Answer:
Remote Angles: ∠STU and ∠UST
Exterior Angle: ∠TUA
x = 11
m∠STU = 65°
m∠TUA = 123°
Step-by-step explanation:
An exterior angle of a triangle, is an angle formed by extending one side of the triangle beyond the vertex. So in this case, the exterior angle is ∠TUA.
The remote angles are the two angles of the triangle that are not adjacent to a specific exterior angle. So in this case, the remote angles are ∠STU and ∠UST.
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Therefore:
m∠TUA = m∠STU + m∠UST
Given that:
- m∠TUA = (11x + 2)°
- m∠STU = (5x + 10)°
- m∠UST = 58°
Substitute the expressions for each angle into the sum equation and solve for x:
m∠TUA = m∠STU + m∠UST
(11x + 2)° = (5x + 10)° +58°
11x + 2 = 5x + 10 + 58
11x + 2 = 5x + 68
6x + 2 = 68
6x = 66
x = 11
Therefore, the value of x is x = 11.
To find the measure of angle STU, substitute x = 11 into the angle expression:
m∠STU = (5(11) + 10)°
m∠STU = (55 + 10)°
m∠STU = 65°
To find the measure of angle TUA, substitute x = 11 into the angle expression:
m∠TUA = (11(11) + 2)°
m∠TUA = (121 + 2)°
m∠TUA = 123°
