Question

1. find the value of x if \\( \overline{qs} \\) bisects \\( \angle pqr \\) and \\( m\angle pqr = 82^{\circ} \\).

Explanation:

Step1: Use the angle - bisector property

Since \( \overline{QS} \) bisects \( \angle PQR \), then \( m\angle PQS=m\angle SQR \) and \( m\angle PQR = 2m\angle PQS \).
Given \( m\angle PQR=82^{\circ} \), so \( 2(10x + 1)=82 \).

Step2: Solve the equation for \( x \)

First, divide both sides of the equation \( 2(10x + 1)=82 \) by 2:
\( 10x+1=\frac{82}{2}=41 \).
Then, subtract 1 from both sides: \( 10x=41 - 1=40 \).
Finally, divide both sides by 10: \( x=\frac{40}{10}=4 \).

Answer:

\( x = 4 \)