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. ∠f
2. ∠g

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

49° + 180° - ∠f + 180° - ∠f + ∠g + 23° = 180° (Angles on a straight line, KT)
49° + 180° + 180° + 23° - ∠f - ∠f + ∠g  = 180°
432° - 2∠f + ∠g = 180°
2∠f - ∠g = 432° - 180° 
2∠f - ∠g = 252°

∠g = 2∠f - 252° --- (1)

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

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

∠g = 4∠f - 540° --- (2)

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

∠f
= 288° ÷ 2
= 144°

(b)
From (1)
∠g
=  2∠f - 252°
= 288° - 252° 
= 36°

Answer(s): (a) 144°; (b) 36°