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