Answers:
- angle STU = 65 degrees
- angle TUA = 123 degrees
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Step-by-step explanation:
Let's find the expression for angle TUS in terms of x
Angle TUA has the expression 11x+2. It will add to angle TUS to get 180 degrees
(angle TUS) + (angle TUA) = 180
angle TUS = 180 - (angle TUA)
angle TUS = 180 - (11x + 2)
angle TUS = 180 - 11x - 2
angle TUS = -11x + 178
Now focus on the interior angles of triangle TUS. We have these three angles
- T = 5x+10
- U = -11x+178
- S = 58
For any triangle, the interior angles always add to 180, so,
T+U+S = 180
(5x+10) + (-11x+178) + (58) = 180
(5x-11x) + (10+178+58) = 180
-6x + 246 = 180
-6x = 180-246
-6x = -66
x = -66/(-6)
x = 11
Use this x value to find the angle measures we're after:
- angle STU = 5x+10 = 5*11+10 = 55+10 = 65 degrees
- angle TUA = 11x+2 = 11*11+2 = 121+2 = 123 degrees
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As an alternative route, you can apply the remote interior angle theorem. This says that adding two interior angles leads to the exterior angle that isn't adjacent to either one.
In this case, we would say
(angle STU) + (angle TSU) = angle TUA
that leads to the equation
(5x+10) + (58) = 11x+2
Solving this should lead to x = 11 to generate the angles mentioned earlier.