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
- ∠c
- ∠d
(a)
∠NVR = 180° - ∠c (Interior angles, VN//UP)
∠SRV = 180° - ∠c (Angles on a straight line)
∠SVR = 180° - ∠c (Isosceles triangle)
50° + 180° - ∠c + 180° - ∠c + ∠d + 20° = 180° (Angles on a straight line, LU)
50° + 180° + 180° + 20° - ∠c - ∠c + ∠d = 180°
430° - 2∠c + ∠d = 180°
2∠c - ∠d = 430° - 180°
2∠c - ∠d = 250°
∠d = 2∠c - 250° --- (1)
∠VST
= ∠VTS
= 2 x (180° - ∠c)
= 360° - 2∠c (Exterior angle of a triangle)
∠d = 180° - (360° - 2∠c) - (360° - 2∠c)
∠d = 180° - 360° + 2∠c - 360° + 2∠c
∠d = 180° - 360° - 360° + 2∠c + 2∠c
∠d = 4∠c - 540° (Angles sum of triangle)
∠d = 4∠c - 540° --- (2)
(2) = (1)
4∠c - 540° = 2∠c - 250°
4∠c - 2∠c= 540° - 250°
2∠c = 290°
∠c
= 290° ÷ 2
= 145°
(b)
From (1)
∠d
= 2∠c - 250°
= 290° - 250°
= 40°
Answer(s): (a) 145°; (b) 40°