Solve Question:

Prove that “the opposite angles of a cyclic quadrilateral are supplementary”

When 4 vertices of a quadrilateral is inside a circle then that is known as cyclic quadrilateral.

Here we have to prove that ∠A + ∠c = 180° &

∠B + ∠D = 180°

As shown in fig. we are dividing the quadrilateral into 2 triangles (Δ ACB, Δ ADB)

Since the angle formed in the same segment are equal.

We can say that ∠ACB = ∠ADB ⇒ 1

Similarly, we can say that ∠BAC = ∠BDC ⇒ 2

From fig.

∠D = ∠ADB + ∠BDC

Adding both the equations

∠ACB + ∠BAC = ∠ADB + ∠BDC

∠ACB + ∠BAC = ∠D

Adding ∠ABC on both sides

∠ACB + ∠BAC + ∠ABC = ∠D + ∠ABC

(∠ACB + ∠BAC + ∠ABC = 180° since they represent all angles in a triangle ACB)

∵ ∠ABC = ∠B

∠B + ∠D = 180° ⇒ 3

We know that sum of all angles in a quadrilateral is 360

∴ ∠A + ∠B + ∠C + ∠D = 360°

Substituting eqn 3 in above eqn.

We get, ∠A + ∠c = 180°

Thus, we have proved the opposite angles of a cyclic quadrilateral are supplementary.