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.)

Explanation:

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

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle MNO\), let the third angle be \(x\). Then \(x + 53^{\circ}+45^{\circ}=180^{\circ}\), so \(x = 180-(53 + 45)=82^{\circ}\).

Step2: Calculate the third angle of triangle \(PQR\)

For \(\triangle PQR\), let the third angle be \(y\). Assume we use the Side - Angle - Side (SAS) similarity criterion. Check the ratio of sides. \(\frac{O N}{R Q}=\frac{15}{30}=\frac{1}{2}\), \(\frac{O M}{R P}=\frac{21}{42}=\frac{1}{2}\), and \(\angle O=\angle R = 53^{\circ}\)

Answer:

The triangles are similar because the ratio of two pairs of corresponding sides is equal (\(\frac{ON}{RQ}=\frac{OM}{RP}=\frac{1}{2}\)) and the included angles (\(\angle O\) and \(\angle R\)) are equal (SAS similarity criterion).