Question

20 cool-down: making angle bisectors triangle a b c is a reflection of triangle abc across line bd. c = c explain how you know that ray bc is the angle bisector of angle aba. notes

Explanation:

Step1: Recall Reflection Properties

Reflection over a line (here, \( CC' \)) maps a figure to its mirror image, so corresponding angles and sides are equal. Thus, \( \angle ABC = \angle A'BC' \) and \( BC = BC' \).

Step2: Analyze Angle Bisector Definition

An angle bisector divides an angle into two equal angles. Since \( \triangle A'BC' \) is the reflection of \( \triangle ABC \) over \( CC' \), \( BC \) (and \( BC' \)) creates \( \angle ABA' \) such that \( \angle ABC = \angle A'BC \). By the definition of reflection (congruent angles) and angle addition, \( BC \) splits \( \angle ABA' \) into two equal angles, so \( BC \) is the angle bisector.

Answer:

Since \( \triangle A'BC' \) is the reflection of \( \triangle ABC \) over \( CC' \), \( \angle ABC = \angle A'BC' \) (reflection preserves angle measure) and \( BC = BC' \). By the angle - bisector definition (a ray that divides an angle into two congruent angles), \( BC \) creates \( \angle ABC=\angle A'BC \), so \( BC \) bisects \( \angle ABA' \).