In the figure, ULNS is a trapezium and triangles USR and RUP are isosceles triangles. TK, TN and KN are straight lines. US = UR = RP. Find
- ∠h
- ∠i
(a)
∠LUP = 180° - ∠h (Interior angles, UL//TN)
∠RPU = 180° - ∠h (Angles on a straight line)
∠RUP = 180° - ∠h (Isosceles triangle)
55° + 180° - ∠h + 180° - ∠h + ∠i + 16° = 180° (Angles on a straight line, KT)
55° + 180° + 180° + 16° - ∠h - ∠h + ∠i = 180°
431° - 2∠h + ∠i = 180°
2∠h - ∠i = 431° - 180°
2∠h - ∠i = 251°
∠i = 2∠h - 251° --- (1)
∠URS
= ∠USR
= 2 x (180° - ∠h)
= 360° - 2∠h (Exterior angle of a triangle)
∠i = 180° - (360° - 2∠h) - (360° - 2∠h)
∠i = 180° - 360° + 2∠h - 360° + 2∠h
∠i = 180° - 360° - 360° + 2∠h + 2∠h
∠i = 4∠h - 540° (Angles sum of triangle)
∠i = 4∠h - 540° --- (2)
(2) = (1)
4∠h - 540° = 2∠h - 251°
4∠h - 2∠h= 540° - 251°
2∠h = 289°
∠h
= 289° ÷ 2
= 144.5°
(b)
From (1)
∠i
= 2∠h - 251°
= 289° - 251°
= 38°
Answer(s): (a) 144.5°; (b) 38°