In the figure, VNPT is a trapezium and triangles VTS and SVR are isosceles triangles. UL, UP and LP are straight lines. VT = VS = SR. Find
- ∠v
- ∠w
(a)
∠NVR = 180° - ∠v (Interior angles, VN//UP)
∠SRV = 180° - ∠v (Angles on a straight line)
∠SVR = 180° - ∠v (Isosceles triangle)
57° + 180° - ∠v + 180° - ∠v + ∠w + 20° = 180° (Angles on a straight line, LU)
57° + 180° + 180° + 20° - ∠v - ∠v + ∠w = 180°
437° - 2∠v + ∠w = 180°
2∠v - ∠w = 437° - 180°
2∠v - ∠w = 257°
∠w = 2∠v - 257° --- (1)
∠VST
= ∠VTS
= 2 x (180° - ∠v)
= 360° - 2∠v (Exterior angle of a triangle)
∠w = 180° - (360° - 2∠v) - (360° - 2∠v)
∠w = 180° - 360° + 2∠v - 360° + 2∠v
∠w = 180° - 360° - 360° + 2∠v + 2∠v
∠w = 4∠v - 540° (Angles sum of triangle)
∠w = 4∠v - 540° --- (2)
(2) = (1)
4∠v - 540° = 2∠v - 257°
4∠v - 2∠v= 540° - 257°
2∠v = 283°
∠v
= 283° ÷ 2
= 141.5°
(b)
From (1)
∠w
= 2∠v - 257°
= 283° - 257°
= 26°
Answer(s): (a) 141.5°; (b) 26°