Question

if \\( \overline { p q } \cong \overline { p s } \\), \\( m \angle p r q = v + 50 ^ { \circ } \\), and \\( m \angle p r s = 6 v - 15 ^ { \circ } \\), what is the value of \\( v \\)?

Explanation:

Step1: Use the Angle Bisector Theorem

Since \( \overline{PQ}\cong\overline{PS}\) and \( \angle PQR = \angle PSR=90^{\circ}\), \(PR\) is the angle - bisector of \( \angle QRS\). So \(m\angle PRQ=m\angle PRS\).

Step2: Set up the equation

Set \(v + 50=6v-15\).

Step3: Solve the equation for \(v\)

Subtract \(v\) from both sides: \(50 = 5v-15\).
Add \(15\) to both sides: \(5v=50 + 15\), so \(5v=65\).
Divide both sides by \(5\): \(v=\frac{65}{5}=13\).

Answer:

\(13\)