In the figure, YRSV is a trapezium and triangles YVU and UYT are isosceles triangles. XP, XS and PS are straight lines. YV = YU = UT. Find
- ∠r
- ∠s
(a)
∠RYT = 180° - ∠r (Interior angles, YR//XS)
∠UTY = 180° - ∠r (Angles on a straight line)
∠UYT = 180° - ∠r (Isosceles triangle)
52° + 180° - ∠r + 180° - ∠r + ∠s + 24° = 180° (Angles on a straight line, PX)
52° + 180° + 180° + 24° - ∠r - ∠r + ∠s = 180°
436° - 2∠r + ∠s = 180°
2∠r - ∠s = 436° - 180°
2∠r - ∠s = 256°
∠s = 2∠r - 256° --- (1)
∠YUV
= ∠YVU
= 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 - 256°
4∠r - 2∠r= 540° - 256°
2∠r = 284°
∠r
= 284° ÷ 2
= 142°
(b)
From (1)
∠s
= 2∠r - 256°
= 284° - 256°
= 28°
Answer(s): (a) 142°; (b) 28°