Question

triangle pqr has two known interior angles of 24° and 100°. triangle rst has two known interior angles of 24° and 56°. what can be determined about whether triangles pqr and rst are similar? all interior angles must be given to determine similarity. similarity cannot be determined from the given information. the triangles are similar. the triangles are not similar.

Explanation:

Step1: Calculate the third angle of triangle PQR

The sum of interior angles of a triangle is \(180^{\circ}\). Let the third angle of \(\triangle PQR\) be \(x\). Then \(x = 180^{\circ}-(24^{\circ}+ 100^{\circ})=56^{\circ}\)

Step2: Calculate the third angle of triangle RST

Let the third angle of \(\triangle RST\) be \(y\). Then \(y=180^{\circ}-(24^{\circ}+56^{\circ}) = 100^{\circ}\)

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

In \(\triangle PQR\) and \(\triangle RST\), we have \(\angle PQR = 24^{\circ}=\angle RST\), \(\angle QPR=100^{\circ}=\angle STR\), \(\angle PRQ = 56^{\circ}=\angle SRT\). By AA similarity criterion (if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar), \(\triangle PQR\sim\triangle RST\)

Answer:

The triangles are similar.