Answer:
See attached for the cyclic quadrilateral
To prove: <BAD + <BCD =180°
Construction: Join B and D to the centre O of circle ABCD
Proof
With the lettering of the attached drawing,
<BOD = 2y (angle at centre is 2 X angle at circumference)
Reflex <BOD = 2x (angle at centre is 2 X angle at circumference)
∴ 2x + 2y = 360° (angle at point)
∴ x + y = 180°
∴ <BAD + <BCD = 180°
Explanation:
The vertices of a cyclic quadrilateral lie on the circumference of the circle and the opposite angles of a cyclic quadrilateral lie in opposite segment of a circle.
The question is to prove that the opposite angles of a cyclic quadrilateral are supplementary that is 180°. Another way of stating this theory is 'Angles in opposite segments are supplementary'.
Note that the sum of supplementary angles is 180°.