Theory question 104
Circle TheoremsIn the diagram BCDE, is a circle with centre A. ∠BCD = (2x + 40°), ∠BAD = (5x − 35)°, ∠BED = (2y + 10°) and ∠ADC = 40°. Find: (a) the values of x and y; (b) ∠ABC
Show the model answer
Model answer
(a) Angle at centre = twice angle at circumference: 5x − 35 = 2(2y + 10) → 5x − 4y = 55
BCDE is cyclic: (2x + 40) + (2y + 10) = 180 → x + y = 65
Solve: 9x = 315 → x = 35, y = 30
(b) ∠BAD = 140°, ∠BCD = 110°
Quadrilateral ABCD: ∠ABC = 360° − (140° + 110° + 40°) = 70°
Explanation
Form two equations: one from the angle at the centre, one from the cyclic quadrilateral.