Question

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

Explanation:

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

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle LMN\), if \(\angle L = 57^{\circ}\) and \(\angle M=61^{\circ}\), then \(\angle N=180^{\circ}-(57^{\circ} + 61^{\circ})=180^{\circ}-118^{\circ} = 62^{\circ}\)

Step2: Check the angle - angle similarity criterion

In \(\triangle LMN\), angles are \(57^{\circ},61^{\circ},62^{\circ}\). In \(\triangle OPQ\), angles are \(57^{\circ},62^{\circ}\). Let's find the third angle of \(\triangle OPQ\). Using the angle - sum property of a triangle (\(180^{\circ}\)), if \(\angle O = 57^{\circ}\) and \(\angle Q = 62^{\circ}\), then \(\angle P=180^{\circ}-(57^{\circ}+62^{\circ})=61^{\circ}\)

Since \(\angle L=\angle O = 57^{\circ}\), \(\angle M=\angle P = 61^{\circ}\) and \(\angle N=\angle Q = 62^{\circ}\), 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), \(\triangle LMN\sim\triangle OPQ\)

Answer:

\(\triangle LMN\) is similar to \(\triangle OPQ\)