Question

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

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

Explanation:

Step1: Use angle - bisector property

Since $\overline{QS}$ bisects $\angle PQT$, we know that $m\angle PQS=m\angle SQT$ and $m\angle PQT = 2m\angle SQT$. So, $9x + 34=2(8x - 25)$.

Step2: Solve the equation for $x$

Expand the right - hand side: $9x + 34=16x-50$.
Move the $x$ terms to one side and constants to the other side: $16x - 9x=34 + 50$.
$7x=84$, so $x = 12$.

Step3: Find $m\angle PQS$

Since $m\angle PQS=m\angle SQT$ and $m\angle SQT=(8x - 25)^{\circ}$, substitute $x = 12$ into the formula. $m\angle PQS=8\times12-25=96 - 25=71^{\circ}$.

Step4: Find $m\angle PQT$

Substitute $x = 12$ into the formula for $m\angle PQT=(9x + 34)^{\circ}$. $m\angle PQT=9\times12 + 34=108+34 = 142^{\circ}$.

Step5: Find $m\angle TQR$

We know that $m\angle SQR = 112^{\circ}$ and $m\angle SQT = 71^{\circ}$. Then $m\angle TQR=m\angle SQR - m\angle SQT$. So $m\angle TQR=112 - 71=41^{\circ}$.

Answer:

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