Question

In a triangle ABC, the external bisector at ZB and the internal bisector of C intersect each other at a point O. Prove that (∠BOC ≅ ∠BAC).

Answer 1

In triangle ABC, the external bisector at ZB and the internal bisector of C intersect at a point O. We can prove that ∠BOC is congruent to ∠BAC using the angle bisector theorem and angle-side-angle congruence.

Step-by-step explanation:

We can prove that ∠BOC is congruent to ∠BAC by using angle bisector theorem and angle-side-angle congruence.

1. From the angle bisector theorem, we know that the ratio of the lengths of the segments formed by the angle bisector in a triangle is equal to the ratio of the lengths of the opposite sides. Therefore, AZ/AC = BZ/BC.
2. Since AO is the external bisector of ∠B, we know that ∠BOC is a linear pair with ∠BAC. Therefore, ∠BOC + ∠BAC = 180 degrees.
3. Using angle-side-angle congruence, we can conclude that ∠BOC is congruent to ∠BAC. This is because ∠BOC is an exterior angle of triangle ABC and ∠ABC is the corresponding interior angle.