Question

Two circles intersect each other at point P and Q. Secants drawn through p and Q intersect the circles at points A,B and D,C
Prove that `: /_ ADC + /_ BCD = 180^(@)`

Answer 1

Draw seg PQ
`square ADQP` is a cyclic quadrialateral
`/_ ADQ + /_ APQ = 180^(@) ` ….( Opposite angles of cyclic quadrilateral are supplementary )
`:. /_ ADC + /_ APQ = 180^(@)` ….( D-Q-C ) …(1)
` /_ APQ ` is an exterior angle of cyclic quadrilateral PQCB
`:. /_ APQ = /_ BCQ ` ....( Measure of an exterior angle of cyclic quadrilateral is equal to the meausre of its interior opposite angle )
`:. /_ APQ = /_ BCD ` ...( D-Q-C ) ...(2)
`:. /_ ADC + /_ BCD = 180^(@)` ......[From (1) and (2) ]