05:55.400 --> 00:05:56.681
how do we prove congruence?

00:05:56.681 --> 00:05:58.930
Because it's cool, because
if you can prove congruence

00:05:58.930 --> 00:06:00.750
of two triangles,
then all of a sudden

00:06:00.750 --> 00:06:02.960
you can make all of
these assumptions.

00:06:02.960 --> 00:06:06.280
And what we're going to find
out, and this is going to be,

00:06:06.280 --> 00:06:08.350
we're going to assume
it for the sake of kind

00:06:08.350 --> 00:06:10.020
of an introductory
geometry course

00:06:10.020 --> 00:06:12.227
that this is an axiom,
or postulate, or just

00:06:12.227 --> 00:06:13.310
something that you assume.

00:06:13.310 --> 00:06:15.410
And let me actually
just write this down.

00:06:15.410 --> 00:06:19.350
So an axiom, very fancy word.

00:06:19.350 --> 00:06:22.340
Postulate, also a
very fancy word.

00:06:22.340 --> 00:06:25.760
It really just means things that
we're going to assume are true.

00:06:25.760 --> 00:06:28.590
An axiom is sometimes, there's
a little bit of distinction

00:06:28.590 --> 00:06:31.120
sometimes, where someone will
say an axiom is something

00:06:31.120 --> 00:06:34.850
that's self-evident, or it
seems like a universal truth

00:06:34.850 --> 00:06:36.090
that is definitely true.

00:06:36.090 --> 00:06:37.423
And we just take it for granted.

00:06:37.423 --> 00:06:38.760
You can't prove an axiom.

00:06:38.760 --> 00:06:41.024
A postulate kind of
has that same role,

00:06:41.024 --> 00:06:42.440
but sometimes you
could say, well,

00:06:42.440 --> 00:06:44.410
let's just assume
this is true and see,

00:06:44.410 --> 00:06:47.070
if we assume that that's true,
what we can derive from it.

00:06:47.070 --> 00:06:49.500
What we can prove if
we assume it's true.

00:06:49.500 --> 00:06:52.210
But for the sake of a
introductory geometry class,

00:06:52.210 --> 00:06:53.680
and really in most
of mathematics

00:06:53.680 --> 00:06:55.970
today, these two words
are used interchangeably.

00:06:55.970 --> 00:06:58.820
An axiom or postulate,
just very fancy words

00:06:58.820 --> 00:07:00.550
for things that we
take as a given.

00:07:00.550 --> 00:07:02.170
Things that we just will assume.

00:07:02.170 --> 00:07:03.150
We won't prove them.

00:07:03.150 --> 00:07:05.170
We will start with
these assumptions,

00:07:05.170 --> 00:07:07.500
and then we're just going
to build up from there.

00:07:07.500 --> 00:07:09.890
And one of the core ones
that we'll see in geometry

00:07:09.890 --> 00:07:14.920
is the axiom, or the postulate,
that if all the sides are

00:07:14.920 --> 00:07:17.710
congruent, or if the lengths of
all the sides of the triangle

00:07:17.710 --> 00:07:21.120
are congruent,
then we are dealing

00:07:21.120 --> 00:07:22.740
with congruent triangles.

00:07:22.740 --> 00:07:26.040
So it's sometimes called
a side-side-side postulate

00:07:26.040 --> 00:07:27.046
or axiom.

00:07:27.046 --> 00:07:28.420
We're not going
to prove it here.

00:07:28.420 --> 00:07:29.630
We're just going to
take it as a given.

00:07:29.630 --> 00:07:31.463
So this literally stands
for side-side-side.

00:07:35.810 --> 00:07:44.286
And what it tells us is, if
we have two triangles-- and so

00:07:44.286 --> 00:07:46.160
let's say that's, and
that's another triangle

00:07:46.160 --> 00:07:49.710
right over there-- and we
know that corresponding sides

00:07:49.710 --> 00:07:50.880
are equal.

00:07:50.880 --> 00:07:54.310
So we know that this side right
over here is equal in length

00:07:54.310 --> 00:07:59.060
to that side, we know that
this side right over here

00:07:59.060 --> 00:08:02.300
is equal in length to that
side right over there,

00:08:02.300 --> 00:08:07.470
and we know that this side
over here is equal in length

00:08:07.470 --> 00:08:09.510
to this side over here.

00:08:09.510 --> 00:08:11.400
Then we know, and
we're just going

00:08:11.400 --> 00:08:15.050
to take this as an assumption,
and can build up off of this,

00:08:15.050 --> 00:08:17.030
we know that they are congruent.

00:08:17.030 --> 00:08:21.384
That these two triangles
are congruent to each other.

00:08:21.384 --> 00:08:23.800
I didn't put any labels there,
so it's kind of hard for me

00:08:23.800 --> 00:08:24.870
to refer to them.

00:08:24.870 --> 00:08:26.660
But these two are
congruent triangles.

00:08:26.660 --> 00:08:28.118
What's powerful
there is if we know

00:08:28.118 --> 00:08:30.030
all the corresponding
sides are equal,

00:08:30.030 --> 00:08:31.270
then we know they're
congruent, and then we

00:08:31.270 --> 00:08:32.978
can make all the other
assumptions, which

00:08:32.978 --> 00:08:36.240
means that the corresponding
angles are also equal.

00:08:36.240 --> 00:08:38.809
So that we know that that's
going to be congruent to that,

00:08:38.809 --> 00:08:40.190
or have the same measure.

00:08:40.190 --> 00:08:42.539
That's going to have the
same measure as that.

00:08:42.539 --> 00:08:45.460
And then that is going
to have the same measure

00:08:45.460 --> 00:08:47.370
as that right over there.

00:08:47.370 --> 00:08:52.110
And to see why that
is a reasonable axiom,

00:08:52.110 --> 00:08:54.830
or a reasonable assumption,
or a reasonable postulate

00:08:54.830 --> 00:08:59.590
to start off with, let's
start with one triangle.

00:08:59.590 --> 00:09:01.720
So let's say I have this
triangle right over here.

00:09:01.720 --> 00:09:04.980
So it has that side, and
then it has that side,

00:09:04.980 --> 00:09:07.630
and then it has that
side right over here.

00:09:07.630 --> 00:09:11.840
And what I'm going to do is see,
if I have another triangle that

00:09:11.840 --> 00:09:15.080
has the exact same side
lengths, is there any way for me

00:09:15.080 --> 00:09:18.250
to construct a triangle with
the same side lengths that

00:09:18.250 --> 00:09:21.110
is different, that can't be
translated to this triangle

00:09:21.110 --> 00:09:23.720
through flipping,
shifting, or rotating.

00:09:23.720 --> 00:09:25.870
So we assume this
other triangle's

00:09:25.870 --> 00:09:28.380
going to have a side that's
the same length as that one

00:09:28.380 --> 00:09:29.480
right over there.

00:09:29.480 --> 00:09:31.580
So I'll try to
draw it like that.

00:09:31.580 --> 00:09:33.100
Roughly the same length.

00:09:33.100 --> 00:09:38.271
We know it's going to have
a side that's that length.

00:09:38.271 --> 00:09:39.770
Let me put on this
side just to make

00:09:39.770 --> 00:09:41.970
it look a little bit
more interesting.

00:09:41.970 --> 00:09:44.760
So we know it's going to
have a side like that.

00:09:44.760 --> 00:09:46.760
So I'm going to draw it
roughly the same length,

00:09:46.760 --> 00:09:48.760
but I'm going to try to
do it a different angle.

00:09:48.760 --> 00:09:52.490
Now we know it's going to have
a side that looks like that.

00:09:55.170 --> 00:09:57.280
And so let me, I'll
put it right over here.

00:09:57.280 --> 00:09:59.447
It's about that length
right over there.

00:09:59.447 --> 00:10:01.030
And so clearly, this
isn't a triangle.

00:10:01.030 --> 00:10:02.529
In order to make
it a triangle, I'll

00:10:02.529 --> 00:10:07.250
have to connect this point to
that point right over there.

00:10:07.250 --> 00:10:09.000
And really there's
only two ways to do it.

00:10:09.000 --> 00:10:12.380
I can rotate it around that
little hinge right over there.

00:10:12.380 --> 00:10:15.690
If I connect them
over here, then I'm

00:10:15.690 --> 00:10:22.520
going to get a triangle
that looks like this, which

00:10:22.520 --> 00:10:26.020
is really just a flipped--
am I visualizing it right?

00:10:26.020 --> 00:10:27.310
Yeah, just a flipped version.

00:10:27.310 --> 00:10:29.480
You can rotate it a
little bit back this way.

00:10:29.480 --> 00:10:31.000
But then you'd have the
magenta on this side

00:10:31.000 --> 00:10:32.166
and the yellow on this side.

00:10:32.166 --> 00:10:34.300
And you could flip
it vertically,

00:10:34.300 --> 00:10:36.140
and then it would look
exactly like this.

00:10:36.140 --> 00:10:39.050
Our other option to make
these two points connect,

00:10:39.050 --> 00:10:41.930
is to rotate them out this way.

00:10:41.930 --> 00:10:46.720
And then the yellow side
is going to be here,

00:10:46.720 --> 00:10:51.300
and then the magenta side is
going to be right over here.

00:10:51.300 --> 00:10:53.120
That's not magenta.

00:10:53.120 --> 00:10:56.934
The magenta side is going
to be just like that.

00:10:56.934 --> 00:10:58.600
And if we do that,
then we actually just

00:10:58.600 --> 00:11:02.792
have to rotate it around
to get that exact triangle.

00:11:02.792 --> 00:11:04.500
So this isn't a proof,
but actually we're

00:11:04.500 --> 00:11:06.806
going to start assuming
that this is an axiom.

00:11:06.806 --> 00:11:08.930
But hopefully you see that
it's a pretty reasonable

00:11:08.930 --> 00:11:12.000
starting point, that if
all of the sides, all

00:11:12.000 --> 00:11:14.440
the corresponding sides,
of two different triangles

00:11:14.440 --> 00:11:19.260
are equal, then we know
that they are congruent.

00:11:19.260 --> 00:11:21.470
We're just going to assume,
it is an axiom that we're

00:11:21.470 --> 00:11:23.386
going to build off of,
that they're congruent.

00:11:23.386 --> 00:11:25.850
And then we also know that
all the corresponding angles

00:11:25.850 --> 00:11:27.924
are going to be equivalent.


Challenging triangle angle problem

WEBVTT
Kind: captions
Language: en

00:00:00.629 --> 00:00:02.420
Now this looks like an
interesting problem.

00:00:02.420 --> 00:00:04.280
We have this polygon.

00:00:04.280 --> 00:00:06.765
It looks like a pentagon right
over here, has five sides.

00:00:06.765 --> 00:00:07.890
It's an irregular pentagon.

00:00:07.890 --> 00:00:10.520
Not all the sides look
to be the same length.

00:00:10.520 --> 00:00:12.950
And the sides are
kind of continued on.

00:00:12.950 --> 00:00:15.850
And we have these
particular exterior angles

00:00:15.850 --> 00:00:17.540
of this pentagon.

00:00:17.540 --> 00:00:19.550
And what we're asked
is, what is the sum

00:00:19.550 --> 00:00:21.422
of all of these exterior angles.

00:00:21.422 --> 00:00:23.630
And it's kind of daunting,
because they don't give us

00:00:23.630 --> 00:00:24.546
any other information.

00:00:24.546 --> 00:00:26.580
They don't even give us
any particular angles.

00:00:26.580 --> 00:00:28.535
They don't start
us off anywhere.

00:00:28.535 --> 00:00:30.160
And so what we can
do, let's just think

00:00:30.160 --> 00:00:33.260
about the step by step, just
based on what we do know.

00:00:33.260 --> 00:00:35.850
Well, we have these
exterior angles.

00:00:35.850 --> 00:00:39.470
And these exterior
angles, they're

00:00:39.470 --> 00:00:42.340
each supplementary to
some interior angle.

00:00:42.340 --> 00:00:44.400
So maybe if we can
express them as a function

00:00:44.400 --> 00:00:47.440
of the interior angles, we
can maybe write this problem

00:00:47.440 --> 00:00:49.930
in a way that seems a
little bit more doable.

00:00:49.930 --> 00:00:54.380
So let's write the
interior angles over here.

00:00:54.380 --> 00:00:56.360
We already got to e.

00:00:56.360 --> 00:00:59.710
So let's call this f,
this interior angle f.

00:00:59.710 --> 00:01:01.250
Let's call this
interior angle g.

00:01:01.250 --> 00:01:03.530
Let's call this
interior angle h.

00:01:03.530 --> 00:01:05.120
Let's call this one i.

00:01:05.120 --> 00:01:07.190
And let's call this one j.

00:01:07.190 --> 00:01:10.170
And so this sum of these
particular exterior angles, a

00:01:10.170 --> 00:01:16.050
is now the same
thing as 180 minus g,

00:01:16.050 --> 00:01:18.850
because a and g
are supplementary.

00:01:18.850 --> 00:01:22.660
So a is 180 minus g.

00:01:22.660 --> 00:01:25.640
And then we have plus b.

00:01:25.640 --> 00:01:28.190
But b we can write in terms
of this interior angle.

00:01:28.190 --> 00:01:31.650
It's going to be 180 minus h,
because these two angles once

00:01:31.650 --> 00:01:33.197
again, are supplementary.

00:01:33.197 --> 00:01:34.280
We do that in a new color.

00:01:34.280 --> 00:01:38.590
So this is going
to be 180 minus h.

00:01:38.590 --> 00:01:41.140
And we could do the same
thing for each of them. c,

00:01:41.140 --> 00:01:46.360
we can write as 180 minus
i, so plus 180 minus i.

00:01:46.360 --> 00:01:55.100
And then d, we can write as 180
minus j, so plus 180 minus j.

00:01:55.100 --> 00:01:58.080
And then finally, e, I'm
running out of colors,

00:01:58.080 --> 00:02:05.890
e, we can write as 180 minus
f, so plus 180 minus f.

00:02:05.890 --> 00:02:08.770
And so, what we're left with,
if we add up all the 180s,

00:02:08.770 --> 00:02:10.550
we have 180 5 times.

00:02:10.550 --> 00:02:17.910
So this is going to be equal to
5 times 180 which is what, 900.

00:02:17.910 --> 00:02:22.520
And then you have minus g, minus
h, minus i, minus j, minus f.

00:02:22.520 --> 00:02:26.100
Or we could write
that as minus-- I'll

00:02:26.100 --> 00:02:30.040
try to do the same
colors-- g plus h--

00:02:30.040 --> 00:02:32.770
I'm kind of factoring out
this negative sign-- g

00:02:32.770 --> 00:02:36.360
plus h-- I'll do
the same color as g,

00:02:36.360 --> 00:02:48.570
that's not the same color-- g
plus h, plus i, plus j, plus f.

00:02:51.130 --> 00:02:52.690
And the whole
reason why did this

00:02:52.690 --> 00:02:54.710
and why this is
interesting now, is

00:02:54.710 --> 00:02:56.270
that we've expressed
this first thing

00:02:56.270 --> 00:02:57.395
that we need to figure out.

00:02:57.395 --> 00:03:01.380
We've expressed it in terms
of sum of the interior angles.

00:03:01.380 --> 00:03:04.230
So it's going to be 900
minus all of this business.

00:03:04.230 --> 00:03:09.190
So this is 900 minus all
of this business, which

00:03:09.190 --> 00:03:11.630
is the sum of the
interior angles.

00:03:11.630 --> 00:03:15.070
So this is the sum of
the interior angles.

00:03:15.070 --> 00:03:17.735
So it seems like we've made a
little progress, at least if we

00:03:17.735 --> 00:03:19.990
can figure out the sum
of the interior angles.

00:03:19.990 --> 00:03:22.350
And to do that part, I'll
show you a little trick.

00:03:22.350 --> 00:03:25.280
What you want to do is
divide this polygon,

00:03:25.280 --> 00:03:27.100
the inside of the
polygon, into three

00:03:27.100 --> 00:03:29.260
non-overlapping triangles.

00:03:29.260 --> 00:03:31.880
And so we could do
that from any side.

00:03:31.880 --> 00:03:34.180
Let's say that
they're all coming out

00:03:34.180 --> 00:03:36.000
of that side right over there.

00:03:36.000 --> 00:03:36.540
So there.

00:03:36.540 --> 00:03:39.830
I have divide it-- let me
do this in a neutral color,

00:03:39.830 --> 00:03:43.260
I'm doing it in
white-- so that's

00:03:43.260 --> 00:03:44.890
one triangle right over here.

00:03:44.890 --> 00:03:47.554
And then let me make another
triangle just like that.

00:03:47.554 --> 00:03:48.220
So there you go.

00:03:48.220 --> 00:03:50.906
I've divided into three
non-overlapping triangles.

00:03:50.906 --> 00:03:52.780
And the reason why I
did that, the reason why

00:03:52.780 --> 00:03:54.154
this is valuable,
is we know what

00:03:54.154 --> 00:03:57.320
the sum of the angles
of a triangle add up to.

00:03:57.320 --> 00:03:59.130
And so to make that
useful, we have

00:03:59.130 --> 00:04:04.514
to express these angles
in terms of angles

00:04:04.514 --> 00:04:06.180
that we can figure
out based on the fact

00:04:06.180 --> 00:04:08.810
that the sums of the
angles, or the measures

00:04:08.810 --> 00:04:11.390
of the angles in a
triangle add up to 180.

00:04:11.390 --> 00:04:15.140
So g is kind of already one
of the angles in the triangle.

00:04:15.140 --> 00:04:17.310
F is made up of two
angles in the triangle.

00:04:17.310 --> 00:04:20.329
So remember, f is this
entire angle right over here.

00:04:20.329 --> 00:04:23.860
So let's divide f
into two other angles,

00:04:23.860 --> 00:04:26.070
or two other measures
of angles, I should say.

00:04:26.070 --> 00:04:32.660
So let's say that f is equal
to-- we've already gone as high

00:04:32.660 --> 00:04:34.310
as, let's see, a,
b, c, d, e, f, g, h,

00:04:34.310 --> 00:04:37.530
i, j-- we haven't used k yet.

00:04:37.530 --> 00:04:40.990
So let's say that f
is equal to k plus l.

00:04:40.990 --> 00:04:43.450
It's equal to the sum of
the measures of these two

00:04:43.450 --> 00:04:45.850
adjacent angles right over here.

00:04:45.850 --> 00:04:48.920
So f is equal to k plus l.

00:04:48.920 --> 00:04:52.350
So that way we've
split it up into angles

00:04:52.350 --> 00:04:54.200
of these other triangles.

00:04:54.200 --> 00:05:00.520
And then we can do
that with j as well,

00:05:00.520 --> 00:05:02.550
because j, once again,
is that whole thing.

00:05:02.550 --> 00:05:06.280
So we could that j is equal to--
let's see, we already used l.

00:05:06.280 --> 00:05:08.785
So let's say j is
equal to m plus n.

00:05:13.450 --> 00:05:16.012
And then finally,
we can split up h.

00:05:16.012 --> 00:05:16.950
h is up here.

00:05:16.950 --> 00:05:18.390
Remember, it's this whole thing.

00:05:18.390 --> 00:05:22.740
Let's say that h is the same
thing as o plus p plus q.

00:05:22.740 --> 00:05:25.120
This is o, this is p, this is q.

00:05:25.120 --> 00:05:27.750
And once again, I wanted to
split up these interior angles

00:05:27.750 --> 00:05:30.830
if they're not already
an angle of a triangle.

00:05:30.830 --> 00:05:32.910
I want to split
them up into angles

00:05:32.910 --> 00:05:34.790
that are parts of
these triangles.

00:05:34.790 --> 00:05:39.349
So we have h is equal
to o, plus p, plus q.

00:05:39.349 --> 00:05:40.890
And the reason why
that's interesting

00:05:40.890 --> 00:05:44.690
is now we can write the sum
of these interior angles

00:05:44.690 --> 00:05:46.690
as the sum of a
bunch of angles that

00:05:46.690 --> 00:05:48.080
are part of these triangles.

00:05:48.080 --> 00:05:51.730
And then we can use the fact
that, for any one triangle,

00:05:51.730 --> 00:05:54.090
they add up to 180 degrees.

00:05:54.090 --> 00:05:54.840
So let's do that.

00:05:54.840 --> 00:05:56.800
So this expression
right over her

00:05:56.800 --> 00:05:59.190
is going to be g. g is
that angle right over here.

00:05:59.190 --> 00:06:01.180
We didn't make
any substitutions.

00:06:01.180 --> 00:06:03.092
So it's going to
be g-- actually,

00:06:03.092 --> 00:06:04.300
let me write the whole thing.

00:06:04.300 --> 00:06:09.235
So we have 900 minus,
and instead of a g, well,

00:06:09.235 --> 00:06:10.860
actually I'm not
making a substitution,

00:06:10.860 --> 00:06:15.880
so I can write g plus,
and instead of an h

00:06:15.880 --> 00:06:18.920
I can write that h
is o plus p plus q.

00:06:22.590 --> 00:06:25.050
And then plus i.

00:06:25.050 --> 00:06:26.690
i is sitting right over there.

00:06:26.690 --> 00:06:27.980
Plus i.

00:06:27.980 --> 00:06:29.430
And then plus j.

00:06:29.430 --> 00:06:32.160
And I kind of messed
up the colors.

00:06:32.160 --> 00:06:33.340
The magenta will go with i.

00:06:33.340 --> 00:06:35.960
And then j is this
expression right over here.

00:06:35.960 --> 00:06:40.000
So j is equal to
m plus n instead

00:06:40.000 --> 00:06:41.560
of writing a j right there.

00:06:41.560 --> 00:06:43.880
And then finally, we have our f.

00:06:43.880 --> 00:06:49.550
And f, we've already seen,
is equal to k plus l.

00:06:49.550 --> 00:06:52.170
So plus k plus l.

00:06:52.170 --> 00:06:54.770
So once again, I just rewrote
this part right over here,

00:06:54.770 --> 00:06:57.080
in terms of these
component angles.

00:06:57.080 --> 00:07:00.000
And now something very
interesting is going to happen,

00:07:00.000 --> 00:07:02.100
because we know what these
sums are going to be.

00:07:02.100 --> 00:07:05.660
Because we know that g plus
k plus o is 180 degrees.

00:07:05.660 --> 00:07:07.270
They are the
measures of the angle

00:07:07.270 --> 00:07:08.770
for this first
triangle over here,

00:07:08.770 --> 00:07:10.630
for this triangle
right over here.

00:07:10.630 --> 00:07:14.320
So g plus o plus
k is 180 degrees.

00:07:14.320 --> 00:07:17.040
So g-- let me do
this in a new color.

00:07:17.040 --> 00:07:20.130
So for this triangle
right over here,

00:07:20.130 --> 00:07:24.200
we know that g plus
o plus k are going

00:07:24.200 --> 00:07:25.840
to be equal to 180 degrees.

00:07:25.840 --> 00:07:30.180
So if we cross those out,
we can write 180 instead.

00:07:30.180 --> 00:07:33.170
And then we also know-- let me
see, I'm definitely running out

00:07:33.170 --> 00:07:36.140
of colors-- we know that
p, for this middle triangle

00:07:36.140 --> 00:07:39.615
right over here, we know that
p plus l plus m is 180 degrees.

00:07:44.770 --> 00:07:46.920
So you take those
out and you know

00:07:46.920 --> 00:07:50.400
that sum is going to be
equal to 180 degrees.

00:07:50.400 --> 00:07:53.270
And then finally, this
is the home stretch here.

00:07:53.270 --> 00:07:57.010
We know that q plus n
plus i is 180 degrees

00:07:57.010 --> 00:07:57.960
in this last triangle.

00:08:05.960 --> 00:08:08.230
Those three are also
going to be 180 degrees.

00:08:08.230 --> 00:08:10.920
And so now we know that the
sum of the interior angles

00:08:10.920 --> 00:08:13.200
for this irregular
pentagon-- it's actually

00:08:13.200 --> 00:08:15.580
going to be true for any
pentagon-- is 180 plus 180

00:08:15.580 --> 00:08:19.630
plus 180, which is 540 degrees.

00:08:19.630 --> 00:08:22.004
So that whole thing
is 540 degrees.

00:08:22.004 --> 00:08:24.170
And if we want to get the
sum of those extra angles,

00:08:24.170 --> 00:08:26.220
we just subtract it from 900.

00:08:26.220 --> 00:08:31.540
So 900 minus 540 is
going to be 360 degrees.

00:08:31.540 --> 00:08:32.730
And we are done.

00:08:32.730 --> 00:08:36.330
This is equal to 360 degrees.


Triangle angle example 3

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.980
So in this diagram over here,
I have this big triangle.

00:00:02.980 --> 00:00:05.000
And then I have all these
other little triangles

00:00:05.000 --> 00:00:06.900
inside of this big triangle.

00:00:06.900 --> 00:00:09.000
And what I want to do is
see if I can figure out

00:00:09.000 --> 00:00:10.780
the measure of this
angle right here.

00:00:10.780 --> 00:00:13.020
And we'll call
that measure theta.

00:00:13.020 --> 00:00:15.030
And they tell us a
few other things.

00:00:15.030 --> 00:00:17.010
You might have seen
this symbol before.

00:00:17.010 --> 00:00:19.150
That means that these
are right angles

00:00:19.150 --> 00:00:21.010
or that they have a
measure of 90 degrees.

00:00:21.010 --> 00:00:24.040
So that's a 90-degree angle,
that is a 90-degree angle,

00:00:24.040 --> 00:00:26.570
and that is a 90-degree
angle over there.

00:00:26.570 --> 00:00:29.914
And they also tell us that this
angle over here is 32 degrees.

00:00:29.914 --> 00:00:31.080
So let's see what we can do.

00:00:31.080 --> 00:00:33.100
And maybe we can solve this
in multiple different ways.

00:00:33.100 --> 00:00:34.600
That's what's really
fun about these

00:00:34.600 --> 00:00:37.730
is there's multiple ways
to solve these problems.

00:00:37.730 --> 00:00:39.870
So if this angle
is theta, we have

00:00:39.870 --> 00:00:44.970
theta is adjacent
to this green angle.

00:00:44.970 --> 00:00:46.470
And if you add them
together, you're

00:00:46.470 --> 00:00:50.110
going to get this right angle.

00:00:50.110 --> 00:00:54.910
So this pink angle, theta,
plus this green angle

00:00:54.910 --> 00:00:58.030
must be equal to 90 degrees.

00:00:58.030 --> 00:01:00.230
When you combine them,
you get a right angle.

00:01:00.230 --> 00:01:02.160
So you could call this
one-- its measure is

00:01:02.160 --> 00:01:04.830
going to be 90 minus beta.

00:01:04.830 --> 00:01:09.840
And now we have three
angles in the triangle,

00:01:09.840 --> 00:01:11.750
and we just have
to solve for theta.

00:01:11.750 --> 00:01:14.560
Because we know this angle
plus this angle plus this angle

00:01:14.560 --> 00:01:16.970
are going to be
equal to 180 degrees.

00:01:16.970 --> 00:01:26.670
So you have 90 minus theta plus
90 degrees plus 32 degrees--

00:01:26.670 --> 00:01:31.840
so I'm going to do that
in a different color--

00:01:31.840 --> 00:01:34.370
is going to be equal
to 180 degrees.

00:01:34.370 --> 00:01:36.950
The sum of the measures of
the angle inside of a triangle

00:01:36.950 --> 00:01:38.000
add up to 180 degrees.

00:01:38.000 --> 00:01:39.790
That's all we're
doing over here.

00:01:39.790 --> 00:01:44.090
And so let's see if we can
simplify this a little bit.

00:01:44.090 --> 00:01:46.130
So these two guys--
90 plus 90's going

00:01:46.130 --> 00:01:50.310
to be 180, so you get
180 minus theta plus 32

00:01:50.310 --> 00:01:53.960
is equal to 180 degrees.

00:01:53.960 --> 00:01:56.580
And then what else do we have?

00:01:56.580 --> 00:01:57.930
We have 180 on both sides.

00:01:57.930 --> 00:01:59.750
We can subtract that
from both sides.

00:01:59.750 --> 00:02:00.830
So that cancels out.

00:02:00.830 --> 00:02:02.160
That goes to 0.

00:02:02.160 --> 00:02:06.190
And then you have negative theta
plus 32 degrees is equal to 0.

00:02:06.190 --> 00:02:08.070
You can add theta to both sides.

00:02:08.070 --> 00:02:11.180
And you get 32 degrees
is equal to theta,

00:02:11.180 --> 00:02:14.220
or theta is equal to 32 degrees.

00:02:14.220 --> 00:02:18.310
So it's going to actually be
the same measure as this angle

00:02:18.310 --> 00:02:19.070
right over here.

00:02:19.070 --> 00:02:20.560
That's one way to
do the problem.

00:02:20.560 --> 00:02:22.900
There's other ways that we
could have done the problem.

00:02:25.147 --> 00:02:27.480
Actually, there's a ton of
ways we could have done this.

00:02:27.480 --> 00:02:29.340
We could have looked at
this big triangle over here.

00:02:29.340 --> 00:02:30.465
And we could've said, look.

00:02:30.465 --> 00:02:33.960
If this is 90 degrees over here,
this is 32 degrees over here,

00:02:33.960 --> 00:02:36.600
this angle up here
is going to be

00:02:36.600 --> 00:02:41.760
180 minus 90 degrees
minus 32 degrees.

00:02:41.760 --> 00:02:45.350
Because they all have to
add up to 180 degrees.

00:02:45.350 --> 00:02:47.886
And I just kind of
skipped a step there.

00:02:47.886 --> 00:02:49.260
Actually, let me
not skip a step.

00:02:49.260 --> 00:02:51.080
Let me call this x.

00:02:51.080 --> 00:02:52.910
If we call the measure
of that angle x,

00:02:52.910 --> 00:02:55.540
we would have x plus 90.

00:02:55.540 --> 00:02:58.230
I'm looking at the biggest
triangle in this diagram right

00:02:58.230 --> 00:03:06.420
here. x plus 90 plus 32 is going
to be equal to 180 degrees.

00:03:06.420 --> 00:03:10.120
And so if you subtract 90
and 32 from both sides.

00:03:10.120 --> 00:03:12.970
So if you subtract 90 from
both sides, you get x plus 32

00:03:12.970 --> 00:03:15.250
is equal to 90.

00:03:15.250 --> 00:03:17.210
And then if you subtract
32 from both sides,

00:03:17.210 --> 00:03:20.485
you get x is equal to--
what is this-- 58 degrees.

00:03:27.340 --> 00:03:28.280
Fair enough.

00:03:28.280 --> 00:03:29.740
Now, what else
can we figure out?

00:03:29.740 --> 00:03:33.814
Well, if this angle over here
is a right angle-- and I'm

00:03:33.814 --> 00:03:35.980
just redoing the problem
over again just to show you

00:03:35.980 --> 00:03:37.940
that there's multiple
ways to get the answer.

00:03:37.940 --> 00:03:39.648
We were given that
this is a right angle.

00:03:39.648 --> 00:03:41.930
If that is 90 degrees,
then this angle over here

00:03:41.930 --> 00:03:47.760
is supplementary to it, and
it also has to be 90 degrees.

00:03:47.760 --> 00:03:53.490
So then we have this angle
plus 90 degrees plus this angle

00:03:53.490 --> 00:03:55.260
have to equal 180.

00:03:55.260 --> 00:03:56.910
Maybe we could call that y.

00:03:56.910 --> 00:04:02.600
So y plus 58 plus
90 is equal to 180.

00:04:02.600 --> 00:04:05.120
You can subtract
90 from both sides.

00:04:05.120 --> 00:04:06.890
Subtract 90 from both sides.

00:04:06.890 --> 00:04:08.240
This will become 90.

00:04:08.240 --> 00:04:11.585
Subtract 58 from both sides, you
get y is equal to 32 degrees.

00:04:14.360 --> 00:04:19.540
Well, if y is 32 degrees,
it is complementary.

00:04:19.540 --> 00:04:22.350
It is complementary to
this angle right over here.

00:04:22.350 --> 00:04:25.411
It is complementary-- let
me do that in a new color,

00:04:25.411 --> 00:04:26.160
not supplementary.

00:04:26.160 --> 00:04:27.380
It is complementary.

00:04:27.380 --> 00:04:29.480
It adds up to 90 degrees.

00:04:29.480 --> 00:04:31.350
It is complementary to
this angle over here.

00:04:31.350 --> 00:04:32.610
We could call it z.

00:04:32.610 --> 00:04:35.820
So these two combined are
going to add up to 90 degrees,

00:04:35.820 --> 00:04:39.750
or z is going to be
equal to 58 degrees.

00:04:39.750 --> 00:04:41.320
And now we're
inside the triangle

00:04:41.320 --> 00:04:43.310
that we care about
to figure out theta,

00:04:43.310 --> 00:04:46.450
theta that we've already figured
out earlier in this video.

00:04:46.450 --> 00:04:48.860
Well, this z is 58 degrees.

00:04:48.860 --> 00:04:51.510
If this angle over here is
90, then this one over here

00:04:51.510 --> 00:04:54.180
is also going to be 90,
because they're supplementary.

00:04:54.180 --> 00:04:57.880
So you have 58 degrees.

00:04:57.880 --> 00:05:00.570
I wanted to do that
in that orange color.

00:05:00.570 --> 00:05:02.520
So if you have 58
degrees, so you

00:05:02.520 --> 00:05:07.820
have 58 plus this 90,
plus 90, plus theta

00:05:07.820 --> 00:05:13.110
now is going to be
equal to 180 degrees.

00:05:13.110 --> 00:05:16.060
You can subtract
90 from both sides.

00:05:16.060 --> 00:05:17.840
That becomes 90,
and then you have

00:05:17.840 --> 00:05:20.980
58 plus theta is equal to 90.

00:05:20.980 --> 00:05:22.630
Subtract 58 from both sides.

00:05:22.630 --> 00:05:25.786
You get theta is equal
to 32 degrees again.

00:05:25.786 --> 00:05:27.160
And so we've got
the same answer.

00:05:27.160 --> 00:05:28.670
I just wanted to
do that, show you

00:05:28.670 --> 00:05:30.980
that there are multiple
ways to do these problems.

00:05:30.980 --> 00:05:33.188
And as long as you're doing
things that are logically

00:05:33.188 --> 00:05:36.250
consistent, you're making
assumptions that you can make

00:05:36.250 --> 00:05:38.520
and then logically
deducing step by step,

00:05:38.520 --> 00:05:41.880
there's multiple ways to
get that right answer.


Finding more angles

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.800
Thought I would do some
more example problems

00:00:02.800 --> 00:00:04.120
involving triangles.

00:00:04.120 --> 00:00:06.810
And so this first one, it says
the measure of the largest

00:00:06.810 --> 00:00:10.990
angle in a triangle
is 4 times the measure

00:00:10.990 --> 00:00:13.360
of the second largest angle.

00:00:13.360 --> 00:00:15.850
The smallest angle
is 10 degrees.

00:00:15.850 --> 00:00:18.677
What are the measures
of all the angles?

00:00:18.677 --> 00:00:19.760
Well, we know one of them.

00:00:19.760 --> 00:00:20.760
We know it's 10 degrees.

00:00:20.760 --> 00:00:23.550
Let's draw an arbitrary
triangle right over here.

00:00:23.550 --> 00:00:25.940
So let's say that
is our triangle.

00:00:25.940 --> 00:00:28.401
We know that the smallest angle
is going to be 10 degrees.

00:00:28.401 --> 00:00:29.900
And I'll just say,
let's just assume

00:00:29.900 --> 00:00:32.570
that this right over here is the
measure of the smallest angle.

00:00:32.570 --> 00:00:34.180
It's 10 degrees.

00:00:34.180 --> 00:00:37.960
Now let's call the
second largest angle--

00:00:37.960 --> 00:00:39.090
let's call that x.

00:00:39.090 --> 00:00:41.960
So the second largest
angle, let's call that x.

00:00:41.960 --> 00:00:43.535
So this is going to be x.

00:00:43.535 --> 00:00:44.910
And then the first
sentence, they

00:00:44.910 --> 00:00:47.130
say the measure of the
largest angle in a triangle

00:00:47.130 --> 00:00:52.340
is 4 times the measure of
the second largest angle.

00:00:52.340 --> 00:00:54.160
So the second
largest angle is x.

00:00:54.160 --> 00:00:57.100
4 times that measure
is going to be 4x.

00:00:57.100 --> 00:01:01.620
So the largest angle
is going to be 4x.

00:01:01.620 --> 00:01:04.269
And so the one thing we
know about the measures

00:01:04.269 --> 00:01:05.980
of the angles
inside of a triangle

00:01:05.980 --> 00:01:08.820
is that they add
up to 180 degrees.

00:01:08.820 --> 00:01:17.200
So we know that 4x
plus x plus 10 degrees

00:01:17.200 --> 00:01:20.590
is going to be equal
to 180 degrees.

00:01:20.590 --> 00:01:22.630
It's going to be equal to 180.

00:01:22.630 --> 00:01:26.100
And 4x plus x, that
just gives us 5x.

00:01:26.100 --> 00:01:30.170
And then we have 5x plus
10 is equal to 180 degrees.

00:01:30.170 --> 00:01:32.050
Subtract 10 from both sides.

00:01:32.050 --> 00:01:36.710
You get 5x is equal to 170.

00:01:36.710 --> 00:01:38.716
And so x is equal to 170/5.

00:01:41.280 --> 00:01:43.860
And let's see, it'll go into
it-- what is that, 34 times?

00:01:43.860 --> 00:01:45.130
Let me verify this.

00:01:45.130 --> 00:01:47.904
So 5 goes into-- yeah,
it should be 34 times

00:01:47.904 --> 00:01:50.320
because it's going to go into
it twice as many times as 10

00:01:50.320 --> 00:01:51.100
would go into it.

00:01:51.100 --> 00:01:53.730
10 would go into 170 17 times.

00:01:53.730 --> 00:01:56.300
5 would go into 170 34 times.

00:01:56.300 --> 00:01:57.420
So we could verify it.

00:01:57.420 --> 00:01:58.345
Go into 170.

00:01:58.345 --> 00:02:00.340
5 goes into 17 three times.

00:02:00.340 --> 00:02:02.230
3 times 5 is 15.

00:02:02.230 --> 00:02:04.310
Subtract, you get 2.

00:02:04.310 --> 00:02:05.320
Bring down the 0.

00:02:05.320 --> 00:02:07.510
5 goes into 20 four
times, and then you're

00:02:07.510 --> 00:02:08.919
not going to have a remainder.

00:02:08.919 --> 00:02:10.430
4 times 5 is 20.

00:02:10.430 --> 00:02:11.090
No remainder.

00:02:11.090 --> 00:02:12.280
So it's 34 times.

00:02:12.280 --> 00:02:14.540
So x is equal to 34.

00:02:14.540 --> 00:02:18.370
So the second largest angle
has a measure of 34 degrees.

00:02:18.370 --> 00:02:21.890
This angle up here is
going to be 4 times that.

00:02:21.890 --> 00:02:24.340
So 4 times 34--
let's see, that's

00:02:24.340 --> 00:02:27.090
going to be 120 degrees
plus 16 degrees.

00:02:27.090 --> 00:02:30.660
This is going to be 136 degrees.

00:02:30.660 --> 00:02:32.320
Is that right?

00:02:32.320 --> 00:02:36.180
4 times 4 is 16, 4 times
3 is 120, 16 plus 120

00:02:36.180 --> 00:02:37.640
is 136 degrees.

00:02:37.640 --> 00:02:38.380
So we're done.

00:02:38.380 --> 00:02:41.630
The three measures, or the
sizes of the three angles,

00:02:41.630 --> 00:02:46.090
are 10 degrees, 34
degrees, and 136 degrees.

00:02:46.090 --> 00:02:47.860
Let's do another one.

00:02:47.860 --> 00:02:48.440
So let's see.

00:02:48.440 --> 00:02:50.650
We have a little bit
of a drawing here.

00:02:50.650 --> 00:02:52.282
And what I want
to do is-- and we

00:02:52.282 --> 00:02:53.740
could think about
different things.

00:02:53.740 --> 00:02:55.420
We could say, let's solve for x.

00:02:55.420 --> 00:02:59.080
I'm assuming that 4x is
the measure of this angle.

00:02:59.080 --> 00:03:02.320
2x is the measure of that
angle right over there.

00:03:02.320 --> 00:03:03.264
We can solve for x.

00:03:03.264 --> 00:03:04.930
And then if we know
x, we can figure out

00:03:04.930 --> 00:03:06.990
what the actual measures
of these angles are,

00:03:06.990 --> 00:03:08.690
assuming that we
can figure out x.

00:03:08.690 --> 00:03:10.231
And the other thing
that they tell us

00:03:10.231 --> 00:03:12.350
is that this line
over here is parallel

00:03:12.350 --> 00:03:13.390
to this line over here.

00:03:13.390 --> 00:03:14.730
And it was very craftily drawn.

00:03:14.730 --> 00:03:17.800
Because it's parallel,
but one stops here,

00:03:17.800 --> 00:03:19.165
and then one starts up there.

00:03:19.165 --> 00:03:20.540
So the first thing
I want to do--

00:03:20.540 --> 00:03:22.730
if they're telling us that
these two lines are parallel,

00:03:22.730 --> 00:03:24.730
there's probably going
to be something involving

00:03:24.730 --> 00:03:25.840
transversals or something.

00:03:25.840 --> 00:03:27.923
It might be something
involving-- the other option

00:03:27.923 --> 00:03:29.840
is something
involving triangles.

00:03:29.840 --> 00:03:32.019
And at first, you might
say, wait, is this angle

00:03:32.019 --> 00:03:33.310
and that angle vertical angles?

00:03:33.310 --> 00:03:34.290
But you have to be very careful.

00:03:34.290 --> 00:03:34.831
They are not.

00:03:34.831 --> 00:03:36.750
This is not the same line.

00:03:36.750 --> 00:03:39.920
This line is parallel
to that line.

00:03:39.920 --> 00:03:42.240
This line, it's bending
right over there,

00:03:42.240 --> 00:03:45.010
so we can't make any type
of assumption like that.

00:03:45.010 --> 00:03:47.390
So the interesting
thing-- and I'm not sure

00:03:47.390 --> 00:03:49.140
if this will lead in
the right direction--

00:03:49.140 --> 00:03:51.940
is to just make it
clear that these two are

00:03:51.940 --> 00:03:53.130
part of parallel lines.

00:03:53.130 --> 00:03:55.330
So I could continue this
line down like this.

00:03:55.330 --> 00:03:58.185
And then I can continue
this line up like that.

00:03:58.185 --> 00:04:00.560
And then that starts to look
a little bit more like we're

00:04:00.560 --> 00:04:02.740
used to when we're dealing
with parallel lines.

00:04:02.740 --> 00:04:05.280
And then this line
segment, BC-- or we

00:04:05.280 --> 00:04:08.160
could even say line BC, if
we were to continue it on.

00:04:08.160 --> 00:04:11.580
If we were to continue it
on and on, even pass D,

00:04:11.580 --> 00:04:13.590
then this is clearly
a transversal

00:04:13.590 --> 00:04:15.550
of those two parallel lines.

00:04:15.550 --> 00:04:17.700
This is clearly a transversal.

00:04:17.700 --> 00:04:22.850
And so if this angle
right over here is 4x,

00:04:22.850 --> 00:04:25.330
it has a corresponding angle.

00:04:25.330 --> 00:04:28.140
Half of the-- or maybe most
of the work on all of these

00:04:28.140 --> 00:04:30.930
is to try to see the parallel
lines and see the transversal

00:04:30.930 --> 00:04:34.970
and see the things that
might be useful for you.

00:04:34.970 --> 00:04:37.660
So that right there
is the transversal.

00:04:37.660 --> 00:04:38.950
These are the parallel lines.

00:04:38.950 --> 00:04:40.150
That's one parallel line.

00:04:40.150 --> 00:04:41.515
That is the other parallel line.

00:04:41.515 --> 00:04:42.950
You can almost try
to zone out all

00:04:42.950 --> 00:04:44.810
of the other stuff
in the diagram.

00:04:44.810 --> 00:04:47.330
And so if this angle
right over here is 4x,

00:04:47.330 --> 00:04:50.930
it has a corresponding
angle where the transversal

00:04:50.930 --> 00:04:52.930
intersects the
other parallel line.

00:04:52.930 --> 00:04:56.190
This right here is its
corresponding angle.

00:04:56.190 --> 00:04:58.640
So let me draw it
in that same yellow.

00:04:58.640 --> 00:05:00.770
This right over here is
a corresponding angle.

00:05:00.770 --> 00:05:04.810
So this will also be 4x.

00:05:04.810 --> 00:05:08.710
And we see that this angle--
this angle and this angle,

00:05:08.710 --> 00:05:11.470
this angle that has measure
4x and this angle that

00:05:11.470 --> 00:05:13.960
measures 2x-- we see that
they're supplementary.

00:05:13.960 --> 00:05:15.330
They're adjacent to each other.

00:05:15.330 --> 00:05:18.269
Their outer sides
form a straight angle.

00:05:18.269 --> 00:05:19.810
So they're supplementary,
which means

00:05:19.810 --> 00:05:22.640
that their measures
add up to 180 degrees.

00:05:22.640 --> 00:05:24.990
They kind of form-- they
go all the way around

00:05:24.990 --> 00:05:27.300
like that if you add the two
adjacent angles together.

00:05:27.300 --> 00:05:33.210
So we know that 4x plus 2x needs
to be equal to 180 degrees,

00:05:33.210 --> 00:05:37.020
or we get 6x is
equal to 180 degrees.

00:05:37.020 --> 00:05:38.670
Divide both sides by 6.

00:05:38.670 --> 00:05:42.580
You get x is equal to 30,
or x is equal to-- well,

00:05:42.580 --> 00:05:45.100
I shouldn't say--
well, x could be 30.

00:05:45.100 --> 00:05:47.880
And then this angle right
over here is 2 times x.

00:05:47.880 --> 00:05:50.600
So it's going to be 60 degrees.

00:05:50.600 --> 00:05:53.480
So this angle right over here
is going to be 60 degrees.

00:05:53.480 --> 00:05:58.320
And this angle right
over here is 4 times x.

00:05:58.320 --> 00:06:04.720
So it is 120 degrees,
and we're done.


Proof: Corresponding angle equivalence implies parallel lines

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.150
We know that if we have two
lines that are parallel-- so

00:00:03.150 --> 00:00:06.560
let me draw those two
parallel lines, l and m.

00:00:06.560 --> 00:00:08.580
So that's line l and line m.

00:00:08.580 --> 00:00:10.990
We know that if
they are parallel,

00:00:10.990 --> 00:00:14.040
then if we were to draw a
transversal that intersects

00:00:14.040 --> 00:00:17.450
both of them, that the
corresponding angles are equal.

00:00:17.450 --> 00:00:20.010
So this is x, and
this is y So we

00:00:20.010 --> 00:00:27.709
know that if l is parallel
to m, then x is equal to y.

00:00:27.709 --> 00:00:29.500
What I want to do in
this video is prove it

00:00:29.500 --> 00:00:30.460
the other way around.

00:00:30.460 --> 00:00:33.830
I want to prove-- So
this is what we know.

00:00:33.830 --> 00:00:35.630
We know this.

00:00:35.630 --> 00:00:42.160
What I want to do is prove
if x is equal to y, then l

00:00:42.160 --> 00:00:43.442
is parallel to m.

00:00:43.442 --> 00:00:44.650
So that we can go either way.

00:00:44.650 --> 00:00:47.080
If they're parallel, then the
corresponding angles are equal.

00:00:47.080 --> 00:00:49.455
And I want to show if the
corresponding angles are equal,

00:00:49.455 --> 00:00:51.500
then the lines are
definitely parallel.

00:00:51.500 --> 00:00:54.890
And what I'm going to do is
prove it by contradiction.

00:00:54.890 --> 00:00:56.820
So let's put this
aside right here.

00:00:56.820 --> 00:00:58.220
This is our goal.

00:00:58.220 --> 00:01:01.650
I'm going to assume
that this isn't true.

00:01:01.650 --> 00:01:03.710
I'm going to assume
that it's not true.

00:01:03.710 --> 00:01:09.730
So I'm going to assume
that x is equal to y

00:01:09.730 --> 00:01:16.980
and l is not parallel to m.

00:01:16.980 --> 00:01:20.010
So let's think about what type
of a reality that would create.

00:01:20.010 --> 00:01:23.310
So if l and m are not parallel,
and they're different lines,

00:01:23.310 --> 00:01:25.490
then they're going to
intersect at some point.

00:01:25.490 --> 00:01:27.960
So let me draw l like this.

00:01:27.960 --> 00:01:29.160
This is line l.

00:01:29.160 --> 00:01:30.346
Let me draw m like this.

00:01:30.346 --> 00:01:31.470
They're going to intersect.

00:01:31.470 --> 00:01:34.020
By definition, if two
lines are not parallel,

00:01:34.020 --> 00:01:35.980
they're going to
intersect each other.

00:01:35.980 --> 00:01:37.560
And that is going to be m.

00:01:37.560 --> 00:01:39.450
And then this thing
that was a transversal,

00:01:39.450 --> 00:01:41.640
I'll just draw it over here.

00:01:41.640 --> 00:01:43.200
So I'll just draw it over here.

00:01:43.200 --> 00:01:45.310
And then this is x.

00:01:45.310 --> 00:01:46.080
This is y.

00:01:46.080 --> 00:01:47.870
And we're assuming
that y is equal to x.

00:01:47.870 --> 00:01:53.970
So we could also call the
measure of this angle x.

00:01:53.970 --> 00:01:55.800
So given all of this
reality, and we're

00:01:55.800 --> 00:01:58.890
assuming in either case
that this is some distance,

00:01:58.890 --> 00:02:01.590
that this line is
not of 0 length.

00:02:01.590 --> 00:02:04.450
And so this line right over here
is not going to be of 0 length.

00:02:04.450 --> 00:02:10.910
Or this line segment
between points A and B.

00:02:10.910 --> 00:02:15.120
I guess we could say that AB,
the length of that line segment

00:02:15.120 --> 00:02:16.900
is greater than 0.

00:02:16.900 --> 00:02:19.830
I think that's a fair
assumption in either case.

00:02:19.830 --> 00:02:23.230
AB is going to be
greater than 0.

00:02:23.230 --> 00:02:26.210
So when we assume that these
two things are not parallel,

00:02:26.210 --> 00:02:28.560
we form ourselves a
nice little triangle

00:02:28.560 --> 00:02:30.470
here, where AB is
one of the sides,

00:02:30.470 --> 00:02:32.610
and the other two
sides are-- I guess

00:02:32.610 --> 00:02:35.280
we could label this point of
intersection C. The other two

00:02:35.280 --> 00:02:39.180
sides are line segment
BC and line segment AC.

00:02:39.180 --> 00:02:42.450
And we know a lot about finding
the angles of triangles.

00:02:42.450 --> 00:02:45.150
So let's just see what
happens when we just

00:02:45.150 --> 00:02:47.070
apply what we already know.

00:02:47.070 --> 00:02:49.820
Well first of all, if
this angle up here is x,

00:02:49.820 --> 00:02:55.090
we know that it is supplementary
to this angle right over here.

00:02:55.090 --> 00:03:00.510
So this angle over here is going
to have measure 180 minus x.

00:03:00.510 --> 00:03:02.700
And then we know that
this angle, this angle

00:03:02.700 --> 00:03:06.320
and this last angle--
let's call it angle

00:03:06.320 --> 00:03:09.560
z-- we know that the sum
of those interior angles

00:03:09.560 --> 00:03:13.190
of a triangle are going to
be equal to 180 degrees.

00:03:13.190 --> 00:03:22.970
So we know that x plus 180
minus x plus 180 minus x plus z

00:03:22.970 --> 00:03:25.990
is going to be equal
to 180 degrees.

00:03:25.990 --> 00:03:27.790
Now these x's cancel out.

00:03:27.790 --> 00:03:30.620
We can subtract 180
degrees from both sides.

00:03:30.620 --> 00:03:35.380
And we are left with
z is equal to 0.

00:03:35.380 --> 00:03:38.300
So if we assume
that x is equal to y

00:03:38.300 --> 00:03:42.660
but that l is not parallel to
m, we get this weird situation

00:03:42.660 --> 00:03:44.860
where we formed this
triangle, and the angle

00:03:44.860 --> 00:03:48.340
at the intersection
of those two lines

00:03:48.340 --> 00:03:50.340
that are definitely not
parallel all of a sudden

00:03:50.340 --> 00:03:54.210
becomes 0 degrees.

00:03:54.210 --> 00:03:55.860
But that's completely
nonsensical.

00:03:55.860 --> 00:03:58.110
If this was 0
degrees, that means

00:03:58.110 --> 00:03:59.620
that this triangle
wouldn't open up

00:03:59.620 --> 00:04:02.200
at all, which means
that the length of AB

00:04:02.200 --> 00:04:04.060
would have to be 0.

00:04:04.060 --> 00:04:05.560
Essentially, you
could call it maybe

00:04:05.560 --> 00:04:06.780
like a degenerate triangle.

00:04:06.780 --> 00:04:08.071
It wouldn't even be a triangle.

00:04:08.071 --> 00:04:08.890
It would be a line.

00:04:08.890 --> 00:04:11.770
These two lines would
have to be the same line.

00:04:11.770 --> 00:04:13.640
They wouldn't even
form a triangle.

00:04:13.640 --> 00:04:15.670
And so this leads us
to a contradiction.

00:04:15.670 --> 00:04:18.040
The contradiction is
that this line segment

00:04:18.040 --> 00:04:21.070
AB would have to be equal to 0.

00:04:21.070 --> 00:04:23.050
It kind of wouldn't be there.

00:04:23.050 --> 00:04:25.810
Or another contradiction
that you could come up with

00:04:25.810 --> 00:04:31.000
would be that these
two lines would

00:04:31.000 --> 00:04:33.380
have to be the same
line because there's

00:04:33.380 --> 00:04:35.436
no kind of opening between them.

00:04:35.436 --> 00:04:37.310
So either way, this
leads to a contradiction.

00:04:40.440 --> 00:04:42.260
And since it leads to
that contradiction,

00:04:42.260 --> 00:04:45.540
since if you assume x equals
y and l is not equal to m,

00:04:45.540 --> 00:04:49.340
you get to something that
makes absolutely no sense.

00:04:49.340 --> 00:04:51.350
You contradict your
initial assumptions.

00:04:51.350 --> 00:04:55.120
Then it essentially proves
that if x is equal to y,

00:04:55.120 --> 00:04:56.917
then l is parallel to m.

00:04:56.917 --> 00:04:58.750
Because we've shown
that if x is equal to y,

00:04:58.750 --> 00:05:03.260
there's no way for l and m
to be two different lines

00:05:03.260 --> 00:05:05.760
and for them not to be parallel.

00:05:05.760 --> 00:05:07.930
And so we have
proven our statement.

00:05:07.930 --> 00:05:09.370
So now we go in both ways.

00:05:09.370 --> 00:05:12.370
If lines are parallel,
corresponding angles are equal.

00:05:12.370 --> 00:05:14.240
If corresponding
angles are equal,

00:05:14.240 --> 00:05:17.079
then the lines are parallel.