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
- ∠d
- ∠e
(a)
∠LUP = 180° - ∠d (Interior angles, UL//TN)
∠RPU = 180° - ∠d (Angles on a straight line)
∠RUP = 180° - ∠d (Isosceles triangle)
51° + 180° - ∠d + 180° - ∠d + ∠e + 23° = 180° (Angles on a straight line, KT)
51° + 180° + 180° + 23° - ∠d - ∠d + ∠e = 180°
434° - 2∠d + ∠e = 180°
2∠d - ∠e = 434° - 180°
2∠d - ∠e = 254°
∠e = 2∠d - 254° --- (1)
∠URS
= ∠USR
= 2 x (180° - ∠d)
= 360° - 2∠d (Exterior angle of a triangle)
∠e = 180° - (360° - 2∠d) - (360° - 2∠d)
∠e = 180° - 360° + 2∠d - 360° + 2∠d
∠e = 180° - 360° - 360° + 2∠d + 2∠d
∠e = 4∠d - 540° (Angles sum of triangle)
∠e = 4∠d - 540° --- (2)
(2) = (1)
4∠d - 540° = 2∠d - 254°
4∠d - 2∠d= 540° - 254°
2∠d = 286°
∠d
= 286° ÷ 2
= 143°
(b)
From (1)
∠e
= 2∠d - 254°
= 286° - 254°
= 32°
Answer(s): (a) 143°; (b) 32°