As it is written this question isn't very clear, and I'm not positive that the entire thing can be answered. I don't know where point R is. Is there a picture that goes with this problem? If so, maybe that would make it more clear. What I can do is show you how to solve for x. While I don't believe this is the final answer, I believe it is a step toward the right answer. We know what ray QS bisects angle PQT. If you draw out angle PQT and ray QS you can see that angle PQS is congruent to angle SQT. Because of the definition of angle bisector. This also means that angle SQT is half of PQT. Because of this we know that we can set the measure of PQT equal to 2 times the measure of SQT. So we can set up and solve the following equation as such:
2(8x-25) = 9x +34
16x - 50 = 9x +34
7x -50 = 34
7x = 84
x = 12
Again I this doesn't seem to be your final answer. To find the answer the measures of each angle you're going to want to plug 12 in for x for one of the angles. So I'll do it for angle SQT.
8(12) - 25
96 -25
71
So the measure of angle SQT should be 71. Therefore since QS bisects PQR, the measure of angle PQS should also be 71. The measure of PQT should be double that which makes it 142. I'm not positive that this is all the problem is looking for, but I think that's most of it. Hope this helps!
