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

Angles shown: PQS, SQT, TQR, PQR

For sake of ease, I’ll solve the angles in this order:

1. SQT

2. PQS

3. TQR

4. PQR

If Ray QS bisects angle PQT

Then, m∠PQT = m∠SQT + m∠PQS

And m∠SQT = m∠PQS

Therefore, m∠PQT = 2m∠SQT = 2m∠PQS

1. Find the measure of angle SQT

Given,

m∠SQT = (8x-25)

m∠ PQT= (9x+34)

Since m∠PQT = 2m∠SQT

9x + 34 = 2 (8x – 25)

9x + 34 = 16x – 50

Add 50 to both sides of the equation

9x + 34 + 50 = 16x – 50 + 50

9x + 84 = 16x

Subtract 9x from both sides of the equation

9x – 9x + 84 = 16x – 9x

84 = 7x

7x = 84

x = 84/7

x = 12

m∠SQT = (8x-25)

m∠SQT = (8*12) – 25

m∠SQT = 96 – 25

m∠SQT = 71

2. Find the measure of angle PQS

m∠SQT = m∠PQS

m∠SQT = 71

Therefore, m∠PQS = 71

3. Find the measure of angle TQR

m∠SQR = m∠SQT + m∠TQR

m∠TQR = m∠SQR – m∠SQT

Given,

m∠SQR=112

m∠SQT = 71

m∠TQR = 112 – 71

m∠TQR = 41

4. Find the measure of angle PQR

m∠SQT + m∠ PQS + m∠ TQR + m∠ PQR = 360

m∠SQT = 71

m∠PQS = 71

m∠TQR = 41

Therefore, 71 + 71 + 41 + m∠ PQR = 360

183 + m∠ PQR = 360

Subtract 183 from both sides of the equation

183 – 183 + m∠ PQR = 360 -183

m∠ PQR = 360 -183

m∠ PQR = 177

Conclusively, each measure is as stated below:

1. m∠SQT = 71

2. m∠PQS = 71

3. m∠TQR = 41

4. m∠PQR = 177