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CCSS Math: HSG.CO.D.12

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.