Question

if $overrightarrow{qs}$ bisects $angle pqr$, $mangle pqs=(7x - 6)^{circ}$, and $mangle sqr=(4x + 15)^{circ}$, find $mangle pqt$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{QS}$ bisects $\angle PQR$, then $m\angle PQS=m\angle SQR$. So we set up the equation $7x - 6=4x + 15$.

Step2: Solve the equation for $x$

Subtract $4x$ from both sides: $7x-4x - 6=4x-4x + 15$, which simplifies to $3x-6 = 15$. Then add 6 to both sides: $3x-6 + 6=15 + 6$, getting $3x=21$. Divide both sides by 3: $x=\frac{21}{3}=7$.

Step3: Find $m\angle PQS$

Substitute $x = 7$ into the expression for $m\angle PQS$: $m\angle PQS=7x-6=7\times7 - 6=49 - 6=43^{\circ}$.

Step4: Find $m\angle PQR$

Since $m\angle PQR=m\angle PQS + m\angle SQR$ and $m\angle PQS=m\angle SQR$, then $m\angle PQR = 2m\angle PQS$. So $m\angle PQR=2\times43^{\circ}=86^{\circ}$. But it seems there is a mis - label in the problem as we are asked for $m\angle PQT$ which is not related to the angle - bisector information given in a clear way. Assuming you meant $m\angle PQR$, we have found it. If we go with the steps based on the angle - bisector info we have:
We know $m\angle PQS = 43^{\circ}$ and $m\angle SQR=43^{\circ}$, so $m\angle PQR=86^{\circ}$

Answer:

There is likely a mis - label in the problem. If we assume you want $m\angle PQR$ based on the angle - bisector work, the answer is $86^{\circ}$. If we consider the values of $m\angle PQS$ which we found as part of the process, it is $43^{\circ}$. But as the problem asks for $m\angle PQT$ which is not clearly related to the given angle - bisector info, if we go by the closest related angle calculations from the given data about the bisected angle, the most relevant value we calculated is $m\angle PQS = 43^{\circ}$ among the options provided. So the answer is $43$.