In the figure, SHKP is a trapezium and triangles SPN and NSL are isosceles triangles. RG, RK and GK are straight lines. SP = SN = NL. Find

1. ∠v
2. ∠w

(a)
∠HSL = 180° - ∠v (Interior angles, SH//RK)
∠NLS = 180° - ∠v (Angles on a straight line)
∠NSL = 180° - ∠v (Isosceles triangle)

48° + 180° - ∠v + 180° - ∠v + ∠w + 19° = 180° (Angles on a straight line, GR)
48° + 180° + 180° + 19° - ∠v - ∠v + ∠w  = 180°
427° - 2∠v + ∠w = 180°
2∠v - ∠w = 427° - 180° 
2∠v - ∠w = 247°

∠w = 2∠v - 247° --- (1)

∠SNP
= ∠SPN
= 2 x (180° - ∠v)
= 360° - 2∠v (Exterior angle of a triangle)

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

∠w = 4∠v - 540° --- (2)

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

∠v
= 293° ÷ 2
= 146.5°

(b)
From (1)
∠w
=  2∠v - 247°
= 293° - 247° 
= 46°

Answer(s): (a) 146.5°; (b) 46°