Question

question
determine if triangle mno and triangle pqr are or are not similar, and, if they are, state how you know. (note that figures are not
necessarily drawn to scale.)
answer attempt 1 out of 3
the triangles ≈ similar.

Explanation:

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

The sum of angles in a triangle is \( 180^{\circ} \).
For \( \triangle PQR \), if \( \angle Q = 52^{\circ} \) and \( \angle R=61^{\circ} \), then \( \angle P=180^{\circ}-(52^{\circ} + 61^{\circ})=180^{\circ}-113^{\circ}=67^{\circ} \).

Step2: Compare angles and side - ratios

In \( \triangle MNO \), we have one angle \( \angle N = 52^{\circ} \).
The ratio of sides adjacent to \( 52^{\circ} \) in \( \triangle MNO \) is \( \frac{MN}{NO}=\frac{11}{13} \).
In \( \triangle PQR \), the ratio of sides adjacent to \( 52^{\circ} \) is \( \frac{PQ}{QR}=\frac{55}{60}=\frac{11}{12} \).
Since the ratios of the sides adjacent to the equal angles (\( 52^{\circ} \)) are not equal (\( \frac{11}{13}
eq\frac{11}{12} \)), and the other angles (\( \angle P = 67^{\circ},\angle M \) and \( \angle O \) which we can't get to match with \( 61^{\circ} \) based on the given non - matching side ratios) don't give an AA (angle - angle) similarity condition.

Answer:

Triangles \( MNO \) and \( PQR \) are not similar. Because the ratios of the sides adjacent to the equal angle (\( 52^{\circ} \)) are not equal (\( \frac{MN}{NO}=\frac{11}{13}\) and \( \frac{PQ}{QR}=\frac{55}{60}=\frac{11}{12}\)), and there is no AA (angle - angle) similarity (the other angles don't match up to satisfy the similarity criteria).