Question

the two triangles below are similar. also, ( mangle b = 15^{circ} ) and ( mangle c = 35^{circ} ) as shown below. find ( mangle p ), ( mangle q ), and ( mangle r ). assume the triangles are accurately drawn.

Explanation:

Step1: Find \(m\angle A\)

In \(\triangle ABC\), use the triangle - angle - sum theorem (\(m\angle A+m\angle B + m\angle C=180^{\circ}\)).
Substitute \(m\angle B = 15^{\circ}\) and \(m\angle C = 35^{\circ}\):
\(m\angle A=180^{\circ}-(15^{\circ}+35^{\circ})=180^{\circ}-50^{\circ}=130^{\circ}\)

Step2: Use the property of similar triangles

Since \(\triangle ABC\sim\triangle PQR\), corresponding angles are equal.
\(\angle A\) corresponds to \(\angle P\), \(\angle B\) corresponds to \(\angle Q\), and \(\angle C\) corresponds to \(\angle R\).
So \(m\angle P=m\angle A = 130^{\circ}\), \(m\angle Q=m\angle B = 15^{\circ}\), \(m\angle R=m\angle C = 35^{\circ}\)

Answer:

\(m\angle P = 130^{\circ}\), \(m\angle Q = 15^{\circ}\), \(m\angle R = 35^{\circ}\)