Question

find the value of x if qs bisects ∠pqr and m∠pqr = 82°.
diagram: point q with rays qp, qs, qr; ∠pqs is (10x + 1)°
options:
○ x = 4
○ x = 5
○ x = 8
○ x = 8.1

Explanation:

Step1: Recall Angle Bisector Definition

An angle bisector divides an angle into two equal parts. So, \( m\angle PQS = m\angle SQR \), and \( m\angle PQR = m\angle PQS + m\angle SQR = 2 \times m\angle PQS \).

Step2: Set Up the Equation

Given \( m\angle PQR = 82^\circ \) and \( m\angle PQS=(10x + 1)^\circ \), we have \( 2(10x + 1)=82 \).

Step3: Solve for x

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

Answer:

\( x = 4 \) (corresponding to the option "x = 4")