Question

explain why the triangles are similar. then find the distance represented by x. why are the triangles similar? choose the correct answer below. a. the corresponding sides of two triangles are proportional, so the triangles are similar by the sss - theorem b. there is a pair of congruent angles and the sides that include the two vertical angles are also congruent, so the triangles are similar by the sas - theorem c. there is a pair of congruent angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the sas - theorem. d. there is a pair of congruent angles and a pair of congruent right angles, so the triangles are similar by the aa - postulate. e. there is a pair of congruent angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the aa - postulate.

Explanation:

Step1: Recall similarity - criteria

The AA (Angle - Angle) postulate states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. If there is a pair of congruent angles and a pair of congruent right - angles, the triangles satisfy the AA postulate.

Step2: Analyze options

Option A requires all three sides to be proportional (SSS - Theorem). Option B and C require congruent sides (SAS - Theorem). Option E has an incorrect combination of rules as AA postulate is about angles only, not about proportional sides including vertical angles. The correct reason for similarity here is the presence of a pair of congruent angles and a pair of congruent right - angles, which is Option D.

Answer:

D. There is a pair of congruent angles and a pair of congruent right angles, so the triangles are similar by the AA - Postulate.