Question

which of these is a valid argument concerning the measures of the corresponding angles of similar triangles?
a. similar triangles must have the same shape, so the measures of the corresponding angles must be equal to each other
b. similar triangles do not have to have the same shape, so the measures of the corresponding angles do not have to be equal to each other
c. similar triangles must be the same size, so the measures of the corresponding angles must be equal to each other
d. similar triangles do not have to be the same size, so the measures of the corresponding angles do not have to be equal to each other

Explanation:

Step1: Recall the definition of similar triangles

Similar triangles have the same shape. By the AA (Angle - Angle) similarity criterion, if two triangles are similar, their corresponding angles are equal.

Step2: Analyze each option

• Option A: Similar triangles must have the same shape, so the measures of the corresponding angles must be equal to each other. This is correct.
• Option B: Similar triangles do have the same shape, and their corresponding angles are equal. So this option is incorrect.
• Option C: Similar triangles do not need to be the same size. Congruent triangles are the same size. So this option is incorrect.
• Option D: Similar triangles' corresponding angles must be equal. So this option is incorrect.

Answer:

A. Similar triangles must have the same shape, so the measures of the corresponding angles must be equal to each other.