Question

what is cyclic quadrilateral. prove that the sum of oppositre angles of cyvlic quadrilateral is180

Answer 1

$\mathcal{\huge{\blue{SOLUTION}}}$

CYCLIC QUADRILATERAL -::

A quadrilateral is said to be cyclic if its all four points lie on a circle.

To prove -: Sum of opposite angles of a cyclic quadrilateral is 180°.

Proof -:

In the above figure,

We have,

Angle ACB = Angle ADB(angle in the same segment)

Angle BAC = Angle BDC (angle in the same segment)

Therefore,

ACB + BAC = ADB + BDC

=> ACB + BAC = ADC

=> ACB + BAC + ABC = ADC + ABC (adding ABC on both sides)

=> ADC + ABC = 180°

=> Angle B + Angle D = 180° -------(i)

Now,

A + B + C + D = 360°(sum of angles of a quadrilateral)

=> A + C = 360°- (B + D)

=> A + C = 360° - 180° = 180° (using (i))

Hence,

Angle A + Angle C = 180°
Angle B + Angle D = 180°...

!!!So proved!!!