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°
