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
- ∠w
- ∠x
(a)
∠RYT = 180° - ∠w (Interior angles, YR//XS)
∠UTY = 180° - ∠w (Angles on a straight line)
∠UYT = 180° - ∠w (Isosceles triangle)
53° + 180° - ∠w + 180° - ∠w + ∠x + 18° = 180° (Angles on a straight line, PX)
53° + 180° + 180° + 18° - ∠w - ∠w + ∠x = 180°
431° - 2∠w + ∠x = 180°
2∠w - ∠x = 431° - 180°
2∠w - ∠x = 251°
∠x = 2∠w - 251° --- (1)
∠YUV
= ∠YVU
= 2 x (180° - ∠w)
= 360° - 2∠w (Exterior angle of a triangle)
∠x = 180° - (360° - 2∠w) - (360° - 2∠w)
∠x = 180° - 360° + 2∠w - 360° + 2∠w
∠x = 180° - 360° - 360° + 2∠w + 2∠w
∠x = 4∠w - 540° (Angles sum of triangle)
∠x = 4∠w - 540° --- (2)
(2) = (1)
4∠w - 540° = 2∠w - 251°
4∠w - 2∠w= 540° - 251°
2∠w = 289°
∠w
= 289° ÷ 2
= 144.5°
(b)
From (1)
∠x
= 2∠w - 251°
= 289° - 251°
= 38°
Answer(s): (a) 144.5°; (b) 38°