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.
In a triangle if you draw in one of your angle bisectors, remember there's three, one for each vertex, you're going to divide the opposite side proportionally. Well, we could write two different ratios here and I'm going to explain what I mean by opposite side. If I look at this as my vertex and I'm bisecting it the side that's opposite to it, is going to be that opposite side the one that contains x and y here.
So I can say that the ratio of a:x so I'm going to write the ratio of a:x has to be equal to the ratio of b:y. Another way of looking at it is what's the relationship between a and b. I can say that a goes to b the way x goes to y.
So when you homework around a test you're going to see a problem where you have a triangle with an angle bisector and you're going to be trying to find some of these missing lengths. Remember that it will divide this opposite side proportionally.