Question

1. find the value of x if qs bisects ∠pqr and m∠pqr = 82°. (10x + 1)°

Explanation:

Step1: Recall angle - bisector property

If a ray bisects an angle, it divides the angle into two equal parts. So, if $\overrightarrow{QS}$ bisects $\angle PQR$, then $m\angle PQS=m\angle SQR$ and $m\angle PQS=\frac{1}{2}m\angle PQR$.
Since $m\angle PQR = 82^{\circ}$, then $m\angle PQS=\frac{82^{\circ}}{2}=41^{\circ}$.

Step2: Set up an equation

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

Step3: Solve the equation for x

Subtract 1 from both sides of the equation: $10x+1 - 1=41 - 1$, which simplifies to $10x=40$.
Then divide both sides by 10: $\frac{10x}{10}=\frac{40}{10}$, so $x = 4$.

Answer:

$x = 4$