If qs bisects pat, sqt= (8x-25), pqt=(9x+34) and sqr= 112 find each measure

Angles provided: PQS, SQT, TQR, PQR

For convenience, I'll determine the angles in this sequence:

1. SQT

2. PQS

3. TQR

4. PQR

If QS bisects angle PQT, then:

m∠PQT = m∠SQT + m∠PQS

Since QS bisects PQT, the two smaller angles are equal:

m∠SQT = m∠PQS

Hence, m∠PQT = 2 × m∠SQT = 2 × m∠PQS

1. Calculate m∠SQT

Given:

m∠SQT = (8x - 25)

m∠PQT = (9x + 34)

Because m∠PQT = 2 × m∠SQT, set up the equation:

9x + 34 = 2(8x - 25)

9x + 34 = 16x - 50

Add 50 to both sides:

9x + 84 = 16x

Subtract 9x:

84 = 7x

Divide by 7:

x = 12

Calculate m∠SQT:

m∠SQT = 8(12) - 25 = 96 - 25 = 71

2. Calculate m∠PQS

Because m∠SQT equals m∠PQS,

m∠PQS = 71

3. Calculate m∠TQR

Since m∠SQR = m∠SQT + m∠TQR,

m∠TQR = m∠SQR - m∠SQT

Given m∠SQR = 112 and m∠SQT = 71,

m∠TQR = 112 - 71 = 41

4. Calculate m∠PQR

The sum of angles around point Q is 360°, so:

m∠SQT + m∠PQS + m∠TQR + m∠PQR = 360

71 + 71 + 41 + m∠PQR = 360

183 + m∠PQR = 360

Subtract 183:

m∠PQR = 177

To summarize the results:

1. m∠SQT = 71°

2. m∠PQS = 71°

3. m∠TQR = 41°

4. m∠PQR = 177°