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