Question

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

Explanation:

Step1: Calculate angles in \(\triangle PQR\)

In \(\triangle PQR\), using the angle - sum property of a triangle (\(\angle Q+\angle P+\angle R = 180^{\circ}\)). Given \(\angle P = 49^{\circ}\) and \(\angle R=21^{\circ}\), then \(\angle Q=180^{\circ}-(49^{\circ}+21^{\circ}) = 110^{\circ}\).

Step2: Calculate angles in \(\triangle LMN\)

In \(\triangle LMN\), using the angle - sum property of a triangle (\(\angle M+\angle N+\angle L = 180^{\circ}\)). Given \(\angle M = 115^{\circ}\) and \(\angle N = 21^{\circ}\), then \(\angle L=180^{\circ}-(115^{\circ}+21^{\circ})=44^{\circ}\).
Since the corresponding angles of \(\triangle LMN\) and \(\triangle PQR\) are not equal (\(115^{\circ}
eq110^{\circ}\), \(44^{\circ}
eq49^{\circ}\), \(21^{\circ} = 21^{\circ}\)), \(\triangle LMN\) is not similar to \(\triangle PQR\) (by the AA (angle - angle) similarity criterion which requires two pairs of equal corresponding angles for two triangles to be similar).

Answer:

\(\triangle LMN\) is not similar to \(\triangle PQR\)