Question

which rule explains why these triangles are similar?

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

Explanation:

Step1: Calculate the third angle of triangle \(QRP\)

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle QRP\), if two angles are \(56^{\circ}\) and \(32^{\circ}\), then the third angle \(\angle Q=180^{\circ}-(56^{\circ} + 32^{\circ})=92^{\circ}\)

Step2: Calculate the third angle of triangle \(VUT\)

For \(\triangle VUT\), if two angles are \(56^{\circ}\) and \(36^{\circ}\), then the third angle \(\angle U=180^{\circ}-(56^{\circ}+36^{\circ}) = 88^{\circ}\)

Step3: Check similarity rules

• \(SSS\) (Side - Side - Side) similarity requires the ratio of all three sides to be equal. No side lengths are given.
• \(SAS\) (Side - Angle - Side) similarity requires two sides in proportion and the included angle equal. No side lengths are given.
• \(AA\) (Angle - Angle) similarity requires two pairs of equal angles. Here, only one pair of angles (\(56^{\circ}\)) is equal.

Answer:

none of the above; the triangles cannot be proven similar