If two corresponding interior angles of two triangles are congruent, how do you know that the triangles are similar?

Question

A. Since two corresponding interior angles are congruent, the sides opposite the angles are congruent. The remaining pair of sides must also be congruent. The triangles are similar by SSS criterion.

B. Since two corresponding interior angles are congruent, the sides opposite the angles are congruent. The triangles are similar by SAS criterion.

C. Since the sum of angle measures in a triangle is 180° , the third angle pair must have the same measure and are thus congruent. Because all three corresponding interior angles of the triangle are congruent, the triangles are similar by the definition of similarity.

D. Since the sum of angle measures in a triangle is 360° , the third angle pair must have the same measure and are thus congruent. Because all three corresponding interior angles of the triangle are congruent, the triangles are similar by the definition of similarity.

Answer 1

The correct explanation is:

C. Since the sum of angle measures in a triangle is 180°, the third angle pair must have the same measure and are thus congruent. Because all three corresponding interior angles of the triangle are congruent, the triangles are similar by the definition of similarity.

This is based on the Angle-Angle (AA) similarity criterion, which states that if two angles in one triangle are congruent to two angles in another triangle, the two triangles will be similar. Since the sum of the angles in a triangle is 180°, if two angles are congruent, the third angle must also be congruent, which leads to all three angles being congruent and thus proving the triangles are similar.