Question

1. if qs bisects ∠pqt, m∠sqt=(8x - 25)°, m∠pqt=(10x + 34)°, and m∠sqr = 122°, find each measure.

x =
m∠pqs=
m∠pqt=
m∠tqr=

Explanation:

Step1: Use angle - bisector property

Since $QS$ bisects $\angle PQT$, we know that $m\angle SQT=\frac{1}{2}m\angle PQT$. So, $8x - 25=\frac{1}{2}(10x + 34)$.

Step2: Solve the equation for $x$

First, multiply both sides of the equation $8x - 25=\frac{1}{2}(10x + 34)$ by 2 to get $2(8x - 25)=10x + 34$.
Expand the left - hand side: $16x-50 = 10x + 34$.
Subtract $10x$ from both sides: $16x-10x-50=10x - 10x+34$, which simplifies to $6x-50 = 34$.
Add 50 to both sides: $6x-50 + 50=34 + 50$, so $6x=84$.
Divide both sides by 6: $x = 14$.

Step3: Find $m\angle PQS$

Since $m\angle PQS=m\angle SQT$ and $m\angle SQT=8x - 25$, substitute $x = 14$ into the formula. $m\angle PQS=8\times14 - 25=112 - 25 = 87^{\circ}$.

Step4: Find $m\angle PQT$

Substitute $x = 14$ into the formula $m\angle PQT=10x + 34$. $m\angle PQT=10\times14+34=140 + 34=174^{\circ}$.

Step5: Find $m\angle TQR$

We know that $m\angle SQR = 122^{\circ}$ and $m\angle SQT = 87^{\circ}$. So, $m\angle TQR=m\angle SQR - m\angle SQT=122-87 = 35^{\circ}$.

Answer:

$x = 14$
$m\angle PQS=87^{\circ}$
$m\angle PQT=174^{\circ}$
$m\angle TQR=35^{\circ}$