In the figure, YRSV is a trapezium and triangles YVU and UYT are isosceles triangles. XP, XS and PS are straight lines. YV = YU = UT. Find

1. ∠x
2. ∠y

(a)
∠RYT = 180° - ∠x (Interior angles, YR//XS)
∠UTY = 180° - ∠x (Angles on a straight line)
∠UYT = 180° - ∠x (Isosceles triangle)

52° + 180° - ∠x + 180° - ∠x + ∠y + 15° = 180° (Angles on a straight line, PX)
52° + 180° + 180° + 15° - ∠x - ∠x + ∠y  = 180°
427° - 2∠x + ∠y = 180°
2∠x - ∠y = 427° - 180° 
2∠x - ∠y = 247°

∠y = 2∠x - 247° --- (1)

∠YUV
= ∠YVU
= 2 x (180° - ∠x)
= 360° - 2∠x (Exterior angle of a triangle)

∠y = 180° - (360° - 2∠x) - (360° - 2∠x)
∠y = 180° - 360° + 2∠x - 360° + 2∠x 
∠y = 180° - 360° - 360° + 2∠x + 2∠x
∠y = 4∠x - 540° (Angles sum of triangle) 

∠y = 4∠x - 540° --- (2)

(2) = (1)
4∠x - 540° = 2∠x - 247°
4∠x - 2∠x= 540° - 247°
2∠x = 293°

∠x
= 293° ÷ 2
= 146.5°

(b)
From (1)
∠y
=  2∠x - 247°
= 293° - 247° 
= 46°

Answer(s): (a) 146.5°; (b) 46°