Question

determine whether the triangles can be proved simila explain your reasoning. 9.

Explanation:

Step1: Analyze triangle \( \triangle MNO \)

Since \( MN = MO \), \( \triangle MNO \) is isosceles. Using the angle - sum property of a triangle (\( \angle N+\angle O+\angle M=180^{\circ} \)) and \( \angle N = \angle O=71^{\circ} \), we find \( \angle M=180^{\circ}-71^{\circ}-71^{\circ}=38^{\circ} \).

Step2: Analyze triangle \( \triangle PCR \)

Since \( PR = CR \), \( \triangle PCR \) is isosceles. Let \( \angle P=\angle C \). Using the angle - sum property (\( \angle P+\angle C+\angle R = 180^{\circ} \)), and \( \angle R = 38^{\circ} \), we get \( \angle P=\angle C=\frac{180^{\circ}-38^{\circ}}{2}=71^{\circ} \).

Step3: Check for similarity

For \( \triangle MNO \) and \( \triangle PCR \), \( \angle M=\angle R = 38^{\circ} \), \( \angle N=\angle C=71^{\circ} \), \( \angle O=\angle P = 71^{\circ} \). 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), \( \triangle MNO\sim\triangle PCR \).

Answer:

The triangles \( \triangle MNO \) and \( \triangle PCR \) are similar. By the AA similarity criterion, since \( \angle M=\angle R = 38^{\circ} \) and \( \angle N=\angle C = 71^{\circ} \) (after calculating angles using the angle - sum property for isosceles triangles), the two triangles are similar.