Question

△pqr and △rst are shown below.
which statement is true?
△pqr is similar to △rst.
△pqr is not similar to △rst.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Find the third angle of \(\triangle PQR\)

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle PQR\), if two angles are \(56^{\circ}\) and \(88^{\circ}\), then the third angle \(\angle QPR=180^{\circ}-(56^{\circ} + 88^{\circ})=36^{\circ}\).

Step2: Compare angles of \(\triangle PQR\) and \(\triangle RST\)

In \(\triangle PQR\), angles are \(36^{\circ},56^{\circ},88^{\circ}\). In \(\triangle RST\), angles are \(36^{\circ},56^{\circ},88^{\circ}\) (since \(180-(36 + 56)=88^{\circ}\)).

Step3: Use 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, \(\angle QPR=\angle SRT = 36^{\circ}\) and \(\angle PQR=\angle RST = 88^{\circ}\) (or \(\angle PRQ=\angle RTS=56^{\circ}\))

Answer:

\(\triangle PQR\) is similar to \(\triangle RST\)