Geometric constructions: congruent angles

- [Instructor] What we're
gonna do in this video

is learn to construct congruent angles,

and we're gonna do it, with of course,

a pen or a pencil here.

I'm gonna use a ruler as a straight edge.

And then I'm gonna use a tool

known as a compass.

Which looks a little bit fancy,

but what it allows us to do

it'll apply using it in a little bit,

is it allows us to draw perfect circles,

or arcs, of a given radius.

You pivot on one point here

and then you use your pen or your pencil

to trace out the arc,

or the circle.

So let's just start with this angle

right over here,

and I'm going to construct an angle

that is congruent to it.

So let me make the
vertex of my second angle

right over there,

and then let me draw one of the rays

that originates at that vertex.

And I'm gonna put this angle

in a different orientation,

just to show that they don't even have

to have the same orientation.

So it's going to look

something like that,
that's one of the rays.

But then we have to figure out

where do we put,

where do we put the other ray

so that the two angles are congruent?

And this is where our compass

is going to be really useful.

So what I'm going to do
is put the pivot point

of a compass, of the compass,

right at the vertex of the first angle,

and I'm going to draw
out an arc like this.

And what's useful about the compass

is you can keep the radius constant,

and you can see it intersects

our first two rays at points,

let's just call this B and C.

And I could call this point A,

right over here.

And so let me,

now that I have my compass with the exact

right radius right now,

let me draw that right over here.

But this alone won't allow us to draw

the angle just yet,

but let me draw it like this,

and that is pretty good.

And let's call this
point right over here D,

and I'll call this one E,

and I wanna figure out where to put

my third point F,

so I can define ray E F,

so that these two angles are congruent.

And what I can do is take my compass again

and get a clear sense of the distance

between C and B,

by adjusting my compass.

So one point is on C,

and my pencil is on B.

So I have, get this right,

so I have this distance right over here.

I know this distance,

and I've adjusted my compass accordingly,

so I can get that same distance

right over there.

And so you can now image

where I'm going to draw that second ray.

That second ray,

if I put point F right over here,

my second ray,

I can just draw between,
starting at point E

right over here,

going through point F.

I could draw a little bit neater,

so it would look like that, my second ray.

Ignore that first little line I drew,

I'm using a pen,

which I don't recommend for you to do it.

I'm doing it so that you can see

it on this video.

Now how do we know that this angle

is now congruent to this angle

right over here?

Well one way to do it, is to think

about triangle B A C,

triangle B A C,

and triangle, let's just call it D F E.

So this triangle right over here.

When we drew that first arc,

we know that the distance between A C

is equivalent to the distance between A B,

and we kept the compass radius the same.

So we know that's also
the distance between E F,

and the distance between E D.

And then the second time,

when we adjusted our compass radius,

we now know that the distance between B C

is the same as the
distance between F and D.

Or the length of B C

is the same as the length of F D.

So it's very clear that we
have congruent triangles.

All of the three sides

have the same measure,

and therefore the corresponding angles

must be congruent as well.