Question

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

\\( x= \\)
\\( m \angle pqt= \\)
\\( m \angle pqs= \\)
\\( m \angle tqr= \\)
7
\\( m \angle abd= \\)
\\( m \angle abc= \\)

Explanation:

Step1: Use the angle - bisector property

Since \( \overline{QS} \) bisects \( \angle PQT \), we have \( m\angle SQT=\frac{1}{2}m\angle PQT \).
Given \( m\angle SQT=(8x - 25)^{\circ} \) and \( m\angle PQT=(9x + 34)^{\circ} \), then \( 8x-25=\frac{1}{2}(9x + 34) \).

Step2: Solve the equation for \( x \)

Multiply both sides of the equation \( 8x-25=\frac{1}{2}(9x + 34) \) by 2 to get \( 16x-50 = 9x+34 \).
Subtract \( 9x \) from both sides: \( 16x-9x-50=9x - 9x+34 \), which simplifies to \( 7x-50 = 34 \).
Add 50 to both sides: \( 7x-50 + 50=34 + 50 \), so \( 7x=84 \).
Divide both sides by 7: \( x=\frac{84}{7}=12 \).

Step3: Find \( m\angle PQT \)

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

Step4: Find \( m\angle PQS \)

Since \( \angle PQS+\angle SQR = 180^{\circ} \) (linear - pair of angles), and we know \( m\angle SQR = 112^{\circ} \).
Then \( m\angle PQS=180^{\circ}-m\angle SQR \).
\( m\angle PQS=180 - 112=68^{\circ} \).

Step5: Find \( m\angle TQR \)

We know that \( m\angle PQT + m\angle TQR=180^{\circ} \) (linear - pair of angles).
Since \( m\angle PQT = 142^{\circ} \), then \( m\angle TQR=180 - 142=38^{\circ} \).

Answer:

\( x = 12\), \(m\angle PQT=142^{\circ}\), \(m\angle PQS = 68^{\circ}\), \(m\angle TQR=38^{\circ}\)