Question

which rule explains why these scalene triangles are similar?
(image of two triangles with angles: angle at s is 35°, angle at q is 35°, angle at r is 50°, and intersection at u)
options:
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Identify Vertical Angles

Angles at \( U \) are vertical angles, so \( \angle TUS \cong \angle RUQ \) (vertical angles are equal).

Step2: Identify Given Angles

We know \( \angle S = 35^\circ \) and \( \angle Q = 35^\circ \), so \( \angle S \cong \angle Q \).

Step3: Apply AA Similarity

AA (Angle - Angle) similarity states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Here, we have two pairs of congruent angles (\( \angle S \cong \angle Q \) and \( \angle TUS \cong \angle RUQ \)), so by AA similarity, the triangles are similar. SSS requires all three sides proportional (not given), SAS requires two sides proportional and included angle (not given), so AA is the rule.

Answer:

AA