Question

sarah constructed ray \\(\overrightarrow{ae}\\) to bisect \\(\angle bac\\).

image of angle bisector construction

which of the following statements best justifies why \\(\angle bae \cong \angle cae\\)?

choose 1 answer:

a) they share a ray, so they must be congruent.

b) they are vertical angles, so they must be congruent.

c) they are corresponding angles created by parallel lines.

d) they are corresponding angles of congruent triangles.

Explanation:

• Option A: Sharing a ray doesn't imply congruence. Angles' congruence depends on measure, not shared rays. Eliminate A.
• Option B: Vertical angles are formed by intersecting lines, opposite each other. ∠BAE and ∠CAE share a common side (ray AE) and are adjacent, not vertical angles. Eliminate B.
• Option C: Corresponding angles with parallel lines require a transversal, but here there's no parallel lines context. Eliminate C.
• Option D: The construction (bisecting ∠BAC) involves creating congruent triangles (e.g., triangles with sides from the arcs, making them congruent by SSS). ∠BAE and ∠CAE are corresponding angles of these congruent triangles, so they are congruent.

Answer:

D. They are corresponding angles of congruent triangles.