a) We were told that RS is congruent to ST. This means that triangle RST is an isosceles triangle. In an isosceles triangle, the base angles are equal. This means that
angle SRT = angle STR
Angle STR and angle STU are linear pairs. They lie on a straight line. The sum of the angles on a straight line is 180 degrees. This means that
angle STR + angle STU = 180
Given that angle STU = 8x, then
angle STR + 8x = 180
Angle STR = 180 - 8x
Given that angle SRT = 5x - 54, then
5x - 54 = 180 - 8x
5x + 8x = 180 + 54
13x = 234
x = 234/13
x = 18
Thus,
b) angle STU = 8 * 18 = 144 degrees
c) Recall, the sum of the angles in a triangle is 180 degrees. This means that
angle S + angle R + angle T = 180
angle T = angle R = 180 - 144 = 36
Thus,
36 + 36 + S = 180
72 + S = 180
S = 180 - 72
Angle S = 108 degrees