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
- ∠r
- ∠s
(a)
∠LUP = 180° - ∠r (Interior angles, UL//TN)
∠RPU = 180° - ∠r (Angles on a straight line)
∠RUP = 180° - ∠r (Isosceles triangle)
48° + 180° - ∠r + 180° - ∠r + ∠s + 21° = 180° (Angles on a straight line, KT)
48° + 180° + 180° + 21° - ∠r - ∠r + ∠s = 180°
429° - 2∠r + ∠s = 180°
2∠r - ∠s = 429° - 180°
2∠r - ∠s = 249°
∠s = 2∠r - 249° --- (1)
∠URS
= ∠USR
= 2 x (180° - ∠r)
= 360° - 2∠r (Exterior angle of a triangle)
∠s = 180° - (360° - 2∠r) - (360° - 2∠r)
∠s = 180° - 360° + 2∠r - 360° + 2∠r
∠s = 180° - 360° - 360° + 2∠r + 2∠r
∠s = 4∠r - 540° (Angles sum of triangle)
∠s = 4∠r - 540° --- (2)
(2) = (1)
4∠r - 540° = 2∠r - 249°
4∠r - 2∠r= 540° - 249°
2∠r = 291°
∠r
= 291° ÷ 2
= 145.5°
(b)
From (1)
∠s
= 2∠r - 249°
= 291° - 249°
= 42°
Answer(s): (a) 145.5°; (b) 42°