Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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An angle is formed by two rays with a common endpoint. The angle bisector is a ray or line segment that bisects the angle, creating two congruent angles. To **construct an angle bisector** you need a compass and straightedge. Bisectors are very important in identifying corresponding parts of similar triangles and in solving proofs.

The key points to an angle bisector

is that it does -- almost lost it

-- it does a couple things.

The first thing is it bisects the

angle, creating two congruent angles.

So if I have an angle that's in blue here

and the red ray is my angle bisector,

then it has created two

congruent angles.

So notice that this red is a ray.

That's another key thing.

Now, it also could be a line segment, if

you're talking about something in an

isosceles triangle perhaps, we could

say line segment, and every point

along this bisector is the same distance

from the two rays that make up

the sides.

But how do we measure distance?

Well, the shortest distance from a point

on this ray to a ray that forms the

angle is along a perpendicular.

So if you're to construct the perpendicular

from the angle bisector to a side,

and if you did that down here, then you

would say that these two segments

are congruent.

So that's the key parts

to an angle bisector.

Is that it bisects the angle creating two

congruent angles, it's a line-- a ray

or line segment, and that every point

on this ray is the same distance

measured along the perpendicular from

the rays that make up your angle.

But how do we actually construct that?

To do that let's grab our compass and our

straight edge and head over to this

angle right here.

So we know that we're going to create a

ray that creates two congruent angles.

So the first thing you're going to do is

you're going to swing an arc just like

if you were duplicating an angle.

So from the vertex I'm going to swing

an arc so that I create two points of

intersection.

Now I want to create a point out here that

is the same distance from these two

points of intersection.

So if you want to, you can change your compass,

but you don't have to, for the

sake of argument I will.

And you're going to swing an arc

from each of these end points.

So there's one arc from this intersection.

Here's another point of intersection

that I'm going to swing an arc from.

Now, this point right here is the same distance

from both of these end points.

So I'm going to connect this point

of intersection with my vertex.

Thereby creating my angle bisector.

So I'm going to draw this, connect my vertex,

and that point of intersection,

and what we've created are two congruent

angles that when they sum you

get the angle that you started with.