Everything Maths Grade 12

9.2 CHAPTER 9. GEOMETRY

O�

A

B

P

Q

1

2

Consider a circle, with centre O. Draw a cyclic quadrilateral ABPQ. Draw AO and PO.

The aim is to prove thatABPˆ +AQPˆ = 180◦andQABˆ +QPBˆ = 180◦.

Oˆ 1 = 2ABPˆ (∠s at centre (Theorem 4))

Oˆ 2 = 2AQPˆ (∠s at centre (Theorem 4))

But,Oˆ 1 +Oˆ 2 = 360◦

∴ 2 ABPˆ + 2AQPˆ = 360◦

∴ABPˆ +AQPˆ = 180◦

Similarly,QABˆ +QPBˆ = 180◦

Theorem 8. (Converse of Theorem7) If the opposite anglesof a quadrilateral are supplementary, then
the quadrilateral is cyclic.

Proof:

A

B

R

Q

P

Consider a quadrilateral ABPQ, such thatABPˆ +AQPˆ = 180◦andQABˆ +QPBˆ = 180◦.

The aim is to prove thatpoints A, B, P and Q lie on the circumference of a circle.

By contradiction. Assume that point P does not lie on a circledrawn through points A, B and Q. Let
the circle cut QP (or QP extended) at point R. Draw BR.

QABˆ +QRBˆ = 180◦(opp.∠s of cyclic quad. (Theorem 7))

butQABˆ +QPBˆ = 180◦(given)

∴QRBˆ = QPBˆ

but this cannot be true sinceQRBˆ = QPBˆ +RBPˆ (exterior∠ of�)