Question

In the picture, A, B, C, D are points on a circle centred at O. The lines AC and BD are extended to meet at P. The line and BC intersect at Q. Prove that the angle which the small are AB makes at O is the sum of the angles it makes at P and Q.

Answer 1

∠DPC + ∠CQD = 180, In quadrilateral PCQD,

180-1/2x+180-1/2x+ ∠DPC + ∠CQD = 360

360 - x + ∠DPC + ∠CQD = 360

∠DPC + ∠CQD = x

∠DPC + ∠AQB = x

(∠CQD = ∠AQB opposite angle)

∴ ∠AOB = ∠AQB + ∠APB