Question

Triangle X has m∠A=30°, m∠B=90°, and m∠C=60° with side lengths of AB=3, BC=4, AC=5. Triangle Y has m∠A=30°, m∠B=90°, m∠C=60° with side lengths of AB=9, BC=12, AC=15. Which statement best describes the relationship between these two triangles?

a) They are similar triangles.
b) They are congruent triangles.
c) They have no geometric relationship.
d) They are complementary triangles.

Answer 1

Triangle X and Triangle Y are similar triangles because they have the same angle measures and their sides are in proportional lengths.

Step-by-step explanation:

The relationship between Triangle X and Triangle Y is that they are similar triangles. This is because they have identical angles, with m∠A=30°, m∠B=90°, m∠C=60° for both triangles, which means they are similar by AA (Angle-Angle) criterion, where two triangles are similar if they have two angles of one triangle congruent to two angles of the other triangle.

Furthermore, the side lengths of Triangle Y are exactly three times the side lengths of Triangle X. This proportional scaling of side lengths (AB=3 to AB=9, BC=4 to BC=12, and AC=5 to AC=15) is an additional indicator that the triangles are similar. Thus, the correct answer to the question is (a) They are similar triangles.