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. ∠m
2. ∠n

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

57° + 180° - ∠m + 180° - ∠m + ∠n + 15° = 180° (Angles on a straight line, PX)
57° + 180° + 180° + 15° - ∠m - ∠m + ∠n  = 180°
432° - 2∠m + ∠n = 180°
2∠m - ∠n = 432° - 180° 
2∠m - ∠n = 252°

∠n = 2∠m - 252° --- (1)

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

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

∠n = 4∠m - 540° --- (2)

(2) = (1)
4∠m - 540° = 2∠m - 252°
4∠m - 2∠m= 540° - 252°
2∠m = 288°

∠m
= 288° ÷ 2
= 144°

(b)
From (1)
∠n
=  2∠m - 252°
= 288° - 252° 
= 36°

Answer(s): (a) 144°; (b) 36°