Question

1. if qs bisects ∠pqt, m∠sqt=(8x - 25)°, m∠pqt=(9x + 34)°, and m∠sqr = 112, find each measure.

Explanation:

Step1: Use angle - bisector property

Since $QS$ bisects $\angle PQT$, we know that $m\angle SQT=\frac{1}{2}m\angle PQT$. So, $2(8x - 25)=9x + 34$.

Step2: Expand the left - hand side

$16x-50 = 9x + 34$.

Step3: Move like terms to one side

$16x-9x=34 + 50$.

Step4: Simplify

$7x=84$.

Step5: Solve for $x$

$x = 12$.

Step6: Find $m\angle PQS$

Since $m\angle PQS=m\angle SQT$ and $m\angle SQT=8x - 25$, substituting $x = 12$, we get $m\angle PQS=8\times12-25=96 - 25=71^{\circ}$.

Step7: Find $m\angle PQT$

$m\angle PQT=9x + 34$, substituting $x = 12$, we get $m\angle PQT=9\times12+34=108 + 34=142^{\circ}$.

Step8: Find $m\angle TQR$

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

Answer:

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