Question

1. if \\( \triangle p q r \cong \triangle s t u \\), find the measure of each part.

\\( s t = \\)
\\( s u = \\)
\\( m \angle r = \\)
\\( m \angle q = \\)

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle PQR\cong\triangle STU\), corresponding sides and angles are equal.

• Corresponding sides: \(PQ = ST\), \(PR=SU\), \(QR = TU\)
• Corresponding angles: \(\angle P=\angle S\), \(\angle Q=\angle U\), \(\angle R=\angle T\)

Step2: Find the lengths of sides

Given \(PQ = 7m\), so \(ST=7m\) (because \(PQ\) and \(ST\) are corresponding sides). Given \(PR = 8m\), so \(SU = 8m\) (because \(PR\) and \(SU\) are corresponding sides)

Step3: Find the measure of angles

First, find \(\angle Q\) in \(\triangle PQR\). Using the angle - sum property of a triangle (\(\angle P+\angle Q+\angle R=180^{\circ}\)). We know \(\angle P = 59^{\circ}\), \(\angle T=\angle R = 75^{\circ}\) (since \(\angle R\) and \(\angle T\) are corresponding angles). Then \(\angle Q=180^{\circ}-(59^{\circ}+75^{\circ})=46^{\circ}\)

Answer:

\(ST = 7m\), \(SU=8m\), \(m\angle R = 75^{\circ}\), \(m\angle Q=46^{\circ}\)