Question

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

Explanation:

Step1: Find \(\angle L\) in \(\triangle LMN\)

In \(\triangle LMN\), using the angle - sum property of a triangle (\(\angle L+\angle M+\angle LNM = 180^{\circ}\)). Given \(\angle M = 91^{\circ}\) and \(\angle LNM=47^{\circ}\). Then \(\angle L=180^{\circ}-(91^{\circ} + 47^{\circ})=42^{\circ}\).

Step2: Find \(\angle O\) in \(\triangle PON\)

In \(\triangle PON\), using the angle - sum property of a triangle (\(\angle O+\angle P+\angle PNO = 180^{\circ}\)). Given \(\angle P = 42^{\circ}\) and \(\angle PNO = 47^{\circ}\). Then \(\angle O=180^{\circ}-(42^{\circ}+47^{\circ}) = 91^{\circ}\).

Step3: Check for similarity

In \(\triangle LMN\) and \(\triangle PON\), \(\angle L=\angle P = 42^{\circ}\), \(\angle M=\angle O=91^{\circ}\), and \(\angle LNM=\angle PNO = 47^{\circ}\). By the AA (angle - angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Answer:

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