In , is an altitude of and also the bisector of . Prove that is isosceles. This was the question I was given. This is my work: is an altitude of . Therefore, it goes straight down from the vertex to be perpendicular with . is also a bisector of . Therefore, it cuts in half. Because cuts in half by going straight down perpendicular to , and are at the same angle to This means the triangle has two congruent angles, making it isosceles. (See Theorem 4.7 and Theorem 4.9).