Two geometry theorems get a workout here:
1) The sum of the angles of any triangle is 180 degrees,
2) If two angles are supplementary, their sum is 180 degrees.
From this we know that ∠RST + ∠STR + ∠TRS = 180. in ΔRST.
Next, look at line QS with points Q, R, and S. A straight line measures 180 degrees. and any two angles created by the line are supplements.
So, ∠QRT + ∠TRS = 180
Since we have two things equal to 180, we can set them equal to one another through transitivity (if a = b and b = c, then a = c).
∠RST + ∠STR + ∠TRS = ∠QRT + ∠TRS
Now we put in values we know for x.
(9 + x) + 5x + ∠TRS = 9x + ∠TRS
∠TRS was not filled in, but that's okay. If we subtract it from both sides, it won't be there regardless of its measure. The rest of this problem plays out like algebra class.
9 + x + 5x = 9x
9 + 6x = 9x
9 = 3x
x = 3
So x = 3, and we can find our angle measures for all angles in the problem.
