Question

prove that the opposite angles of cycle quadrilateral are suppleatary or sum is 180.​

Answer 1

Answer:

Given : A cyclic quadrilateral ABCD.

To Prove : ∠A+∠C=180o

∠B+∠D=180o

Construction : Let O be the centre of the circle. Join Oto B and D. Then let the angle subtended by the minor arc and the major arc at the centre be xo and yo respectively.

Proof : xo=2∠C [Angle at centre theorem] ...(i)

yo=2∠A ...(ii)

Adding (i) and (ii), we get

xo+yo=2∠C+2∠A ...(iii)

But, xo+yo=360o ....(iv)

From (iii) and (iv), we get

2∠C+2∠A=360o

⇒ ∠C+∠A=180o

But we know that angle sum property of quadrilateral

∠A+∠B+∠C+∠D=360o

∠B+∠D+180o=360o

∠B+∠D=180o

Hence proved.

Step-by-step explanation:

This may help

Answer 2

solution....

The properties of a cyclic quadrilateral are as follows: If one side of the cyclic quadrilateral is produced, then the exterior angle so formed is equal to the interior opposite angle. The sum of the opposite angles of a cyclic quadrilateral is supplementary.

Theorem :

Opposite angles of a cyclic quadrilateral are supplementary

(or)

The sum of opposite angles of a cyclic quadrilateral is 180°

Given : O is the centre of circle. ...

To prove : <BAD + <BCD = 180°, <ABC + <ADC = 180°

.....If the sum of a pair of opposite angles of a quadrilateral is 180^0,

.....the quadrilateral is cyclic.

Thus, ∠BED=∠C ∠ B E D = ∠ C .

...However, this is not possible, since ∠C (being the exterior angle) .

must be larger than∠BED ∠ B E D .

... or.. attachment pic

i hope it helpfull to you...