Question

determine if triangle opq and triangle rst 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 \(OPQ\)

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle of \(\triangle OPQ\) be \(x\). Then \(x + 49^{\circ}+82^{\circ}=180^{\circ}\). So \(x=180^{\circ}-(49^{\circ} + 82^{\circ})=49^{\circ}\)

Step2: Compare the angles of the two triangles

In \(\triangle OPQ\), the angles are \(49^{\circ},49^{\circ},82^{\circ}\). In \(\triangle RST\), the angles are \(49^{\circ},49^{\circ}\). Let the third angle of \(\triangle RST\) be \(y\). Then \(y+49^{\circ}+49^{\circ}=180^{\circ}\), so \(y = 82^{\circ}\)

Since the three angles of \(\triangle OPQ\) are equal to the three angles of \(\triangle RST\) (by 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), the two triangles are similar.

Answer:

The triangles \(OPQ\) and \(RST\) are similar.