Question

are these triangles similar?

Explanation:

Step1: Find the third angle of triangle \(KIJ\)

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle of \(\triangle KIJ\) be \(x\). Then \(x + 93^{\circ}+32^{\circ}=180^{\circ}\), so \(x = 180^{\circ}-(93^{\circ}+32^{\circ})=55^{\circ}\).

Step2: Find the third angle of triangle \(TRS\)

Let the third angle of \(\triangle TRS\) be \(y\). Then \(y + 32^{\circ}+93^{\circ}=180^{\circ}\), so \(y = 180^{\circ}-(93^{\circ}+32^{\circ})=55^{\circ}\).

Step3: Check the AA (Angle - Angle) similarity criterion

In \(\triangle KIJ\) and \(\triangle TRS\), \(\angle I = \angle R=93^{\circ}\) and \(\angle J=\angle T = 32^{\circ}\). By the AA similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), the two triangles are similar.

Answer:

YES.