Question

1. (mangle pqr, mangle tqs, mangle pqt, mangle rqs)

2x + 65
7x - 110

Explanation:

Step1: Set up equation

Vertical - angles are equal. So, $2x + 65=7x - 110$.

Step2: Solve for x

Subtract $2x$ from both sides: $65 = 5x-110$. Then add 110 to both sides: $175 = 5x$. Divide both sides by 5, we get $x = 35$.

Step3: Find $m\angle TQS$ and $m\angle PQR$

Since $m\angle TQS=2x + 65$, substitute $x = 35$: $m\angle TQS=2\times35 + 65=70 + 65 = 135^{\circ}$. And $m\angle PQR$ is vertical - angle to $m\angle TQS$, so $m\angle PQR = 135^{\circ}$.

Step4: Find $m\angle PQT$ and $m\angle RQS$

$m\angle PQT$ and $m\angle TQS$ are supplementary. So $m\angle PQT=180 - 135=45^{\circ}$. $m\angle RQS$ is vertical - angle to $m\angle PQT$, so $m\angle RQS = 45^{\circ}$.

Answer:

$m\angle PQR = 135^{\circ}$, $m\angle TQS = 135^{\circ}$, $m\angle PQT = 45^{\circ}$, $m\angle RQS = 45^{\circ}$