Question

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

Explanation:

Step1: Find the measure of \(\angle PON\)

In \(\triangle PNO\), we know that the sum of angles in a triangle is \(180^{\circ}\). Given \(\angle PNO = 35^{\circ}\) and \(\angle NPO=180 - 105=75^{\circ}\) (linear - pair of angles). Using the angle - sum property of a triangle \(\angle PNO+\angle NPO+\angle PON = 180^{\circ}\). So, \(\angle PON=180-(35 + 75)=70^{\circ}\)

Step2: Find the measure of \(\angle PRQ\)

In \(\triangle PRQ\), using the angle - sum property of a triangle. \(\angle QPR = 105^{\circ}\) (vertically - opposite angles to the angle in \(\triangle PNO\) at \(P\)). Let's assume we want to check similarity. If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
We know that \(\angle NPO=\angle QPR = 75^{\circ}\) (vertically - opposite angles).
In \(\triangle PRQ\), \(\angle PRQ=180-(105 + 35)=40^{\circ}\) (This is wrong. Wait, no. Wait, \(\angle NPO\) and \(\angle QPR\) are vertical angles (\(\angle NPO=\angle QPR = 75^{\circ}\)). Also, \(\angle PNO\) and \(\angle PRQ\) are alternate - interior angles (if \(NO\parallel QR\), but we can also use the AA (angle - angle) similarity criterion.
Wait, another approach:
We know that \(\angle NPO=\angle QPR\) (vertically - opposite angles).
In \(\triangle PNO\), \(\angle PNO = 35^{\circ}\), \(\angle NPO=75^{\circ}\), \(\angle PON = 70^{\circ}\)
In \(\triangle PRQ\), \(\angle QPR = 75^{\circ}\) (vertically - opposite to \(\angle NPO\)), \(\angle RQP=\angle PON\) (if \(NO\parallel QR\) by alternate - interior angles) or we can calculate angles.
Since \(\angle NPO=\angle QPR\) (vertically - opposite angles) and \(\angle PNO=\angle PRQ\) (alternate - interior angles if \(NO\parallel QR\) or by the fact that \(\triangle PNO\) and \(\triangle PRQ\) have two pairs of equal angles.
\(\angle NPO=\angle QPR\) (vertically - opposite angles) and \(\angle PON=\angle PQR\) (alternate - interior angles if \(NO\parallel QR\)).
By the AA (angle - angle) similarity criterion, \(\triangle PNO\sim\triangle PRQ\)

Answer:

\(\triangle PNO\) is similar to \(\triangle PRQ\)