Geometric constructions: angle bisector

We're asked to construct an angle bisector for the given angle. So this is the angle they're talking about. And they want us to make a line that goes right in between that angle, that divides that angle into two angles that have equal measure, that have half the measure of the first angle. So let's first find two points that are equidistant from this point right over here on each of these rays. So to do that, let's draw one circle here. And I can make this of any radius. Wherever this intersects with the rays, that's where I'm going to put a point. So let's say, here and here. Notice both of these points, since they're both on this circle, are going to be equidistant from this point, which is the center of the circle. Now, what I want to do is construct a line that is equidistant from both of these points. And we've done that already when we looked at perpendicular bisectors for lines in this construction module. So let's do that. So let's add their compass. And so what I want to do, this circle is centered at this point. And it has a radius equal to the distance between this point and that point. And then I do that again. So this circle is centered at this point and has a radius equal to the distance between that point and that point. And then the two places where they intersect are equidistant to both of these points. And so we can now draw our angle bisector, just like that. And you might say, well, how do we really know that this angle is equal to this angle? Well, there's a couple ways we can tell. We know this distance right over here is equal to this distance right over there. We know that this distance over here is equal to this distance over here. And both of these triangles share this line. So essentially, if you look at this point, this point, and this point, that forms a triangle. And if you look at this point, this point, and this point, that forms a triangle. We know those two triangles are congruent, so this angle must be equal to this angle. These are the corresponding angles. So they're going to be congruent. This is an angle bisector.