To solve this problem, we need to apply some angles theorems and definitions.
We know that angle MTN is 90°, by given.
Angle STM = 2x+8, by given.
Angle STM = Angle OTP, by vertical angles. Remember that vertical angles only have the vertex in common, and they are always equal.
So, by substitution, we have that:
Angle OTP = 2x+8
On the other hand, we know that the maximum angle around vertex T is 360°, which is form by the sum of all angles we know so far:
STM + 90 + NTP + OTP + STQ = 360°
Now, we substitute all given data:
2x + 8 + 90 + 71 - x + 2x + 8 + STQ = 360°
Then, we reduce like terms:
3x + 177 + STQ = 360°
STQ = 360 - 177 - 3x = 183 - 3x
As you can observe, we need to know the value of the variable x to find the angle STQ.
Angles STM, MTN, NTP are supplementary, by definition, that means they are equal to 180°:
2x + 8 + 90 + 71 - x = 180
We solve the equation for x:
x = 180 - 169 = 11
Now, we use the value of the variable to find the angle STQ:
STQ = 183 - 3x = 183 - 3(11) = 183 - 33 = 150°
Therefore, angle STQ is equal to 150°.