In the figure, ULNS is a trapezium and triangles USR and RUP are isosceles triangles. TK, TN and KN are straight lines. US = UR = RP. Find

1. ∠q
2. ∠r

(a)
∠LUP = 180° - ∠q (Interior angles, UL//TN)
∠RPU = 180° - ∠q (Angles on a straight line)
∠RUP = 180° - ∠q (Isosceles triangle)

50° + 180° - ∠q + 180° - ∠q + ∠r + 23° = 180° (Angles on a straight line, KT)
50° + 180° + 180° + 23° - ∠q - ∠q + ∠r  = 180°
433° - 2∠q + ∠r = 180°
2∠q - ∠r = 433° - 180° 
2∠q - ∠r = 253°

∠r = 2∠q - 253° --- (1)

∠URS
= ∠USR
= 2 x (180° - ∠q)
= 360° - 2∠q (Exterior angle of a triangle)

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

∠r = 4∠q - 540° --- (2)

(2) = (1)
4∠q - 540° = 2∠q - 253°
4∠q - 2∠q= 540° - 253°
2∠q = 287°

∠q
= 287° ÷ 2
= 143.5°

(b)
From (1)
∠r
=  2∠q - 253°
= 287° - 253° 
= 34°

Answer(s): (a) 143.5°; (b) 34°