Question

1. if $overrightarrow{qs}$ bisects $angle pqt$, $mangle sqt=(8x - 25)^{circ}$, $mangle pqt=(9x + 34)^{circ}$, and $mangle sqr = 112^{circ}$, find each measure.

$x=$____
$mangle pqs=$____
$mangle pqt=$____
$mangle tqr=$____

Explanation:

Step1: Use angle - bisector property

Since $QS$ bisects $\angle PQT$, then $m\angle SQT=m\angle PQS$ and $m\angle PQT = 2m\angle SQT$. So $9x + 34=2(8x - 25)$.

Step2: Solve the equation for $x$

$9x+34 = 16x-50$, $16x - 9x=34 + 50$, $7x=84$, $x = 12$.

Step3: Find $m\angle PQS$

$m\angle PQS=m\angle SQT=8x - 25=8\times12 - 25=96 - 25 = 71^{\circ}$.

Step4: Find $m\angle PQT$

$m\angle PQT=9x + 34=9\times12+34=108 + 34=142^{\circ}$.

Step5: Find $m\angle TQR$

$m\angle TQR=m\angle SQR - m\angle SQT=112-71 = 41^{\circ}$.

Answer:

$x = 12$
$m\angle PQS = 71^{\circ}$
$m\angle PQT = 142^{\circ}$
$m\angle TQR = 41^{\circ}$