Question

which rule explains why these scalene triangles are similar?

image of two triangles with angles labeled: triangle u-r-v with angle at u 35°, angle at r 57°; triangle s-t-v with angle at s 35°

options:

• sss
• sas
• aa
• none of the above; the triangles cannot be proven similar

Explanation:

Step1: Identify Vertical Angles

Angles at \( V \) are vertical angles, so they are equal. Let's call this angle \( \angle RVU \) and \( \angle SVT \), so \( \angle RVU = \angle SVT \).

Step2: Analyze Given Angles

In \( \triangle RUV \), we have \( \angle U = 35^\circ \) and \( \angle R = 57^\circ \). In \( \triangle STV \), \( \angle S = 35^\circ \). We already know vertical angles are equal, and we have another pair of equal angles (\( \angle U = \angle S = 35^\circ \)). By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Here, we have two pairs of equal angles (the vertical angles and \( \angle U = \angle S \)), so AA applies.

Answer:

AA