Triangle angle example 2

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.090
We're given a
bunch of lines here

00:00:02.090 --> 00:00:05.540
that intersect in all different
ways and form triangles.

00:00:05.540 --> 00:00:07.490
And what I want to
do in this video,

00:00:07.490 --> 00:00:10.550
we've been given the measures of
some of the angles, this angle,

00:00:10.550 --> 00:00:12.200
that angle, and that angle.

00:00:12.200 --> 00:00:13.700
And what we want
to do in this video

00:00:13.700 --> 00:00:16.110
is figure out what the
measure of this angle is.

00:00:16.110 --> 00:00:18.500
And we're going to
call that measure x.

00:00:18.500 --> 00:00:21.760
And so I encourage you to pause
the video right now and try

00:00:21.760 --> 00:00:22.780
it yourself.

00:00:22.780 --> 00:00:24.990
And then I'm going to
give you the solution.

00:00:24.990 --> 00:00:26.556
So I'm assuming
you've unpaused it.

00:00:26.556 --> 00:00:28.180
And you've solved it
or you've given it

00:00:28.180 --> 00:00:29.210
at least a good shot of it.

00:00:29.210 --> 00:00:29.980
So let's try to do it.

00:00:29.980 --> 00:00:31.290
And what's fun about
these is there's

00:00:31.290 --> 00:00:32.610
multiple ways to solve these.

00:00:32.610 --> 00:00:34.030
And you kind of
just have to keep

00:00:34.030 --> 00:00:35.880
figuring out what
you can figure out.

00:00:35.880 --> 00:00:38.520
So let's say you start on the
left-hand side right over here.

00:00:38.520 --> 00:00:42.150
If this is 121 degrees,
then you'd say, well look,

00:00:42.150 --> 00:00:46.740
this angle right over here is
supplementary to this angle

00:00:46.740 --> 00:00:48.000
right over there.

00:00:48.000 --> 00:00:53.880
So this is 121 degrees
plus this green angle, that

00:00:53.880 --> 00:00:57.190
has to be equal to 180 degrees.

00:00:57.190 --> 00:01:01.630
So this is going to
be 180 minus 121.

00:01:01.630 --> 00:01:04.266
Let's see, that's the
same thing as 80 minus 21.

00:01:04.266 --> 00:01:05.900
80 minus 20 would be 60.

00:01:05.900 --> 00:01:07.770
So that's going
to be 59 degrees.

00:01:07.770 --> 00:01:08.970
So let me write that down.

00:01:08.970 --> 00:01:12.930
That's going to be 59 degrees.

00:01:12.930 --> 00:01:15.920
Now we see that we have
two angles of a triangle.

00:01:15.920 --> 00:01:17.642
If you have two
angles of a triangle,

00:01:17.642 --> 00:01:19.100
you can figure out
the third angle,

00:01:19.100 --> 00:01:21.150
because they need
to add up to 180.

00:01:21.150 --> 00:01:24.220
Or you could say that this
angle right over here--

00:01:24.220 --> 00:01:25.890
so we'll call that
question mark--

00:01:25.890 --> 00:01:30.320
we know that 59 plus
29 plus question mark

00:01:30.320 --> 00:01:33.760
needs to be equal
to 180 degrees.

00:01:33.760 --> 00:01:36.360
And if we subtract the 15 out
of the 29 from both sides,

00:01:36.360 --> 00:01:43.680
we get question mark is equal to
180 minus 59 minus 29 degrees.

00:01:43.680 --> 00:01:48.120
So that is going to be 180
minus 59 minus 29, let's see,

00:01:48.120 --> 00:01:53.600
180 minus 59, we
already know, is 121.

00:01:53.600 --> 00:01:57.050
And then 121 minus 29.

00:01:57.050 --> 00:02:00.920
So if you subtract
just 20, you get 101.

00:02:00.920 --> 00:02:03.720
You subtract another
9, you get 92.

00:02:03.720 --> 00:02:06.090
So that's going to be
equal to 92 degrees.

00:02:06.090 --> 00:02:08.789
This is equal to 92 degrees.

00:02:08.789 --> 00:02:11.150
Well, this right here
is equal to 92 degrees.

00:02:11.150 --> 00:02:13.740
This angle right here is
vertical with that angle.

00:02:13.740 --> 00:02:17.452
So it is also going to
be equal to 92 degrees.

00:02:17.452 --> 00:02:18.910
And now we're
getting pretty close.

00:02:18.910 --> 00:02:21.237
We can zoom in on this
triangle down here.

00:02:21.237 --> 00:02:22.570
And let me save some space here.

00:02:22.570 --> 00:02:24.550
So let me just say
that that over there

00:02:24.550 --> 00:02:27.120
is also going to be 92 degrees.

00:02:27.120 --> 00:02:28.730
And at this triangle
down here, we

00:02:28.730 --> 00:02:30.800
have two of the sides
of the triangle.

00:02:30.800 --> 00:02:32.660
We just have to
figure out the third.

00:02:32.660 --> 00:02:34.909
And actually, we don't even
have to do much math here,

00:02:34.909 --> 00:02:39.960
because we have two of the
angles of this triangle.

00:02:39.960 --> 00:02:41.770
We have to figure
out the third angle.

00:02:41.770 --> 00:02:44.930
So over here, we have one angle
that's 92, one angle that's 29.

00:02:44.930 --> 00:02:48.369
The other one will be
180 minus 92 minus 29.

00:02:48.369 --> 00:02:50.160
And we don't even have
to do any math here,

00:02:50.160 --> 00:02:52.870
because essentially, this is the
exact same angles that we have

00:02:52.870 --> 00:02:54.710
in this triangle
right over here.

00:02:54.710 --> 00:02:57.190
We have a 92 degree angle,
we have a 29 degree angle,

00:02:57.190 --> 00:02:59.960
and the other one is 59 degrees.

00:02:59.960 --> 00:03:01.850
So in this case,
it has to be also

00:03:01.850 --> 00:03:05.430
59 degrees, because over
here they added up to 180.

00:03:05.430 --> 00:03:08.020
So over here, they'll
also add up to 180.

00:03:08.020 --> 00:03:10.510
So that will also
get us to 59 degrees.

00:03:10.510 --> 00:03:14.180
We could just get that by
taking 180, subtracting 29,

00:03:14.180 --> 00:03:15.730
subtracting 92.

00:03:15.730 --> 00:03:18.130
And then if this is 59
degrees, then this angle

00:03:18.130 --> 00:03:21.250
is also going to be 59
degrees, because they

00:03:21.250 --> 00:03:22.570
are vertical angles.

00:03:22.570 --> 00:03:27.694
So we're done. x is
equal to 59 degrees.

00:03:27.694 --> 00:03:29.110
Now there's multiple
ways that you

00:03:29.110 --> 00:03:31.640
could have reasoned
through this problem.

00:03:31.640 --> 00:03:35.130
You could have immediately
said-- so let me start over,

00:03:35.130 --> 00:03:36.780
actually.

00:03:36.780 --> 00:03:38.300
Maybe a faster way,
but you wouldn't

00:03:38.300 --> 00:03:40.466
have been able to do kind
of this basic steps there,

00:03:40.466 --> 00:03:45.510
is you said, look, this is an
exterior angle right over here.

00:03:45.510 --> 00:03:48.690
It is equal to the sum of
the remote interior angles.

00:03:48.690 --> 00:03:53.250
So 121 is going to be 29 plus
this thing right over here.

00:03:53.250 --> 00:03:56.130
And we ended up doing that when
I did it step-by-step before.

00:03:56.130 --> 00:03:59.340
But here, we're just
using kind of a few things

00:03:59.340 --> 00:04:01.590
that we know about
triangles ahead of time

00:04:01.590 --> 00:04:02.840
to maybe skip a step or two.

00:04:02.840 --> 00:04:05.298
Although I like to do it the
other way just so we make sure

00:04:05.298 --> 00:04:07.550
we don't do anything weird.

00:04:07.550 --> 00:04:10.600
So anyway, this
is going to be 129

00:04:10.600 --> 00:04:14.340
minus 29, which
is going to be 92.

00:04:14.340 --> 00:04:17.670
And if this is 92, then
this is also going to be 92.

00:04:17.670 --> 00:04:20.670
And then, if this is x, then
this is also going to be x.

00:04:20.670 --> 00:04:27.560
And you could say x plus 92
plus 29 is equal to 180 degrees.

00:04:27.560 --> 00:04:31.540
And then you'd say
x plus 92 plus 29 is

00:04:31.540 --> 00:04:33.960
going to be 121 degrees.

00:04:33.960 --> 00:04:35.570
We already knew that before.

00:04:35.570 --> 00:04:41.390
And so that is going
to equal 180 degrees.

00:04:41.390 --> 00:04:47.640
And so x is equal to 59 degrees.

00:04:47.640 --> 00:04:51.360
So there's a ton
of ways that you

00:04:51.360 --> 00:04:54.650
could have thought
about this problem.


Triangle angle example 1

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.380
What I want to do
now is just a series

00:00:02.380 --> 00:00:05.570
of problems that really make
sure that we know what we're

00:00:05.570 --> 00:00:09.530
doing with parallel lines and
triangles and all the rest.

00:00:09.530 --> 00:00:12.410
And what we have right here
is a fairly classic problem.

00:00:12.410 --> 00:00:15.070
And what I want to do is I
want to figure out, just given

00:00:15.070 --> 00:00:16.570
the information
here-- so obviously

00:00:16.570 --> 00:00:18.310
I have a triangle here.

00:00:18.310 --> 00:00:20.380
I have another
triangle over here.

00:00:20.380 --> 00:00:24.020
We were given some of the angles
inside of these triangles.

00:00:24.020 --> 00:00:26.400
Given the information
over here, I

00:00:26.400 --> 00:00:29.400
want to figure out what
the measure of this angle

00:00:29.400 --> 00:00:30.850
is right over there.

00:00:30.850 --> 00:00:33.616
I need to figure out what
that question mark is.

00:00:33.616 --> 00:00:35.490
And so you might want
to give a go at it just

00:00:35.490 --> 00:00:39.570
knowing what you know about
the sums of the measures

00:00:39.570 --> 00:00:42.130
of the angles inside of
a triangle, and maybe

00:00:42.130 --> 00:00:44.630
a little bit of what you know
about supplementary angles.

00:00:44.630 --> 00:00:47.740
So you might want to pause
it and give it a try yourself

00:00:47.740 --> 00:00:50.214
because I'm about to
give you the solution.

00:00:50.214 --> 00:00:51.630
So the first thing
you might say--

00:00:51.630 --> 00:00:53.330
and this is a
general way to think

00:00:53.330 --> 00:00:55.510
about a lot of these problems
where they give you some angles

00:00:55.510 --> 00:00:57.300
and you have to figure out
some other angles based

00:00:57.300 --> 00:00:59.630
on the sum of angles and
a triangle equaling 180,

00:00:59.630 --> 00:01:01.850
or this one doesn't have
parallel lines on it.

00:01:01.850 --> 00:01:04.080
But you might see some
with parallel lines

00:01:04.080 --> 00:01:06.440
and supplementary lines
and complementary lines--

00:01:06.440 --> 00:01:09.100
is to just fill in everything
that you can figure out,

00:01:09.100 --> 00:01:11.010
and one way or
another, you probably

00:01:11.010 --> 00:01:14.290
would be able to figure out
what this question mark is.

00:01:14.290 --> 00:01:16.310
So the first thing that
kind of pops out to me

00:01:16.310 --> 00:01:18.810
is we have one triangle
right over here.

00:01:18.810 --> 00:01:21.030
We have this
triangle on the left.

00:01:21.030 --> 00:01:22.440
And on this triangle
on the left,

00:01:22.440 --> 00:01:24.254
we're given 2 of the angles.

00:01:24.254 --> 00:01:26.170
And if you have 2 of the
angles in a triangle,

00:01:26.170 --> 00:01:28.130
you can always figure
out the third angle

00:01:28.130 --> 00:01:31.650
because they're going to
add up to 180 degrees.

00:01:31.650 --> 00:01:38.550
So if you call that x, we
know that x plus 50 plus 64

00:01:38.550 --> 00:01:41.300
is going to be equal
to 180 degrees.

00:01:41.300 --> 00:01:45.860
Or we could say, x
plus, what is this, 114.

00:01:45.860 --> 00:01:49.240
X plus 114 is equal
to 180 degrees.

00:01:49.240 --> 00:01:53.160
We could subtract 114 from
both sides of this equation,

00:01:53.160 --> 00:01:59.220
and we get x is equal
to 180 minus 114.

00:01:59.220 --> 00:02:01.186
So 80 minus 14.

00:02:01.186 --> 00:02:05.550
80 minus 10 would be 70,
minus another 4 is 66.

00:02:05.550 --> 00:02:08.690
So x is 66 degrees.

00:02:08.690 --> 00:02:11.307
Now, if x is 66
degrees, I think you

00:02:11.307 --> 00:02:13.140
might find that there's
another angle that's

00:02:13.140 --> 00:02:14.580
not too hard to figure out.

00:02:14.580 --> 00:02:16.360
So let me write it like this.

00:02:19.830 --> 00:02:22.080
Let me write x is
equal to 66 degrees.

00:02:25.099 --> 00:02:26.890
Well if we know this
angle right over here,

00:02:26.890 --> 00:02:29.630
if we know the measure of
this angle is 66 degrees,

00:02:29.630 --> 00:02:32.970
we know that that
angle is supplementary

00:02:32.970 --> 00:02:34.730
with this angle right over here.

00:02:34.730 --> 00:02:37.260
Their outer sides
form a straight angle,

00:02:37.260 --> 00:02:38.640
and they are adjacent.

00:02:38.640 --> 00:02:40.710
So if we call this
angle right over here,

00:02:40.710 --> 00:02:44.170
y, we know that
y plus x is going

00:02:44.170 --> 00:02:47.750
to be equal to 180 degrees.

00:02:47.750 --> 00:02:50.860
And we know x is
equal to 66 degrees.

00:02:50.860 --> 00:02:52.120
So this is 66.

00:02:52.120 --> 00:02:55.540
And so we can subtract
66 from both sides,

00:02:55.540 --> 00:03:00.900
and we get y is equal to--
these cancel out-- 180 minus 66

00:03:00.900 --> 00:03:02.760
is 114.

00:03:02.760 --> 00:03:05.110
And that number might look
a little familiar to you.

00:03:05.110 --> 00:03:08.340
Notice, this 114 was
the exact same sum

00:03:08.340 --> 00:03:10.790
of these 2 angles over here.

00:03:10.790 --> 00:03:12.260
And that's actually
a general idea,

00:03:12.260 --> 00:03:15.270
and I'll do it on the side
here just to prove it to you.

00:03:15.270 --> 00:03:18.124
If I have, let's say that
these 2 angles-- let's

00:03:18.124 --> 00:03:19.540
say that the measure
of that angle

00:03:19.540 --> 00:03:21.490
is a, the measure
of that angle is

00:03:21.490 --> 00:03:23.340
b, the measure of
this angle we know

00:03:23.340 --> 00:03:26.800
is going to be 180
minus a minus b.

00:03:26.800 --> 00:03:29.250
That's this angle
right over here.

00:03:29.250 --> 00:03:31.290
And then this angle,
which is considered

00:03:31.290 --> 00:03:32.990
to be an exterior angle.

00:03:32.990 --> 00:03:35.440
So in this example, y
is an exterior angle.

00:03:35.440 --> 00:03:38.220
In this example, that
is our exterior angle.

00:03:38.220 --> 00:03:40.090
That is going to be
supplementary to 180

00:03:40.090 --> 00:03:41.650
minus a minus b.

00:03:41.650 --> 00:03:45.230
So this angle plus
180 minus a minus b

00:03:45.230 --> 00:03:46.730
is going to be equal to 180.

00:03:46.730 --> 00:03:48.920
So if you call this
angle y, you would

00:03:48.920 --> 00:03:57.730
have y plus 180 minus a
minus b is equal to 180.

00:03:57.730 --> 00:03:59.990
You could subtract
180 from both sides.

00:03:59.990 --> 00:04:02.160
You could add a plus
b to both sides.

00:04:02.160 --> 00:04:06.080
So plus a plus b.

00:04:06.080 --> 00:04:08.050
Running out of space
on the right hand side.

00:04:08.050 --> 00:04:10.820
And then you're left
with-- these cancel out.

00:04:10.820 --> 00:04:12.680
On the left hand side,
you're left with y.

00:04:12.680 --> 00:04:15.230
On the right hand side
is equal to a plus b.

00:04:15.230 --> 00:04:17.089
So this is just a
general property.

00:04:17.089 --> 00:04:19.430
You can just reason
it through yourself

00:04:19.430 --> 00:04:23.680
just with the sum of the
measures of the angles inside

00:04:23.680 --> 00:04:26.020
of a triangle add
up to 180 degrees,

00:04:26.020 --> 00:04:28.200
and then you have a
supplementary angles

00:04:28.200 --> 00:04:28.950
right over here.

00:04:28.950 --> 00:04:31.940
Or you could just say, look,
if I have the exterior angles

00:04:31.940 --> 00:04:35.220
right over here,
it's equal to the sum

00:04:35.220 --> 00:04:36.972
of the remote interior angles.

00:04:36.972 --> 00:04:39.180
That's just a little
terminology you could see there.

00:04:39.180 --> 00:04:41.220
So y is equal to a plus b.

00:04:41.220 --> 00:04:44.240
114 degrees, we've already
shown to ourselves,

00:04:44.240 --> 00:04:47.380
is equal to 64 plus 50 degrees.

00:04:47.380 --> 00:04:49.005
But anyway, regardless
of how we do it,

00:04:49.005 --> 00:04:50.670
if we just reason
it out step by step

00:04:50.670 --> 00:04:53.090
or if we just knew this
property from the get go,

00:04:53.090 --> 00:04:54.987
if we know that y is
equal to 114 degrees--

00:04:54.987 --> 00:04:57.320
and I like to reason it out
every time just to make sure

00:04:57.320 --> 00:05:00.290
I'm not jumping to conclusions.

00:05:00.290 --> 00:05:03.050
So if y is 114 degrees,
now we know this angle.

00:05:03.050 --> 00:05:05.192
We were given this
angle in the beginning.

00:05:05.192 --> 00:05:07.150
Now we just have to figure
out this third angle

00:05:07.150 --> 00:05:08.920
in this triangle.

00:05:08.920 --> 00:05:12.930
So if we call this z, if we
call this question mark is

00:05:12.930 --> 00:05:20.710
equal to z, we know
that z plus 114 plus 31

00:05:20.710 --> 00:05:23.170
is equal to 180 degrees.

00:05:23.170 --> 00:05:26.460
The sums of the measures of
the angle inside of a triangle

00:05:26.460 --> 00:05:27.770
add up to 180 degrees.

00:05:27.770 --> 00:05:30.290
That's the only property
we're using in this step.

00:05:30.290 --> 00:05:38.520
So we get z plus, what is
this, 145 is equal to 180.

00:05:38.520 --> 00:05:39.560
Did I do that right?

00:05:39.560 --> 00:05:42.962
We have a 15, then a 30.

00:05:42.962 --> 00:05:45.350
Yep, 145 is equal to 180.

00:05:45.350 --> 00:05:49.070
Subtract 145 from both
sides of this equation,

00:05:49.070 --> 00:05:56.840
and we are left with z is equal
to 80 minus 45 is equal to 35.

00:05:56.840 --> 00:06:02.580
So z is equal to 35
degrees, and we are done.


Angles formed by parallel lines and transversals

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.440
Let's say we have
two lines over here.

00:00:02.440 --> 00:00:06.160
Let's call this line
right over here line AB.

00:00:06.160 --> 00:00:08.600
So A and B both
sit on this line.

00:00:08.600 --> 00:00:10.600
And let's say we have
this other line over here.

00:00:10.600 --> 00:00:12.410
We'll call this line CD.

00:00:12.410 --> 00:00:15.622
So it goes through point C
and it goes through point D.

00:00:15.622 --> 00:00:18.230
And it just keeps
on going forever.

00:00:18.230 --> 00:00:22.020
And let's say that these lines
both sit on the same plane.

00:00:22.020 --> 00:00:23.930
And in this case, the
plane is our screen,

00:00:23.930 --> 00:00:25.596
or this little piece
of paper that we're

00:00:25.596 --> 00:00:27.020
looking at right over here.

00:00:27.020 --> 00:00:30.880
And they never intersect.

00:00:30.880 --> 00:00:33.230
So they're on the same
plane, but they never

00:00:33.230 --> 00:00:36.180
intersect each other.

00:00:36.180 --> 00:00:38.680
If those two things are
true, and when they're not

00:00:38.680 --> 00:00:40.400
the same line, they
never intersect

00:00:40.400 --> 00:00:42.320
and they can be
on the same plane,

00:00:42.320 --> 00:00:44.995
then we say that these
lines are parallel.

00:00:48.500 --> 00:00:51.400
They're moving in the
same general direction,

00:00:51.400 --> 00:00:53.160
in fact, the exact
same general direction.

00:00:53.160 --> 00:00:55.535
If we were looking at it from
an algebraic point of view,

00:00:55.535 --> 00:00:57.550
we would say that they
have the same slope,

00:00:57.550 --> 00:00:59.200
but they have
different y-intercepts.

00:00:59.200 --> 00:01:01.020
They involve different points.

00:01:01.020 --> 00:01:02.520
If we drew our
coordinate axes here,

00:01:02.520 --> 00:01:04.599
they would intersect that
at a different point,

00:01:04.599 --> 00:01:07.584
but they would have
the same exact slope.

00:01:07.584 --> 00:01:09.250
And what I want to
do is think about how

00:01:09.250 --> 00:01:11.380
angles relate to parallel lines.

00:01:11.380 --> 00:01:13.870
So right over here, we have
these two parallel lines.

00:01:13.870 --> 00:01:24.240
We can say that line AB
is parallel to line CD.

00:01:24.240 --> 00:01:25.700
Sometimes you'll
see it specified

00:01:25.700 --> 00:01:28.130
on geometric drawings like this.

00:01:28.130 --> 00:01:29.920
They'll put a little
arrow here to show

00:01:29.920 --> 00:01:31.457
that these two
lines are parallel.

00:01:31.457 --> 00:01:33.290
And if you've already
used the single arrow,

00:01:33.290 --> 00:01:36.040
they might put a double arrow to
show that this line is parallel

00:01:36.040 --> 00:01:38.250
to that line right over there.

00:01:38.250 --> 00:01:40.420
Now with that out of the
way, what I want to do

00:01:40.420 --> 00:01:44.750
is draw a line that intersects
both of these parallel lines.

00:01:44.750 --> 00:01:48.100
So here's a line that
intersects both of them.

00:01:48.100 --> 00:01:49.866
Let me draw a little
bit neater than that.

00:01:49.866 --> 00:01:51.615
So let me draw that
line right over there.

00:01:54.930 --> 00:01:57.040
Well, actually, I'll do
some points over here.

00:01:57.040 --> 00:01:58.950
Well, I'll just
call that line l.

00:01:58.950 --> 00:02:02.030
And this line that intersects
both of these parallel lines,

00:02:02.030 --> 00:02:04.370
we call that a transversal.

00:02:04.370 --> 00:02:06.050
This is a transversal line.

00:02:06.050 --> 00:02:09.970
It is transversing both
of these parallel lines.

00:02:09.970 --> 00:02:12.410
This is a transversal.

00:02:12.410 --> 00:02:15.790
And what I want to think about
is the angles that are formed,

00:02:15.790 --> 00:02:17.210
and how they relate
to each other.

00:02:17.210 --> 00:02:19.290
The angles that are
formed at the intersection

00:02:19.290 --> 00:02:22.950
between this transversal line
and the two parallel lines.

00:02:22.950 --> 00:02:24.600
So we could, first
of all, start off

00:02:24.600 --> 00:02:27.820
with this angle right over here.

00:02:27.820 --> 00:02:29.320
And we could call
that angle-- well,

00:02:29.320 --> 00:02:31.530
if we made some labels
here, that would

00:02:31.530 --> 00:02:34.099
be D, this point, and
then something else.

00:02:34.099 --> 00:02:36.140
But I'll just call it this
angle right over here.

00:02:36.140 --> 00:02:39.480
We know that that's going to be
equal to its vertical angles.

00:02:39.480 --> 00:02:42.040
So this angle is
vertical with that one.

00:02:42.040 --> 00:02:45.160
So it's going to be equal to
that angle right over there.

00:02:45.160 --> 00:02:48.080
We also know that this
angle, right over here,

00:02:48.080 --> 00:02:53.120
is going to be equal to its
vertical angle, or the angle

00:02:53.120 --> 00:02:54.820
that is opposite
the intersection.

00:02:54.820 --> 00:02:55.830
So it's going to
be equal to that.

00:02:55.830 --> 00:02:57.371
And sometimes you'll
see it specified

00:02:57.371 --> 00:03:00.520
like this, where you'll see a
double angle mark like that.

00:03:00.520 --> 00:03:02.070
Or sometimes you'll
see someone write

00:03:02.070 --> 00:03:04.760
this to show that
these two are equal

00:03:04.760 --> 00:03:07.655
and these two are
equal right over here.

00:03:07.655 --> 00:03:09.030
Now the other
thing we know is we

00:03:09.030 --> 00:03:10.870
could do the exact
same exercise up here,

00:03:10.870 --> 00:03:13.050
that these two are going
to be equal to each other

00:03:13.050 --> 00:03:15.133
and these two are going
to be equal to each other.

00:03:15.133 --> 00:03:16.310
They're all vertical angles.

00:03:16.310 --> 00:03:24.130
What's interesting here is
thinking about the relationship

00:03:24.130 --> 00:03:27.520
between that angle right
over there, and this angle

00:03:27.520 --> 00:03:29.270
right up over here.

00:03:29.270 --> 00:03:31.770
And if you just look at
it, it is actually obvious

00:03:31.770 --> 00:03:34.000
what that relationship
is-- that they

00:03:34.000 --> 00:03:37.547
are going to be the same
exact angle, that if you

00:03:37.547 --> 00:03:39.130
put a protractor
here and measured it,

00:03:39.130 --> 00:03:42.060
you would get the exact
same measure up here.

00:03:42.060 --> 00:03:43.800
And if I drew
parallel lines-- maybe

00:03:43.800 --> 00:03:45.341
I'll draw it straight
left and right,

00:03:45.341 --> 00:03:47.650
it might be a little
bit more obvious.

00:03:47.650 --> 00:03:50.320
So if I assume that these
two lines are parallel,

00:03:50.320 --> 00:03:52.990
and I have a transversal
here, what I'm saying

00:03:52.990 --> 00:03:54.970
is that this angle
is going to be

00:03:54.970 --> 00:03:58.050
the exact same measure
as that angle there.

00:03:58.050 --> 00:04:00.860
And to visualize that, just
imagine tilting this line.

00:04:00.860 --> 00:04:02.920
And as you take
different-- so it

00:04:02.920 --> 00:04:04.420
looks like it's the
case over there.

00:04:04.420 --> 00:04:07.110
If you take the line like this
and you look at it over here,

00:04:07.110 --> 00:04:09.106
it's clear that this
is equal to this.

00:04:09.106 --> 00:04:10.730
And there's actually
no proof for this.

00:04:10.730 --> 00:04:13.160
This is one of those
things that a mathematician

00:04:13.160 --> 00:04:15.425
would say is intuitively
obvious, that if you look

00:04:15.425 --> 00:04:16.800
at it, as you tilt
this line, you

00:04:16.800 --> 00:04:19.870
would say that these
angles are the same.

00:04:19.870 --> 00:04:21.579
Or think about putting
a protractor here

00:04:21.579 --> 00:04:23.530
to actually measure
these angles.

00:04:23.530 --> 00:04:26.490
If you put a
protractor here, you'd

00:04:26.490 --> 00:04:28.730
have one side of the
angle at the zero degree,

00:04:28.730 --> 00:04:31.820
and the other side would
specify that point.

00:04:31.820 --> 00:04:35.594
And if you put the
protractor over here,

00:04:35.594 --> 00:04:37.010
the exact same
thing would happen.

00:04:37.010 --> 00:04:39.760
One side would be on
this parallel line,

00:04:39.760 --> 00:04:42.910
and the other side would
point at the exact same point.

00:04:42.910 --> 00:04:46.250
So given that, we
know that not only is

00:04:46.250 --> 00:04:48.610
this side equivalent
to this side,

00:04:48.610 --> 00:04:51.634
it is also equivalent
to this side over here.

00:04:51.634 --> 00:04:53.050
And that tells us
that that's also

00:04:53.050 --> 00:04:55.260
equivalent to that
side over there.

00:04:55.260 --> 00:04:57.720
So all of these things
in green are equivalent.

00:04:57.720 --> 00:05:02.760
And by the same exact
argument, this angle

00:05:02.760 --> 00:05:05.150
is going to have the same
measure as this angle.

00:05:05.150 --> 00:05:07.830
And that's going to be
the same as this angle,

00:05:07.830 --> 00:05:11.220
because they are opposite,
or they're vertical angles.

00:05:11.220 --> 00:05:12.767
Now the important
thing to realize

00:05:12.767 --> 00:05:14.100
is just what we've deduced here.

00:05:14.100 --> 00:05:18.330
The vertical angles are equal
and the corresponding angles

00:05:18.330 --> 00:05:21.245
at the same points of
intersection are also equal.

00:05:21.245 --> 00:05:22.620
And so that's a
new word that I'm

00:05:22.620 --> 00:05:24.150
introducing right over here.

00:05:24.150 --> 00:05:26.330
This angle and this
angle are corresponding.

00:05:26.330 --> 00:05:28.680
They represent kind of
the top right corner,

00:05:28.680 --> 00:05:30.450
in this example, of
where we intersected.

00:05:30.450 --> 00:05:34.240
Here they represent still, I
guess, the top or the top right

00:05:34.240 --> 00:05:35.530
corner of the intersection.

00:05:35.530 --> 00:05:38.030
This would be the
top left corner.

00:05:38.030 --> 00:05:41.180
They're always going to be
equal, corresponding angles.

00:05:41.180 --> 00:05:43.730
And once again,
really, it's, I guess,

00:05:43.730 --> 00:05:46.410
for lack of a better
word, it is a bit obvious.

00:05:46.410 --> 00:05:48.300
Now on top of that,
there are other words

00:05:48.300 --> 00:05:49.330
that people will see.

00:05:49.330 --> 00:05:52.820
We've essentially just proven
that not only is this angle

00:05:52.820 --> 00:05:54.730
equivalent to this
angle, but it's also

00:05:54.730 --> 00:05:57.330
equivalent to this
angle right over here.

00:05:57.330 --> 00:06:00.760
And these two angles--
let me label them

00:06:00.760 --> 00:06:02.839
so that we can make
some headway here.

00:06:02.839 --> 00:06:04.380
So I'm going to use
lowercase letters

00:06:04.380 --> 00:06:05.910
for the angles themselves.

00:06:05.910 --> 00:06:09.410
So let's call this lowercase
a, lowercase b, lowercase c.

00:06:09.410 --> 00:06:12.590
So lowercase c for the
angle, lowercase d,

00:06:12.590 --> 00:06:16.750
and then let me call
this e, f, g, h.

00:06:16.750 --> 00:06:20.320
So we know from vertical
angles that b is equal to c.

00:06:20.320 --> 00:06:22.450
But we also know
that b is equal to f

00:06:22.450 --> 00:06:25.620
because they are
corresponding angles.

00:06:25.620 --> 00:06:27.930
And that f is equal to g.

00:06:27.930 --> 00:06:29.760
So vertical angles
are equivalent,

00:06:29.760 --> 00:06:31.670
corresponding angles
are equivalent,

00:06:31.670 --> 00:06:34.050
and so we also know, obviously,
that b is equal to g.

00:06:34.050 --> 00:06:37.032
And so we say that alternate
interior angles are equivalent.

00:06:37.032 --> 00:06:38.990
So you see that they're
kind of on the interior

00:06:38.990 --> 00:06:39.640
of the intersection.

00:06:39.640 --> 00:06:41.400
They're between the
two lines, but they're

00:06:41.400 --> 00:06:43.830
on all opposite sides
of the transversal.

00:06:43.830 --> 00:06:46.240
Now you don't have to
know that fancy word,

00:06:46.240 --> 00:06:48.460
alternate interior
angles, you really just

00:06:48.460 --> 00:06:50.350
have to deduce what
we just saw over here.

00:06:50.350 --> 00:06:52.310
Know that vertical angles
are going to be equal

00:06:52.310 --> 00:06:54.344
and corresponding angles
are going to be equal.

00:06:54.344 --> 00:06:56.010
And you see it with
the other ones, too.

00:06:56.010 --> 00:06:58.750
We know that a is going
to be equal to d, which

00:06:58.750 --> 00:07:06.397
is going to be equal to h,
which is going to be equal to e.


Proof: Vertical angles are equal

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.320
What I want to do in this video is prove to ourselves

00:00:03.320 --> 00:00:06.989
that vertical angles really are equal to each other,

00:00:06.989 --> 00:00:09.218
their measures are really equal to each other.

00:00:09.218 --> 00:00:14.234
So let's have a line here and let's say that I have another line over there,

00:00:14.234 --> 00:00:19.389
and let's call this point A, let's call this point B, point C,

00:00:19.389 --> 00:00:23.568
let's call this D, and let's call this right over there E.

00:00:23.568 --> 00:00:26.494
And so I'm just going to pick an arbitrary angle over here,

00:00:26.494 --> 00:00:31.231
let's say angle CB --what is this, this looks like an F--

00:00:31.231 --> 00:00:35.132
angle CBE. What I want to do is if I can prove that angle

00:00:35.132 --> 00:00:39.126
CBE is always going to be equal to its vertical angle

00:00:39.126 --> 00:00:42.655
--so, angle DBA-- then I'd prove that vertical angles

00:00:42.655 --> 00:00:45.395
are always going to be equal, because this is just a generalilzable

00:00:45.395 --> 00:00:47.299
case right over here.

00:00:47.299 --> 00:00:56.076
So what I want to prove here is angle CBE is equal to, I could say

00:00:56.076 --> 00:00:58.909
the measure of angle CBE --you will see it in different ways--

00:00:58.909 --> 00:01:00.395
actually this time let me write it without measure

00:01:00.395 --> 00:01:02.485
so that you get used to the different notations.

00:01:02.485 --> 00:01:07.349
I will just say prove angle CBE is equal to angle DBA.

00:01:07.349 --> 00:01:12.455
Is equal to angle DBA.

00:01:12.455 --> 00:01:15.627
So the first thing we know...the first thing we know

00:01:15.627 --> 00:01:17.392
so what do we know?

00:01:17.392 --> 00:01:24.600
We know that angle CBE, and we know that angle DBC are supplementary

00:01:24.600 --> 00:01:29.000
they are adjacent angles and their outer sides, both angles,

00:01:29.000 --> 00:01:31.867
form a straight angle over here.

00:01:31.867 --> 00:01:37.533
So we know that angle CBE and angle --so this is CBE--

00:01:37.533 --> 00:01:45.133
and angle DBC are supplementary. I will just write "sup" for that.

00:01:45.133 --> 00:01:46.754
They are supplementary.

00:01:46.754 --> 00:01:53.375
Which means that angle CBE plus angle DBC

00:01:53.375 --> 00:01:57.152
is equal to 180 degrees. Fair enough.

00:01:57.182 --> 00:02:01.298
We also know --so let me see this is CBE, this is what we care about

00:02:01.298 --> 00:02:03.800
and we want to prove that this is equal to that--

00:02:03.800 --> 00:02:06.733
we also know that angle DBA

00:02:06.810 --> 00:02:09.000
--we know that this is DBA right over here--

00:02:09.000 --> 00:02:19.933
we also know that angle DBA and angle DBC are supplementary

00:02:19.933 --> 00:02:22.867
this angle and this angle are supplementary,

00:02:22.867 --> 00:02:26.333
their outer sides form a straight angle, they are adjacent

00:02:26.333 --> 00:02:28.133
so they are supplementary

00:02:28.133 --> 00:02:33.733
which tells us that angle DBA, this angle right over here,

00:02:33.733 --> 00:02:39.000
plus angle DBC, this angle over here, is going to be

00:02:39.000 --> 00:02:42.467
equal to 180 degrees.

00:02:42.467 --> 00:02:47.800
Now, from this top one, this top statement over here,

00:02:47.800 --> 00:02:51.133
we can subtract angle DBC from both sides

00:02:51.133 --> 00:02:59.933
and we get angle CBE is equal to 180 degrees minus angle DBC

00:02:59.933 --> 00:03:02.067
that's this information right over here, I just put

00:03:02.067 --> 00:03:04.000
the angle DBC on the right side

00:03:04.000 --> 00:03:07.200
or subtracted it from both sides of the equation

00:03:07.200 --> 00:03:11.333
and this right over here, if I do the exact same thing,

00:03:11.333 --> 00:03:14.400
subtract angle DBC from both sides of the equation,

00:03:14.400 --> 00:03:19.467
I get angle DBA is equal to 180 degrees

00:03:19.467 --> 00:03:21.133
--let me scroll over to the right a little bit--

00:03:21.133 --> 00:03:26.400
is equal to 180 degrees minus angle DBC.

00:03:26.400 --> 00:03:32.200
So clearly, angle CBE is equal to 180 degrees minus angle DBC

00:03:32.200 --> 00:03:37.000
angle DBA is equal to 180 degrees minus angle DBC

00:03:37.000 --> 00:03:38.800
so they are equal to each other!

00:03:38.800 --> 00:03:41.200
They are both equal to the same thing

00:03:41.200 --> 00:03:43.800
so we get, which is what we wanted to get,

00:03:43.800 --> 00:03:50.067
angle CBE is equal to angle DBA.

00:03:50.067 --> 00:03:53.667
Angle CBE, which is this angle right over here,

00:03:53.667 --> 00:03:56.267
is equal to angle DBA

00:03:56.267 --> 00:03:58.867
and sometimes you might see that shown like this;

00:03:58.867 --> 00:04:01.533
so angle CBE, that's its measure, and you would say that

00:04:01.533 --> 00:04:04.733
this measure right over here is the exact same amount.

00:04:04.733 --> 00:04:06.087
And we have other vertical angles

00:04:06.087 --> 00:04:08.754
whatever this measure is, and sometimes you will

00:04:08.754 --> 00:04:10.733
see it with a double line like that,

00:04:10.733 --> 00:04:12.267
that you can say that THAT is going to be the same

00:04:12.267 --> 00:04:15.133
as whatever this angle right over here is.

00:04:15.133 --> 00:04:17.667
You will see it written like that sometimes, I like to use colors

00:04:17.667 --> 00:04:20.200
but not all books have the luxury of colors,

00:04:20.200 --> 00:04:21.933
or sometimes you will even see it written like this

00:04:21.933 --> 00:04:23.733
to show that they are the same angle;

00:04:23.733 --> 00:04:25.333
this angle and this angle

00:04:25.333 --> 00:04:26.333
--to show that these are different--

00:04:26.333 --> 00:04:29.667
sometimes they will say that they are the same in this way.

00:04:29.667 --> 00:04:33.498
This angle is equal to this vertical angle, is equal to its vertical

00:04:33.498 --> 00:04:34.785
angle right over here

00:04:34.785 --> 00:04:37.400
and that this angle is equal to this angle that is opposite

00:04:37.400 --> 00:04:39.800
the intersection right over here.

00:04:39.800 --> 00:04:41.333
What we have proved is the general case

00:04:41.333 --> 00:04:44.133
because all I did here is I just did two general intersecting lines

00:04:44.133 --> 00:04:47.800
I picked a random angle, and then I proved that it is equal

00:04:47.800 --> 00:04:50.452
to the angle that is vertical to it.


Introduction to vertical angles

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.990
Let's say I have two
intersecting line segments.

00:00:02.990 --> 00:00:05.552
So let's call that segment AB.

00:00:09.510 --> 00:00:12.610
And then I have segment CD.

00:00:12.610 --> 00:00:17.390
So that is C and that is D. And
they intersect right over here

00:00:17.390 --> 00:00:21.440
at point E. And let's
say we know, we're given,

00:00:21.440 --> 00:00:25.170
that this angle right over here,
that the measure of angle--

00:00:25.170 --> 00:00:28.370
That B is kind of, I don't know
why I wrote it so far away.

00:00:28.370 --> 00:00:30.690
So let me make that
a little bit closer.

00:00:30.690 --> 00:00:33.740
Let me make that B
a little bit closer.

00:00:33.740 --> 00:00:36.610
So let's say-- I'll
do that in yellow.

00:00:36.610 --> 00:00:41.240
Let's say that we know that
the measure of this angle

00:00:41.240 --> 00:00:44.950
right over here,
angle BED, let's

00:00:44.950 --> 00:00:48.920
say that we know that
measure is 70 degrees.

00:00:48.920 --> 00:00:51.260
Given that information,
what I want to do,

00:00:51.260 --> 00:00:53.300
based only on what
we know so far

00:00:53.300 --> 00:00:55.760
and not using a protractor,
what I want to do

00:00:55.760 --> 00:00:58.420
is figure out what the other
angles in this picture are.

00:00:58.420 --> 00:01:03.210
So what's the measure of angle
CEB, the measure of angle AEC,

00:01:03.210 --> 00:01:06.410
and the measure of angle AED?

00:01:06.410 --> 00:01:07.840
So the first thing
that you might

00:01:07.840 --> 00:01:09.030
notice when you
look at this, I've

00:01:09.030 --> 00:01:10.863
already told you that
this is a line segment

00:01:10.863 --> 00:01:12.460
and that this is a line segment.

00:01:12.460 --> 00:01:24.460
You see that angle BED and
angle CEB are adjacent.

00:01:28.220 --> 00:01:31.500
And we also see that if
you take the outer sides

00:01:31.500 --> 00:01:33.980
of those angles, it
forms a straight angle.

00:01:33.980 --> 00:01:39.080
And we also see that angle
CED is a straight angle.

00:01:44.180 --> 00:01:48.840
So we know that these two angles
must also be supplementary.

00:01:48.840 --> 00:01:51.580
They're next to each other
and they form a straight angle

00:01:51.580 --> 00:01:53.240
when you take their outer sides.

00:01:53.240 --> 00:02:02.230
So we know that angle BED and
angle CEB are supplementary,

00:02:02.230 --> 00:02:04.970
which means they add up to,
or that their measures add up

00:02:04.970 --> 00:02:06.380
to 180 degrees.

00:02:09.470 --> 00:02:10.930
Supplementary angles.

00:02:13.650 --> 00:02:15.940
Which tells us that
the measure of angle

00:02:15.940 --> 00:02:19.897
BED plus the measure
of angle CEB--

00:02:19.897 --> 00:02:21.230
and I keep writing measure here.

00:02:21.230 --> 00:02:22.930
Sometimes you'll just
see people write,

00:02:22.930 --> 00:02:28.520
angle BED plus angle CEB
is equal to 180 degrees.

00:02:28.520 --> 00:02:32.780
Now we already know the measure
of angle BED is 70 degrees.

00:02:32.780 --> 00:02:35.100
So we already know that
this thing right over here

00:02:35.100 --> 00:02:36.630
is 70 degrees.

00:02:36.630 --> 00:02:39.460
And so 70 degrees plus
the measure of angle CEB

00:02:39.460 --> 00:02:40.620
is 180 degrees.

00:02:40.620 --> 00:02:43.180
You subtract 70 from
both sides, and we

00:02:43.180 --> 00:02:50.730
get the measure of angle
CEB is equal to 110 degrees.

00:02:50.730 --> 00:02:53.180
I just subtracted 70
from both sides of that.

00:02:53.180 --> 00:02:58.979
So we figured out that this
right over here is 110 degrees.

00:02:58.979 --> 00:03:00.020
Well, that's interesting.

00:03:00.020 --> 00:03:01.640
And I went through
more steps than you

00:03:01.640 --> 00:03:03.020
would if you were doing
this problem quickly.

00:03:03.020 --> 00:03:04.540
If you did this problem quickly
in your head, you'd say,

00:03:04.540 --> 00:03:07.280
look this is 70 degrees,
this angle plus this angle

00:03:07.280 --> 00:03:10.910
would be 180 degrees, so
this has to be 110 degrees.

00:03:10.910 --> 00:03:12.920
So now let's use the
same logic to figure out

00:03:12.920 --> 00:03:15.790
what angle CEA is.

00:03:15.790 --> 00:03:19.170
So now we care about the
measure of angle CEA.

00:03:21.760 --> 00:03:25.300
And we can use the exact same
logic that we used over here.

00:03:25.300 --> 00:03:31.210
Angle CEA and angle
CEB, they are adjacent.

00:03:31.210 --> 00:03:34.160
They form a straight angle,
if you look at their outsides,

00:03:34.160 --> 00:03:35.939
so they must be supplementary.

00:03:35.939 --> 00:03:37.730
They form a straight
angle right over here.

00:03:37.730 --> 00:03:38.771
So they're supplementary.

00:03:38.771 --> 00:03:41.410
So they must add
up to 180 degrees.

00:03:41.410 --> 00:03:45.030
So the measure of angle CEA
plus the measure of angle CEB,

00:03:45.030 --> 00:03:50.660
which is 110 degrees, must
be equal to 180 degrees.

00:03:50.660 --> 00:03:53.330
So once again, subtract
110 from both sides.

00:03:53.330 --> 00:04:01.100
You get the measure of angle
CEA is equal to 70 degrees.

00:04:01.100 --> 00:04:04.027
So this one right over
here is also 70 degrees.

00:04:04.027 --> 00:04:05.610
And what we'll learn
in the next video

00:04:05.610 --> 00:04:07.210
is that this is no coincidence.

00:04:07.210 --> 00:04:11.890
These two angles, angle
CEA and angle BED,

00:04:11.890 --> 00:04:14.912
sometimes they're called
opposite angles-- well,

00:04:14.912 --> 00:04:16.620
I have often called
them opposite angles,

00:04:16.620 --> 00:04:20.399
but the more correct term
for them is vertical angles.

00:04:20.399 --> 00:04:21.712
And we haven't proved it.

00:04:21.712 --> 00:04:23.170
We've just seen a
special case here

00:04:23.170 --> 00:04:24.753
where these vertical
angles are equal.

00:04:24.753 --> 00:04:27.190
But it actually turns out that
vertical angles are always

00:04:27.190 --> 00:04:27.820
equal.

00:04:27.820 --> 00:04:31.240
But we haven't proved it to
ourselves for the general case.

00:04:31.240 --> 00:04:33.276
But let me just
write down this word

00:04:33.276 --> 00:04:34.400
since it's a nice new word.

00:04:34.400 --> 00:04:44.130
So angle CEA and angle
BED are vertical.

00:04:46.762 --> 00:04:49.220
And you might say, wait, they
look like they're horizontal,

00:04:49.220 --> 00:04:50.000
they're next to each other.

00:04:50.000 --> 00:04:51.150
And the vertical
really just means

00:04:51.150 --> 00:04:52.649
that they're across
from each other,

00:04:52.649 --> 00:04:54.650
across an intersection
from each other.

00:04:54.650 --> 00:04:58.497
Angle CEB and angle
AED are also vertical.

00:04:58.497 --> 00:04:59.580
So let me write that down.

00:04:59.580 --> 00:05:08.839
Angle CEB and angle
AED are also vertical.

00:05:08.839 --> 00:05:10.880
And that might even make
a little bit more sense,

00:05:10.880 --> 00:05:13.870
because it literally is, one
is on top and one is on bottom.

00:05:13.870 --> 00:05:16.100
They're kind of vertically
opposite from each other.

00:05:16.100 --> 00:05:18.790
But these horizontally
opposite angles

00:05:18.790 --> 00:05:21.200
are also called vertical angles.

00:05:21.200 --> 00:05:26.309
So now we have one angle left
to figure out, angle AED.

00:05:26.309 --> 00:05:27.850
And based on what
I already told you,

00:05:27.850 --> 00:05:31.400
vertical angles tend to be,
or they are always, equal.

00:05:31.400 --> 00:05:33.990
But we haven't proven that to
ourselves yet so we can't just

00:05:33.990 --> 00:05:36.930
use that property to say
that this is 110 degrees.

00:05:36.930 --> 00:05:39.950
So what we're going to do
is use the exact same logic.

00:05:39.950 --> 00:05:44.240
CEA and AED are
clearly supplementary.

00:05:44.240 --> 00:05:46.830
Their outsides form
a straight angle.

00:05:46.830 --> 00:05:50.880
They're clearly
supplementary, so CEA and AED

00:05:50.880 --> 00:05:53.160
must add up to 180 degrees.

00:05:53.160 --> 00:05:57.740
Or we could say the
measure of angle AED

00:05:57.740 --> 00:06:05.790
plus the measure of angle CEA
must be equal to 180 degrees.

00:06:05.790 --> 00:06:10.950
We know the measure
of CEA is 70 degrees.

00:06:10.950 --> 00:06:13.010
We know it is 70 degrees.

00:06:13.010 --> 00:06:15.080
So you subtract 70
from both sides.

00:06:15.080 --> 00:06:21.160
You get the measure of angle
AED is equal to 110 degrees.

00:06:21.160 --> 00:06:24.600
So we got the exact
result that we expected.

00:06:24.600 --> 00:06:27.720
So this angle right over
here is 110 degrees.

00:06:27.720 --> 00:06:30.260
And so if you take any
of the adjacent angles

00:06:30.260 --> 00:06:33.070
that their outer sides
form a straight angle,

00:06:33.070 --> 00:06:34.200
you see they add up to 180.

00:06:34.200 --> 00:06:35.890
This one and that
one add up to 180.

00:06:35.890 --> 00:06:37.461
This one and that
one add up to 180.

00:06:37.461 --> 00:06:38.960
This one and that
one add up to 180.

00:06:38.960 --> 00:06:41.000
And this one and that
one add up to 180.

00:06:41.000 --> 00:06:42.960
If you go all the way
around the circle,

00:06:42.960 --> 00:06:45.140
you'll see that they
add up to 360 degrees.

00:06:45.140 --> 00:06:48.630
Because you literally are
going all the way around.

00:06:48.630 --> 00:06:52.710
So 70 plus 110 is
180, plus 70 is 250,

00:06:52.710 --> 00:06:56.366
plus 110 is 360 degrees.

00:06:56.366 --> 00:06:57.240
I'll leave you there.

00:06:57.240 --> 00:06:58.614
This is the first
time that we've

00:06:58.614 --> 00:07:01.600
kind of found some interesting
results using the tool

00:07:01.600 --> 00:07:03.165
kit that we've built up so far.

00:07:03.165 --> 00:07:04.540
In the next video,
we'll actually

00:07:04.540 --> 00:07:07.742
prove to ourselves using pretty
much the exact same logic here,

00:07:07.742 --> 00:07:09.700
but we'll just do it with
generalized numbers--

00:07:09.700 --> 00:07:15.030
we won't use 70
degrees-- to prove

00:07:15.030 --> 00:07:18.720
that the measure of
vertical angles are equal.


Complementary and supplementary angles

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.750
Let's say I have an
angle ABC, and it

00:00:03.750 --> 00:00:05.400
looks something like this.

00:00:05.400 --> 00:00:09.670
So its vertex is going
to be at B. Maybe

00:00:09.670 --> 00:00:14.890
A sits right over here, and
C sits right over there.

00:00:14.890 --> 00:00:22.620
And then also let's say that we
have another angle called DBA.

00:00:22.620 --> 00:00:26.210
I want to have the vertex
once again at B. So let's

00:00:26.210 --> 00:00:27.420
say it looks like this.

00:00:27.420 --> 00:00:33.200
So this right over here is our
point D. That is our point D.

00:00:33.200 --> 00:00:39.000
And let's say that we know
that the measure of angle DBA

00:00:39.000 --> 00:00:41.010
is equal to 40 degrees.

00:00:41.010 --> 00:00:43.050
So this angle right
over here, its measure

00:00:43.050 --> 00:00:45.340
is equal to 40 degrees.

00:00:45.340 --> 00:00:50.160
And let's say that we know
that the measure of angle ABC

00:00:50.160 --> 00:00:51.635
is equal to 50 degrees.

00:00:55.447 --> 00:00:57.530
So there's a bunch of
interesting things happening

00:00:57.530 --> 00:00:58.060
here.

00:00:58.060 --> 00:01:00.220
The first interesting thing
that you might realize

00:01:00.220 --> 00:01:03.360
is that both of these
angles share a side.

00:01:03.360 --> 00:01:06.230
If you view these as rays-- they
could be lines, line segments,

00:01:06.230 --> 00:01:08.210
or rays-- but if you
view them as rays,

00:01:08.210 --> 00:01:11.320
they both share this ray BA.

00:01:11.320 --> 00:01:13.380
And when you have
two angles like this

00:01:13.380 --> 00:01:16.900
that share the same side, these
are called adjacent angles.

00:01:16.900 --> 00:01:21.650
Because the word "adjacent"
literally means next to.

00:01:21.650 --> 00:01:23.910
These are adjacent.

00:01:23.910 --> 00:01:26.342
They are adjacent angles.

00:01:26.342 --> 00:01:27.800
Now there's something
else that you

00:01:27.800 --> 00:01:29.380
might notice that's
interesting here.

00:01:29.380 --> 00:01:32.770
We know that the measure
of angle DBA is 40 degrees

00:01:32.770 --> 00:01:35.540
and the measure of
angle ABC is 50 degrees.

00:01:35.540 --> 00:01:40.351
And you might be able to guess
what the measure of angle DBC

00:01:40.351 --> 00:01:40.850
is.

00:01:44.000 --> 00:01:46.029
If we drew a
protractor over here--

00:01:46.029 --> 00:01:47.070
I'm not going to draw it.

00:01:47.070 --> 00:01:49.270
It will make my
drawing all messy.

00:01:49.270 --> 00:01:51.089
Well, maybe I'll
draw it really fast.

00:01:51.089 --> 00:01:52.880
So if you had a protractor
right over here,

00:01:52.880 --> 00:01:55.190
clearly this is opening
up to 50 degrees.

00:01:55.190 --> 00:01:57.290
And this is going
another 40 degrees.

00:01:57.290 --> 00:02:01.820
So if you wanted to say what
the measure of angle DBC is,

00:02:01.820 --> 00:02:04.982
it would essentially be the sum
of 40 degrees and 50 degrees.

00:02:04.982 --> 00:02:06.440
And let me delete
all of this stuff

00:02:06.440 --> 00:02:07.773
right here to keep things clean.

00:02:07.773 --> 00:02:13.260
So the measure of angle DBC
would be equal to 90 degrees.

00:02:13.260 --> 00:02:16.010
And we already know that 90
degrees is a special angle.

00:02:16.010 --> 00:02:17.325
This is a right angle.

00:02:22.180 --> 00:02:26.810
There's also a word for
two angles whose sum add up

00:02:26.810 --> 00:02:30.860
to 90 degrees, and
that is complementary.

00:02:30.860 --> 00:02:38.900
So we can also say that
angles DBA and angles

00:02:38.900 --> 00:02:40.900
ABC are complementary.

00:02:46.400 --> 00:02:50.490
And that is because their
measures add up to 90 degrees.

00:02:50.490 --> 00:02:57.300
So the measure of angle DBA
plus the measure of angle ABC

00:02:57.300 --> 00:03:00.410
is equal to 90 degrees.

00:03:00.410 --> 00:03:03.300
They form a right angle
when you add them up.

00:03:03.300 --> 00:03:05.140
And just as another
point of terminology

00:03:05.140 --> 00:03:09.710
that's kind of related
to right angles,

00:03:09.710 --> 00:03:12.550
when a right angle is
formed, the two rays

00:03:12.550 --> 00:03:14.936
that form the right angle
or the two lines that

00:03:14.936 --> 00:03:17.060
form that right angle or
the two line segments that

00:03:17.060 --> 00:03:19.570
form that right angle
are called perpendicular.

00:03:19.570 --> 00:03:23.420
So because we know that measure
of angle DBC is 90 degrees

00:03:23.420 --> 00:03:28.600
or that angle DBC is a
right angle, this tells us,

00:03:28.600 --> 00:03:36.130
we know that the
line segment DB is

00:03:36.130 --> 00:03:46.650
perpendicular to
line segment BC.

00:03:46.650 --> 00:03:53.790
Or we could even say ray BD
is-- instead of using the word

00:03:53.790 --> 00:03:55.822
perpendicular, there's
sometimes this symbol

00:03:55.822 --> 00:03:58.030
right here, which really
just shows two perpendicular

00:03:58.030 --> 00:04:02.890
lines-- perpendicular to BC.

00:04:02.890 --> 00:04:06.040
So all of these are
true statements here.

00:04:06.040 --> 00:04:08.000
And these come out of
the fact that the angle

00:04:08.000 --> 00:04:14.320
formed between DB and BC,
that is a 90-degree angle.

00:04:14.320 --> 00:04:17.970
Now, we have other words
when our two angles add up

00:04:17.970 --> 00:04:19.110
to other things.

00:04:19.110 --> 00:04:22.740
So let's say, for example,
I have one angle over here.

00:04:31.240 --> 00:04:33.780
Let me put some letters
here so we can specify it.

00:04:33.780 --> 00:04:37.470
So let's say this
is X, Y, and Z.

00:04:37.470 --> 00:04:40.470
And let's say that the
measure of angle XYZ

00:04:40.470 --> 00:04:45.250
is equal to 60 degrees.

00:04:45.250 --> 00:04:48.920
And let's say that you
have another angle that

00:04:48.920 --> 00:04:49.650
looks like this.

00:04:52.570 --> 00:04:56.824
And I'll call this,
let's say, maybe MNO.

00:05:02.240 --> 00:05:07.330
And let's say that the measure
of angle MNO is 120 degrees.

00:05:07.330 --> 00:05:10.160
So if you were to add the two
measures of these-- so let

00:05:10.160 --> 00:05:11.280
me write this down.

00:05:11.280 --> 00:05:25.770
The measure of angle MNO plus
the measure of angle XYZ,

00:05:25.770 --> 00:05:30.430
this is going to be equal to 120
degrees plus 60 degrees, which

00:05:30.430 --> 00:05:32.930
is equal to 180 degrees.

00:05:32.930 --> 00:05:35.110
So if you add these
two things up,

00:05:35.110 --> 00:05:38.270
you essentially are able to go
all halfway around the circle.

00:05:38.270 --> 00:05:41.470
Or you could go throughout
the entire half circle

00:05:41.470 --> 00:05:43.720
or semicircle for a protractor.

00:05:43.720 --> 00:05:47.260
And when you have two angles
that add up to 180 degrees,

00:05:47.260 --> 00:05:49.250
we call them supplementary.

00:05:49.250 --> 00:05:51.450
I know it's a little hard
to remember sometimes.

00:05:51.450 --> 00:05:52.960
90 degrees is complementary.

00:05:52.960 --> 00:05:54.670
They're just
complementing each other.

00:05:54.670 --> 00:05:56.750
And then if you add
up to 180 degrees,

00:05:56.750 --> 00:05:58.690
you have supplementary.

00:05:58.690 --> 00:06:00.315
You have supplementary angles.

00:06:03.630 --> 00:06:05.700
And if you have two
supplementary angles that

00:06:05.700 --> 00:06:08.850
are adjacent so that they
share a common side-- so let

00:06:08.850 --> 00:06:10.990
me draw that over here.

00:06:10.990 --> 00:06:13.760
So let's say you have one
angle that looks like this.

00:06:13.760 --> 00:06:15.620
And that you have another angle.

00:06:15.620 --> 00:06:18.290
So let me put some
letters here again.

00:06:18.290 --> 00:06:19.890
And I'll start reusing letters.

00:06:19.890 --> 00:06:21.500
So let's say that this is ABC.

00:06:23.820 --> 00:06:25.820
And you have another angle
that looks like this.

00:06:31.270 --> 00:06:36.570
I already used C.

00:06:36.570 --> 00:06:39.820
Once again, let's say
that this is 50 degrees.

00:06:39.820 --> 00:06:42.950
And let's say that this right
over here is 130 degrees.

00:06:42.950 --> 00:06:49.120
Clearly, angle DBA plus angle
ABC, if you add them together,

00:06:49.120 --> 00:06:52.560
you get 130 degrees plus 50
degrees, which is 180 degrees.

00:06:52.560 --> 00:06:53.930
So they are supplementary.

00:06:53.930 --> 00:06:55.320
So let me write that down.

00:06:55.320 --> 00:07:04.660
Angle DBA and angle
ABC are supplementary.

00:07:04.660 --> 00:07:07.990
They add up to 180 degrees.

00:07:07.990 --> 00:07:10.675
But they are also
adjacent angles.

00:07:14.400 --> 00:07:16.050
And because they're
supplementary

00:07:16.050 --> 00:07:19.340
and they're adjacent, if you
look at the broader angle,

00:07:19.340 --> 00:07:22.170
the angle used from the sides
that they don't have in common.

00:07:22.170 --> 00:07:29.180
If you look at
angle DBC, this is

00:07:29.180 --> 00:07:31.330
going to be essentially
a straight line, which

00:07:31.330 --> 00:07:37.560
we can call a straight angle.

00:07:37.560 --> 00:07:39.490
So I introduced you to
a bunch of words here.

00:07:39.490 --> 00:07:41.680
And now I think we have
all of the tools we

00:07:41.680 --> 00:07:45.050
need to start doing
some interesting proofs.

00:07:45.050 --> 00:07:50.545
And just to review here,
we talked about any angles

00:07:50.545 --> 00:07:55.160
that add up to 90 degrees are
considered to be complementary.

00:07:55.160 --> 00:07:56.830
This is adding up to 90 degrees.

00:07:56.830 --> 00:08:00.160
If they happen to be adjacent,
then the two outside sides

00:08:00.160 --> 00:08:03.390
will form a right angle.

00:08:03.390 --> 00:08:06.370
When you have a right angle,
the two sides of a right angle

00:08:06.370 --> 00:08:09.470
are considered to
be perpendicular.

00:08:09.470 --> 00:08:12.760
And then if you have two angles
that add up to 180 degrees,

00:08:12.760 --> 00:08:14.770
they're considered
supplementary.

00:08:14.770 --> 00:08:16.690
And then if they
happen to be adjacent,

00:08:16.690 --> 00:08:18.680
they will form a straight angle.

00:08:18.680 --> 00:08:21.440
Or another way, if you said,
if you have a straight angle

00:08:21.440 --> 00:08:24.910
and you have one of the
angles, the other angle

00:08:24.910 --> 00:08:26.540
is going to be
supplementary to it.

00:08:26.540 --> 00:08:28.710
They're going to add
up to 180 degrees.

00:08:28.710 --> 00:08:30.710
So I'll leave you there.


Measuring angles in degrees

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.370
Now that we know
what an angle is,

00:00:02.370 --> 00:00:04.260
let's think about how
we can measure them.

00:00:04.260 --> 00:00:06.810
And we already hinted
at one way to think

00:00:06.810 --> 00:00:09.550
about the measure of angle in
the last video where we said,

00:00:09.550 --> 00:00:14.500
look, this angle XYZ seems
more open than angle BAC.

00:00:14.500 --> 00:00:16.920
So maybe the
measure of angle XYZ

00:00:16.920 --> 00:00:20.440
should be larger than
the angle of the BAC,

00:00:20.440 --> 00:00:22.340
and that is exactly
the way we think

00:00:22.340 --> 00:00:23.642
about the measures of angles.

00:00:23.642 --> 00:00:25.100
But what I want to
do in this video

00:00:25.100 --> 00:00:27.790
is come up with an exact
way to measure an angle.

00:00:27.790 --> 00:00:31.890
So what I've drawn over here is
a little bit of a half-circle,

00:00:31.890 --> 00:00:33.650
and it looks very
similar to a tool

00:00:33.650 --> 00:00:36.620
that you can buy at your
local school supplies store

00:00:36.620 --> 00:00:37.882
to measure angles.

00:00:37.882 --> 00:00:40.465
So this is actually a little bit
of a drawing of a protractor.

00:00:43.630 --> 00:00:45.650
And what we do in something
like a protractor--

00:00:45.650 --> 00:00:48.030
you could even construct
one with a piece of paper--

00:00:48.030 --> 00:00:50.120
is we've taken a
half-circle right here,

00:00:50.120 --> 00:00:52.880
and we've divided it
into a 180 sections,

00:00:52.880 --> 00:00:56.390
and each of these marks
marks 10 of those sections.

00:00:56.390 --> 00:00:58.710
And what you do
for any given angle

00:00:58.710 --> 00:01:00.830
is you put one of the
sides of the angle.

00:01:00.830 --> 00:01:02.600
So each of the rays
of an angle are

00:01:02.600 --> 00:01:04.160
considered one of its sides.

00:01:04.160 --> 00:01:08.520
So you put the
vertex of the angle

00:01:08.520 --> 00:01:12.102
at the center of
this half-circle--

00:01:12.102 --> 00:01:14.060
or if you're dealing with
an actual protractor,

00:01:14.060 --> 00:01:16.110
at the center of
that protractor--

00:01:16.110 --> 00:01:19.820
and then you put one
side along the 0 mark.

00:01:22.570 --> 00:01:26.546
So I'm going redraw this
angle right over here

00:01:26.546 --> 00:01:27.920
at the center of
this protractor.

00:01:27.920 --> 00:01:31.480
So if we said this is Y, then
the Z goes right over here.

00:01:31.480 --> 00:01:38.500
And then the other ray, ray
YX in this circumstance,

00:01:38.500 --> 00:01:41.830
will go roughly
in that direction.

00:01:41.830 --> 00:01:47.150
And so it is pointing on the
protractor to the-- let's see.

00:01:47.150 --> 00:01:49.160
This looks like this
is the 70th of section.

00:01:49.160 --> 00:01:50.410
This is the 80th section.

00:01:50.410 --> 00:01:54.640
So maybe this is, I would
guess, the 77th section.

00:01:54.640 --> 00:01:57.550
So this is pointing
to 77 right over here.

00:02:01.550 --> 00:02:04.070
Assuming that I drew it the
right way right over here,

00:02:04.070 --> 00:02:10.380
we could say the measure of
angle XYZ-- sometimes they'll

00:02:10.380 --> 00:02:12.900
just say angle XYZ is equal
to, but this is a little bit

00:02:12.900 --> 00:02:16.586
more formal-- the measure
of angle XYZ is equal to 77.

00:02:19.500 --> 00:02:22.370
Each of these little sections,
we call them "degrees."

00:02:22.370 --> 00:02:26.010
So it's equal to
77-- sometimes it's

00:02:26.010 --> 00:02:29.890
written like that, the same
way you would write "degrees"

00:02:29.890 --> 00:02:31.125
for the temperature outside.

00:02:31.125 --> 00:02:33.750
So you could write "77 degrees"
like that or you could actually

00:02:33.750 --> 00:02:35.427
write out the word
right over there.

00:02:35.427 --> 00:02:37.010
So each of these
sections are degrees,

00:02:37.010 --> 00:02:38.260
so we're measuring in degrees.

00:02:38.260 --> 00:02:41.290
And I want to be clear,
degrees aren't the only way

00:02:41.290 --> 00:02:42.070
to measure angles.

00:02:42.070 --> 00:02:44.290
Really, anything that
measures the openness.

00:02:44.290 --> 00:02:46.212
So when you go
into trigonometry,

00:02:46.212 --> 00:02:47.920
you'll learn that you
can measure angles,

00:02:47.920 --> 00:02:49.790
not only in degrees,
but also using

00:02:49.790 --> 00:02:51.000
something called "radians."

00:02:51.000 --> 00:02:53.190
But I'll leave that
to another day.

00:02:53.190 --> 00:02:57.640
So let's measure this
other angle, angle BAC.

00:02:57.640 --> 00:03:00.150
So once again, I'll
put A at the center,

00:03:00.150 --> 00:03:07.460
and then AC I'll put along the 0
degree edge of this half-circle

00:03:07.460 --> 00:03:08.920
or of this protractor.

00:03:08.920 --> 00:03:11.181
And then I'll point
AB in the-- well,

00:03:11.181 --> 00:03:13.430
assuming that I'm drawing
it exactly the way that it's

00:03:13.430 --> 00:03:13.860
over there.

00:03:13.860 --> 00:03:15.443
Normally, instead
of moving the angle,

00:03:15.443 --> 00:03:17.740
you could actually move the
protractor to the angle.

00:03:17.740 --> 00:03:20.460
So it looks something
like that, and you

00:03:20.460 --> 00:03:27.230
could see that it's pointing to
right about the 30 degree mark.

00:03:27.230 --> 00:03:30.480
So we could say that
the measure of angle BAC

00:03:30.480 --> 00:03:32.700
is equal to 30 degrees.

00:03:32.700 --> 00:03:34.200
And so you can look
just straight up

00:03:34.200 --> 00:03:38.230
from evaluating these
numbers that 77 degrees is

00:03:38.230 --> 00:03:43.020
clearly larger than 30
degrees, and so it is a larger

00:03:43.020 --> 00:03:48.080
angle, which makes sense
because it is a more open angle.

00:03:48.080 --> 00:03:50.780
And in general, there's a
couple of interesting angles

00:03:50.780 --> 00:03:51.720
to think about.

00:03:51.720 --> 00:03:53.640
If you have a 0 degree
angle, you actually

00:03:53.640 --> 00:03:56.300
have something that's
just a closed angled.

00:03:56.300 --> 00:03:58.890
It really is just a
ray at that point.

00:03:58.890 --> 00:04:02.310
As you get larger and larger or
as you get more and more open,

00:04:02.310 --> 00:04:04.880
you eventually get to a
point where one of the rays

00:04:04.880 --> 00:04:07.300
is completely
straight up and down

00:04:07.300 --> 00:04:08.980
while the other one
is left to right.

00:04:08.980 --> 00:04:10.570
So you could imagine
an angle that

00:04:10.570 --> 00:04:17.670
looks like this where one ray
goes straight up down like that

00:04:17.670 --> 00:04:20.860
and the other ray goes
straight right and left.

00:04:20.860 --> 00:04:23.050
Or you could imagine
something like an angle that

00:04:23.050 --> 00:04:26.480
looks like this where, at least,
the way you're looking at it,

00:04:26.480 --> 00:04:27.900
one doesn't look
straight up down

00:04:27.900 --> 00:04:29.670
or one does it look
straight right left.

00:04:29.670 --> 00:04:33.520
But if you rotate it, it would
look just like this thing

00:04:33.520 --> 00:04:36.440
right over here where one is
going straight up and down

00:04:36.440 --> 00:04:38.700
and one is going
straight right and left.

00:04:38.700 --> 00:04:42.080
And you can see from our
measure right over here

00:04:42.080 --> 00:04:45.330
that that gives us
a 90 degree angle.

00:04:45.330 --> 00:04:47.110
It's a very interesting angle.

00:04:47.110 --> 00:04:51.410
It shows up many, many times
in geometry and trigonometry,

00:04:51.410 --> 00:04:54.370
and there's a special word
for a 90 degree angle.

00:04:54.370 --> 00:04:56.440
It is called a "right angle."

00:04:56.440 --> 00:04:58.350
So this right over
here, assuming

00:04:58.350 --> 00:05:00.490
if you rotate it around,
would look just like this.

00:05:00.490 --> 00:05:06.200
We would call this
a "right angle."

00:05:06.200 --> 00:05:09.180
And there is a notation to
show that it's a right angle.

00:05:09.180 --> 00:05:12.090
You draw a little part of
a box right over there,

00:05:12.090 --> 00:05:15.370
and that tells us that this
is, if you were to rotate it,

00:05:15.370 --> 00:05:18.160
exactly up and down while
this is going exactly

00:05:18.160 --> 00:05:19.940
right and left, if
you were to rotate it

00:05:19.940 --> 00:05:22.240
properly, or vice versa.

00:05:22.240 --> 00:05:23.750
And then, as you
go even wider, you

00:05:23.750 --> 00:05:25.960
get wider and wider
and wider and wider

00:05:25.960 --> 00:05:31.120
until you get all the way to
an angle that looks like this.

00:05:31.120 --> 00:05:33.690
So you could imagine
an angle where

00:05:33.690 --> 00:05:38.300
the two rays in that
angle form a line.

00:05:38.300 --> 00:05:42.900
So let's say this is
point X. This is point Y,

00:05:42.900 --> 00:05:48.410
and this is point Z. You
could call this angle ZXY,

00:05:48.410 --> 00:05:50.050
but it's really
so open that it's

00:05:50.050 --> 00:05:51.970
formed an actual line here.

00:05:51.970 --> 00:05:56.310
Z, X, and Y are collinear.

00:05:56.310 --> 00:06:04.560
This is a 180 degree angle where
we see the measure of angle ZXY

00:06:04.560 --> 00:06:07.140
is 180 degrees.

00:06:07.140 --> 00:06:08.730
And you can actually
go beyond that.

00:06:08.730 --> 00:06:11.660
So if you were to go all
the way around the circle

00:06:11.660 --> 00:06:14.617
so that you would get back
to 360 degrees and then

00:06:14.617 --> 00:06:16.700
you could keep going around
and around and around,

00:06:16.700 --> 00:06:19.120
and you'll start to
see a lot more of that

00:06:19.120 --> 00:06:22.370
when you enter a
trigonometry class.

00:06:22.370 --> 00:06:25.010
Now, there's two
last things that I

00:06:25.010 --> 00:06:28.226
want to introduce in this video.

00:06:28.226 --> 00:06:29.600
There are special
words, and I'll

00:06:29.600 --> 00:06:32.580
talk about more types of
angles in the next video.

00:06:32.580 --> 00:06:37.410
But if an angle is
less than 90 degrees,

00:06:37.410 --> 00:06:39.830
so, for example, both of
these angles that we started

00:06:39.830 --> 00:06:42.460
our discussion with are
less than 90 degrees,

00:06:42.460 --> 00:06:48.250
we call them "acute angles."

00:06:48.250 --> 00:06:50.090
So this is acute.

00:06:50.090 --> 00:06:53.230
So that is an acute
angle, and that

00:06:53.230 --> 00:06:55.210
is an acute angle
right over here.

00:06:55.210 --> 00:06:57.330
They are less than 90 degrees.

00:06:57.330 --> 00:06:59.822
What does a non-acute
angle look like?

00:06:59.822 --> 00:07:01.780
And there's a word for
it other than non-acute.

00:07:01.780 --> 00:07:03.530
Well, it would be
more than 90 degrees.

00:07:03.530 --> 00:07:06.200
So, for example-- let
me do this in a color

00:07:06.200 --> 00:07:09.839
I haven't used-- an angle
that looks like this,

00:07:09.839 --> 00:07:11.880
and let me draw it a little
bit better than that.

00:07:11.880 --> 00:07:13.430
An angle it looks like this.

00:07:13.430 --> 00:07:16.960
So that's one side of the
angle or one of the rays

00:07:16.960 --> 00:07:19.220
and then I'll put the other
one on the baseline right

00:07:19.220 --> 00:07:19.720
over here.

00:07:19.720 --> 00:07:22.150
Clearly, this is
larger than 90 degrees.

00:07:22.150 --> 00:07:23.990
If I were to
approximate, let's see,

00:07:23.990 --> 00:07:27.640
that's 100, 110,
120, almost 130.

00:07:27.640 --> 00:07:31.520
So let's call that maybe
a 128-degree angle.

00:07:31.520 --> 00:07:37.641
We call this an "obtuse angle."

00:07:37.641 --> 00:07:40.140
The way I remember it as acute,
it's kind of "a cute" angle.

00:07:40.140 --> 00:07:42.230
It's nice and small.

00:07:42.230 --> 00:07:46.760
I believe acute in either Latin
or Greek or maybe both means

00:07:46.760 --> 00:07:48.600
something like "pin" or "sharp."

00:07:48.600 --> 00:07:50.800
So that's one way
to think about it.

00:07:50.800 --> 00:07:53.050
An acute angle
seems much sharper.

00:07:53.050 --> 00:07:55.400
Obtuse, I kind of
imagine something

00:07:55.400 --> 00:07:58.590
that's kind of
lumbering and large.

00:07:58.590 --> 00:08:00.200
Or you could think
it's not acute.

00:08:00.200 --> 00:08:01.991
It's not nice and
small and pointy.

00:08:01.991 --> 00:08:03.490
So that's one way
to think about it,

00:08:03.490 --> 00:08:06.530
but this is just
general terminology

00:08:06.530 --> 00:08:08.340
for different types of angles.

00:08:08.340 --> 00:08:11.080
Less than 90 degrees,
you have an acute angle.

00:08:11.080 --> 00:08:13.120
At 90 degrees, you
have a right angle.

00:08:13.120 --> 00:08:16.180
Larger than 90 degrees,
you have an obtuse angle.

00:08:16.180 --> 00:08:18.510
And then, if you get all
the way to 180 degrees,

00:08:18.510 --> 00:08:21.468
your angle actually
forms a line.


Measuring angles using a protractor

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.700
What I have right in front
of me is the Khan Academy

00:00:03.700 --> 00:00:06.930
measuring angles exercise.

00:00:06.930 --> 00:00:09.440
I have a small part of it in
this screen right over here.

00:00:09.440 --> 00:00:11.190
And it's a pretty cool
exercise because it

00:00:11.190 --> 00:00:12.900
has this little
virtual protractor

00:00:12.900 --> 00:00:15.909
that we can use to
actually measure angles.

00:00:15.909 --> 00:00:17.700
And I want to give
credit to the person who

00:00:17.700 --> 00:00:19.390
built this protractor, because
I think it's pretty neat.

00:00:19.390 --> 00:00:22.040
Omar Rizwan, who was actually
a high school intern,

00:00:22.040 --> 00:00:23.762
made this pretty neat module.

00:00:23.762 --> 00:00:25.970
And so in general, when you
want to measure an angle,

00:00:25.970 --> 00:00:27.710
what you want to do
is you want to put

00:00:27.710 --> 00:00:30.650
the center of your
protractor at the center of--

00:00:30.650 --> 00:00:33.170
or I should say, at the
vertex of the angle.

00:00:33.170 --> 00:00:35.170
Or you want to put the
vertex of the angle

00:00:35.170 --> 00:00:37.030
at the center of the protractor.

00:00:37.030 --> 00:00:40.230
And then what you want to do
is either rotate the angle

00:00:40.230 --> 00:00:41.370
or rotate the protractor.

00:00:41.370 --> 00:00:43.610
In this case, we're going
to rotate the protractor.

00:00:43.610 --> 00:00:45.140
You want to rotate
the protractor

00:00:45.140 --> 00:00:48.570
so that the 0 angle,
or kind of the 0 mark,

00:00:48.570 --> 00:00:51.850
is at one of the
sides of the angle.

00:00:51.850 --> 00:00:54.590
And the other side of the
angle is within the protractor.

00:00:54.590 --> 00:00:55.590
So let's try to do that.

00:00:55.590 --> 00:00:59.240
So maybe if we want to
do that, this 0 side

00:00:59.240 --> 00:01:02.200
should be at this
side of the angle.

00:01:02.200 --> 00:01:07.120
So let's rotate it that way.

00:01:07.120 --> 00:01:09.837
Let me just keep rotating it.

00:01:09.837 --> 00:01:11.170
If I could just keep it pressed.

00:01:11.170 --> 00:01:12.560
That's better.

00:01:12.560 --> 00:01:14.720
All right.

00:01:14.720 --> 00:01:16.210
That looks about right.

00:01:16.210 --> 00:01:18.030
So one side is at the 0 mark.

00:01:18.030 --> 00:01:20.540
And then my angle,
my other side--

00:01:20.540 --> 00:01:22.630
or if this was a
ray, it points to,

00:01:22.630 --> 00:01:26.120
looks like, pretty close
to the 20 degree mark.

00:01:26.120 --> 00:01:28.060
So I will type that
in off the screen.

00:01:28.060 --> 00:01:29.170
You don't see that.

00:01:29.170 --> 00:01:30.780
And that is the right answer.

00:01:30.780 --> 00:01:32.400
And then we can
get another angle.

00:01:32.400 --> 00:01:34.920
So let's try to measure
this one right over here.

00:01:34.920 --> 00:01:37.980
So once again, place the
center of the protractor

00:01:37.980 --> 00:01:41.620
at the center, at the
vertex, of our angle.

00:01:41.620 --> 00:01:45.170
We can place the 0 degree,
the base of the protractor,

00:01:45.170 --> 00:01:47.340
at this side of the angle.

00:01:47.340 --> 00:01:50.920
So let's just rotate it a
little bit, maybe one more time.

00:01:50.920 --> 00:01:52.270
That looks about right.

00:01:52.270 --> 00:01:55.390
And then the angle is now
opening up-- let's see,

00:01:55.390 --> 00:01:58.200
the other side is
pointing to 110 degrees.

00:01:58.200 --> 00:01:59.690
So this is larger
than 90 degrees.

00:01:59.690 --> 00:02:01.730
It's also an obtuse angle.

00:02:01.730 --> 00:02:03.260
The last one was an acute angle.

00:02:03.260 --> 00:02:05.530
This is obtuse, 110 degrees.

00:02:05.530 --> 00:02:06.790
More than 90 degrees.

00:02:06.790 --> 00:02:08.570
So let me type it in.

00:02:08.570 --> 00:02:10.020
I got the right answer.

00:02:10.020 --> 00:02:11.790
Let's do a couple more of these.

00:02:11.790 --> 00:02:14.265
So once again, put the
center of the protractor

00:02:14.265 --> 00:02:17.540
at the vertex of our angle.

00:02:17.540 --> 00:02:20.480
And now, I want to rotate it.

00:02:20.480 --> 00:02:22.130
There we go.

00:02:22.130 --> 00:02:27.570
And this looks like roughly
an 80 degree angle, not quite.

00:02:27.570 --> 00:02:29.200
If I have to be
really precise, it

00:02:29.200 --> 00:02:31.310
looks like it's maybe
81 or 82 degrees.

00:02:31.310 --> 00:02:35.312
But I'll just go with
80 as my best guess.

00:02:35.312 --> 00:02:36.270
I got the right answer.

00:02:36.270 --> 00:02:37.640
Let's do one more of these.

00:02:37.640 --> 00:02:41.100
So once again,
vertex of my angle

00:02:41.100 --> 00:02:44.050
at the center of my protractor.

00:02:44.050 --> 00:02:47.484
And then I want to put one side
of the angle at the 0 degree.

00:02:47.484 --> 00:02:49.650
And I want to show you,
there's two ways to do that.

00:02:49.650 --> 00:02:51.310
You could do this.

00:02:51.310 --> 00:02:53.710
You could do just this.

00:02:53.710 --> 00:02:56.180
But this isn't too helpful,
because the angle is now

00:02:56.180 --> 00:02:57.120
outside.

00:02:57.120 --> 00:02:59.300
The other side sits
outside of the protractor.

00:02:59.300 --> 00:03:01.350
So you want the 0
degrees on the side,

00:03:01.350 --> 00:03:03.270
so the other side is
within the protractor.

00:03:03.270 --> 00:03:05.990
So let's keep rotating it.

00:03:05.990 --> 00:03:07.020
There we go.

00:03:07.020 --> 00:03:09.480
And then our other
side opens up or you

00:03:09.480 --> 00:03:11.620
could say points to 70 degrees.

00:03:11.620 --> 00:03:13.540
So this is an acute
angle right over here.

00:03:13.540 --> 00:03:15.430
So it is 70 degrees.

00:03:15.430 --> 00:03:16.970
So I'll leave you with that.

00:03:16.970 --> 00:03:19.370
Oh look, I'm ready to move
on, the exercise tells me.

00:03:19.370 --> 00:03:23.130
And now we can start talking
about more things about angles

00:03:23.130 --> 00:03:25.650
now that we know
how to measure them.


Angle basics

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.560
Let's say we have one ray over
here that starts at point A

00:00:03.560 --> 00:00:06.240
and then goes through
point B. And so we

00:00:06.240 --> 00:00:10.030
could call this
ray-- let me draw

00:00:10.030 --> 00:00:14.340
that a little bit straighter--
we could call this ray AB.

00:00:14.340 --> 00:00:17.890
Ray AB starts at A,
or has a vertex at A.

00:00:17.890 --> 00:00:21.120
And let's say that
there's also a ray AC.

00:00:21.120 --> 00:00:24.540
So let's say that C is
sitting right over there.

00:00:24.540 --> 00:00:28.510
And then I can draw another
ray that goes through C.

00:00:28.510 --> 00:00:31.360
So this is ray AC.

00:00:31.360 --> 00:00:33.930
And what's interesting
about these two rays

00:00:33.930 --> 00:00:36.440
is that they have the
exact same vertex.

00:00:36.440 --> 00:00:39.550
They have the exact
same vertex at A.

00:00:39.550 --> 00:00:41.450
And in general,
what we have when

00:00:41.450 --> 00:00:44.150
we have two rays that have
the exact same vertex,

00:00:44.150 --> 00:00:46.220
you have an angle.

00:00:46.220 --> 00:00:48.930
And you're probably
already reasonably familiar

00:00:48.930 --> 00:00:51.250
with the concept
of an angle, which

00:00:51.250 --> 00:00:54.200
I believe comes from the Latin
for corner, which makes sense.

00:00:54.200 --> 00:00:55.990
This looks like a
little bit of a corner

00:00:55.990 --> 00:00:59.020
right over here that
we see at point A.

00:00:59.020 --> 00:01:01.765
But the geometric
definition, or the one

00:01:01.765 --> 00:01:03.600
that you're most
likely to see, is

00:01:03.600 --> 00:01:07.550
when two rays share
a common vertex.

00:01:07.550 --> 00:01:09.160
And that common
vertex is actually

00:01:09.160 --> 00:01:11.090
called the vertex of the angle.

00:01:11.090 --> 00:01:14.410
So A is the vertex.

00:01:14.410 --> 00:01:17.620
Not only is it the vertex of
each of these rays, ray AB

00:01:17.620 --> 00:01:22.475
and ray AC, it is also
the vertex of the angle.

00:01:22.475 --> 00:01:24.100
So the next thing I
want to think about

00:01:24.100 --> 00:01:27.670
is how do we label an angle.

00:01:27.670 --> 00:01:30.280
You might be tempted to
just label it angle A.

00:01:30.280 --> 00:01:33.090
But I'll show you in a
second why that's not

00:01:33.090 --> 00:01:35.710
going to be so clear
to someone, based

00:01:35.710 --> 00:01:39.140
on where our angle
is actually sitting.

00:01:39.140 --> 00:01:40.920
So the way that you
specify an angle--

00:01:40.920 --> 00:01:43.060
and hopefully this will
make sense in a second--

00:01:43.060 --> 00:01:46.432
is that you say angle-- this
is the symbol for angle,

00:01:46.432 --> 00:01:48.640
and it actually looks
strangely similar to this angle

00:01:48.640 --> 00:01:49.680
right over here.

00:01:49.680 --> 00:01:51.290
But this little
pointy thing, or it

00:01:51.290 --> 00:01:53.120
almost looks like
a less-than sign.

00:01:53.120 --> 00:01:53.940
But it's not quite.

00:01:53.940 --> 00:01:55.606
It's flat on the
bottom right over here.

00:01:55.606 --> 00:01:57.200
This is the symbol for angle.

00:01:57.200 --> 00:01:58.771
You'd say angle BAC.

00:02:01.980 --> 00:02:04.610
Or you could say angle CAB.

00:02:08.110 --> 00:02:11.039
In either case, they're kind
of specifying this corner.

00:02:11.039 --> 00:02:13.390
Or sometimes you could
view it as this opening

00:02:13.390 --> 00:02:14.394
right over here.

00:02:14.394 --> 00:02:15.810
And the important
thing to realize

00:02:15.810 --> 00:02:19.034
is that you have the vertex
in the middle of the letters.

00:02:19.034 --> 00:02:20.450
Now you might be
saying, wait, why

00:02:20.450 --> 00:02:22.252
go through the trouble
of listing all three

00:02:22.252 --> 00:02:22.960
of these letters.

00:02:22.960 --> 00:02:25.980
Why can't I just
call this angle A?

00:02:25.980 --> 00:02:29.730
And to see that, let me
show you another diagram.

00:02:29.730 --> 00:02:33.560
And although the geometric
definition of an angle

00:02:33.560 --> 00:02:37.560
involves two rays that have
the same vertex, in practice,

00:02:37.560 --> 00:02:39.170
you're going to see
many angles that

00:02:39.170 --> 00:02:41.627
are made up of lines
and line segments.

00:02:41.627 --> 00:02:43.960
And you could imagine that
you could continue those line

00:02:43.960 --> 00:02:45.837
segments on and on
in one direction.

00:02:45.837 --> 00:02:47.170
And then they would become rays.

00:02:47.170 --> 00:02:48.586
So in that way,
they're consistent

00:02:48.586 --> 00:02:50.030
with this definition.

00:02:50.030 --> 00:02:52.510
But let's say I have one line
segment that looks like that.

00:02:52.510 --> 00:02:53.790
Let me label some points here.

00:02:53.790 --> 00:02:55.070
So we've already used ABC.

00:02:55.070 --> 00:02:57.370
So I'm going to
call this D and E,

00:02:57.370 --> 00:03:00.990
points D and E. So this
is line segment DE.

00:03:00.990 --> 00:03:07.196
And let's say I also
have a line segment FG.

00:03:11.390 --> 00:03:13.340
And let's say this point
where these two line

00:03:13.340 --> 00:03:20.800
segments intersect, let's
call that point point H. Now

00:03:20.800 --> 00:03:24.280
how could we specify this
angle right over here?

00:03:24.280 --> 00:03:26.600
Can we just call that angle H?

00:03:26.600 --> 00:03:27.100
Well, no.

00:03:27.100 --> 00:03:31.070
Because if we just said angle H,
the angle that has a vertex H,

00:03:31.070 --> 00:03:33.100
it could be this
angle right over here.

00:03:33.100 --> 00:03:35.800
Or it could be this
angle right over here.

00:03:35.800 --> 00:03:37.080
Let me draw it this way.

00:03:37.080 --> 00:03:38.205
You could view it that way.

00:03:38.205 --> 00:03:40.090
Or it could be that
angle over there.

00:03:40.090 --> 00:03:42.040
It could be this
angle over here.

00:03:42.040 --> 00:03:43.990
It could be this
angle over here.

00:03:43.990 --> 00:03:46.820
Or it could be that
angle over there.

00:03:46.820 --> 00:03:49.000
And so the only way
to really specify

00:03:49.000 --> 00:03:52.720
which angle you're talking about
well, is to give three letters.

00:03:52.720 --> 00:03:58.680
So if you really did want
to talk about that angle

00:03:58.680 --> 00:04:02.540
right over there, you
would call that angle EHG.

00:04:02.540 --> 00:04:06.310
So that is angle EHG.

00:04:06.310 --> 00:04:10.521
Or you could actually
call that angle GHE.

00:04:15.526 --> 00:04:21.440
If you wanted to specify
this angle right over here,

00:04:21.440 --> 00:04:25.980
the one made up of, if you
imagine that ray and that ray,

00:04:25.980 --> 00:04:28.530
if you were to keep on
going past those points,

00:04:28.530 --> 00:04:36.260
then you could call that
angle DHG, or angle GHD.

00:04:39.767 --> 00:04:40.850
I think you get the point.

00:04:40.850 --> 00:04:44.960
This angle up here
could be FHD or EHF.

00:04:44.960 --> 00:04:48.000
And this one could
be FHD or DHF.

00:04:48.000 --> 00:04:49.930
And when you do
it this way, it's

00:04:49.930 --> 00:04:53.710
very clear what angle
you are referring to.

00:04:53.710 --> 00:04:56.220
So now that we have a general
idea of what an angle is,

00:04:56.220 --> 00:04:59.212
and kind of how do we
denote it with symbols,

00:04:59.212 --> 00:05:00.920
the next thing you
might be curious about

00:05:00.920 --> 00:05:04.180
is, it doesn't look like all
angles are kind of the same.

00:05:04.180 --> 00:05:06.330
It seems like some
angles open up

00:05:06.330 --> 00:05:08.302
or are more open than others.

00:05:08.302 --> 00:05:10.510
And some are a little bit
more closed in than others.

00:05:10.510 --> 00:05:12.430
And that actually is the case.

00:05:12.430 --> 00:05:17.450
So for example, let's
take two angles here.

00:05:17.450 --> 00:05:19.930
So let's say I have one
angle that looks like that.

00:05:19.930 --> 00:05:22.350
So I'll started reusing letters.

00:05:22.350 --> 00:05:28.990
So let's say that this is A, B
and C. I could make these rays.

00:05:28.990 --> 00:05:31.150
I could keep on going and
make them rays if I like.

00:05:31.150 --> 00:05:33.220
Or I could just keep
them as line segments.

00:05:33.220 --> 00:05:37.430
So right over here,
I have angle BAC.

00:05:37.430 --> 00:05:41.630
And let's say over
here, I have angle-- so

00:05:41.630 --> 00:05:48.950
let me draw another one-- and
let's say this is angle XYZ.

00:05:48.950 --> 00:05:51.010
And once again, I
could draw them as rays

00:05:51.010 --> 00:05:52.980
if I like, to go
on and on and on.

00:05:52.980 --> 00:05:57.362
So it's angle XY and Z.

00:05:57.362 --> 00:05:58.820
And so when you
just look at these,

00:05:58.820 --> 00:06:01.190
you just eyeball
these two angles,

00:06:01.190 --> 00:06:03.740
it looks like this
one is more open.

00:06:03.740 --> 00:06:07.990
So this one looks more open.

00:06:07.990 --> 00:06:13.610
While this one over here looks
more closed, at least relative

00:06:13.610 --> 00:06:14.110
to this one.

00:06:16.850 --> 00:06:19.200
So maybe when we
measure angles, we

00:06:19.200 --> 00:06:22.490
should measure it based on
how open or closed they are.

00:06:22.490 --> 00:06:24.420
And that actually is the case.

00:06:24.420 --> 00:06:26.810
And so without even telling
you how we measure an angle,

00:06:26.810 --> 00:06:30.500
you could say that the measure
of angle XYZ, the measure

00:06:30.500 --> 00:06:33.390
of this angle, is greater
than the measure of this angle

00:06:33.390 --> 00:06:34.390
right over here.

00:06:34.390 --> 00:06:37.490
And any convention we use for
measuring angles is essentially

00:06:37.490 --> 00:06:40.250
going to be a measure
of how open or how

00:06:40.250 --> 00:06:42.736
closed an angle actually is.

00:06:42.736 --> 00:06:44.360
And I'll take that
up in the next video

00:06:44.360 --> 00:06:48.330
where we'll see how to
actually measure an angle.