Construct an Angle Bisector

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.