In the figure, XPRU is a trapezium and triangles XUT and TXS are isosceles triangles. VN, VR and NR are straight lines. XU = XT = TS. Find
- ∠s
- ∠t
(a)
∠PXS = 180° - ∠s (Interior angles, XP//VR)
∠TSX = 180° - ∠s (Angles on a straight line)
∠TXS = 180° - ∠s (Isosceles triangle)
57° + 180° - ∠s + 180° - ∠s + ∠t + 17° = 180° (Angles on a straight line, NV)
57° + 180° + 180° + 17° - ∠s - ∠s + ∠t = 180°
434° - 2∠s + ∠t = 180°
2∠s - ∠t = 434° - 180°
2∠s - ∠t = 254°
∠t = 2∠s - 254° --- (1)
∠XTU
= ∠XUT
= 2 x (180° - ∠s)
= 360° - 2∠s (Exterior angle of a triangle)
∠t = 180° - (360° - 2∠s) - (360° - 2∠s)
∠t = 180° - 360° + 2∠s - 360° + 2∠s
∠t = 180° - 360° - 360° + 2∠s + 2∠s
∠t = 4∠s - 540° (Angles sum of triangle)
∠t = 4∠s - 540° --- (2)
(2) = (1)
4∠s - 540° = 2∠s - 254°
4∠s - 2∠s= 540° - 254°
2∠s = 286°
∠s
= 286° ÷ 2
= 143°
(b)
From (1)
∠t
= 2∠s - 254°
= 286° - 254°
= 32°
Answer(s): (a) 143°; (b) 32°