Question

you are given two triangles and the information that the three pairs of corresponding angles are congruent. what other information would guarantee that the triangles are congruent?
a the triangles are right.
b all corresponding sides are proportional.
c one pair of corresponding sides is proportional.
d one pair of corresponding sides has the same measure.

Explanation:

Step1: Recall triangle congruence criteria

For two triangles, if three pairs of corresponding angles are congruent (AAA - which only gives similarity in general), but for congruence, we need side - length information.

Step2: Analyze each option

• Option A: Just being right - angled (one angle is \(90^{\circ}\)) along with AAA (already known angles) is not sufficient for congruence.
• Option B: All corresponding sides proportional gives similarity (by SSS - similarity criterion), not congruence.
• Option C: One pair of corresponding sides is proportional, combined with AAA (from given) still gives similarity (by AA - similarity if two angles are known, but with three angles known and one side proportional - similarity).
• Option D: If three pairs of corresponding angles are congruent (given) and one pair of corresponding sides has the same measure (by AAS or ASA or SSS congruence criteria, depending on the side - angle arrangement, but with three angles congruent and one side equal, the triangles are congruent).

Answer:

D. One pair of corresponding sides has the same measure.