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
- ∠a
- ∠b
(a)
∠LUP = 180° - ∠a (Interior angles, UL//TN)
∠RPU = 180° - ∠a (Angles on a straight line)
∠RUP = 180° - ∠a (Isosceles triangle)
54° + 180° - ∠a + 180° - ∠a + ∠b + 22° = 180° (Angles on a straight line, KT)
54° + 180° + 180° + 22° - ∠a - ∠a + ∠b = 180°
436° - 2∠a + ∠b = 180°
2∠a - ∠b = 436° - 180°
2∠a - ∠b = 256°
∠b = 2∠a - 256° --- (1)
∠URS
= ∠USR
= 2 x (180° - ∠a)
= 360° - 2∠a (Exterior angle of a triangle)
∠b = 180° - (360° - 2∠a) - (360° - 2∠a)
∠b = 180° - 360° + 2∠a - 360° + 2∠a
∠b = 180° - 360° - 360° + 2∠a + 2∠a
∠b = 4∠a - 540° (Angles sum of triangle)
∠b = 4∠a - 540° --- (2)
(2) = (1)
4∠a - 540° = 2∠a - 256°
4∠a - 2∠a= 540° - 256°
2∠a = 284°
∠a
= 284° ÷ 2
= 142°
(b)
From (1)
∠b
= 2∠a - 256°
= 284° - 256°
= 28°
Answer(s): (a) 142°; (b) 28°