LNP is an equilateral triangle, EFGH is a rectangle and NPJK is a trapezium. LNK and LPJ are straight lines. If ∠v = 83°, find

1. ∠NKJ
2. Sum of ∠r, ∠s, ∠t and ∠u.

(a)
∠LNP = 60° (Equilateral triangle)

∠NKJ = ∠LNP = 60° (Corresponding angles)

(b)
∠EFG = 90° (Right angle)

∠FEH = 90° (Right angle)

∠w + ∠x 
= 180° - 90°
= 90° (Angles sum of triangle)

∠y + ∠z
= 180° - 90°
= 90° (Angles sum of triangle)

∠r + ∠v + ∠w = 180° (Angles on a straight line)

∠s + ∠x = 180° (Angles on a straight line)

∠t + ∠y = 180° (Angles on a straight line)

∠u + ∠z = 180° (Angles on a straight line)

Sum of ∠r, ∠s, ∠t, ∠u and ∠v
= (∠r + ∠s + ∠t + ∠u + ∠v + ∠w + ∠x + ∠y + ∠z) - (∠w + ∠x  + ∠y + ∠z)
= (4 x 180°) - (2 x 90°)
= 720° - 180°
= 540°

Sum of ∠r, ∠s, ∠t and ∠u
= 540° - 83° 
= 457°

Answer(s): (a) 60°; (b) 457°