Question

are the triangles similar? if so, which similarity theorem is used?
yes, △stu ~ △xyz by the sas similarity theorem.
yes, △stu ~ △xyz by the aa similarity theorem.
no, the triangles are not similar.
yes, △stu ~ △xyz by the sss similarity theorem.

Explanation:

Step1: Check the ratio of sides

For \(\triangle STU\) and \(\triangle XYZ\), we have \(\frac{SU}{XZ}=\frac{5}{20}=\frac{1}{4}\), \(\frac{ST}{XY}=\frac{8}{32}=\frac{1}{4}\)

Step2: Check the included angle

The included angle \(\angle S = \angle X=48^{\circ}\)

Since the ratio of two pairs of corresponding sides is equal and the included angles are equal, by the SAS (Side - Angle - Side) similarity theorem, \(\triangle STU\sim\triangle XYZ\)

Answer:

Yes, \(\triangle STU\sim\triangle XYZ\) by the SAS Similarity Theorem.