Question

1. find the value of x.
2. find the value of x if $overrightarrow{qs}$ bisects $angle pqr$ and $mangle pqr = 82^{circ}$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{QS}$ bisects $\angle PQR$, then $m\angle PQS=m\angle SQR$ and $m\angle PQR = 2m\angle PQS$. Given $m\angle PQR = 82^{\circ}$, so $m\angle PQS=\frac{m\angle PQR}{2}=\frac{82^{\circ}}{2} = 41^{\circ}$.

Step2: Set up equation

We know that $m\angle PQS=(10x + 1)^{\circ}$. Set up the equation $10x+1 = 41$.

Step3: Solve the equation

Subtract 1 from both sides: $10x=41 - 1=40$. Then divide both sides by 10: $x=\frac{40}{10}=4$.

Answer:

$x = 4$