Question

2.3. (B) Solve any Two of the following:
(OG
1) Prove that, opposite angles of a cyclic quadrilateral are suppliementary.
A​

Answer 1

Answer:

Step-by-step explanation:

in a cyclic Quadilateral abcd

<A+<B+<C+<D =360degree

Given : A cyclic quadrilateral ABCD.

To Prove : ∠A+∠C=180o

                ∠B+∠D=180o

Construction : Let O be the centre of the circle. Join O to 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=180

Answer 2

Step-by-step explanation:

To Prove, In a cyclic quadrilateral A B C D,

                 ∠A+∠C=180°   &    ∠B+∠D=180°

Proof:

let O be the center of the circle. Join O to B&D

Then let the angle subtended by the minor arc and the major arc at the center be x and y resply.,  (refer attachment)

we know x = 2∠C  (by center angle theorem)     -------- (1)

        also y = 2∠A                                                 ----------(2)

(1) + (2) =>  x + y = 2∠C + 2∠A                               ----------(3)

 from fig.,  x + y = 360°

From (iii) and (iv), we get

           2∠C+2∠A=360°

           ∠C+∠A=180°

But we know that sum of all angles in a quadrilateral = 180°

∠A+∠B+∠C+∠D=360°

∠B+∠D+180°=360°

∠B+∠D=180°

Hence proved.