Sum of opposite angles of a quadrilateral is 180

Asked by admin @ 20/08/2021 in Math viewed by 309 People

If the sum of pair opposite angles of a quadrilateral is 180 ,than the quadrilateral is cyclic proof

Answered by admin @ 20/08/2021

Answer:

The proof is explained below.

Step-by-step explanation:

Given the sum of pair of opposite angles of a quadrilateral is 180° we have to prove that the quadrilateral is cyclic.

Let us assume that the quadrilateral ABCD is not cyclic i.e Let the point D does not lie on the circle which makes the quadrilateral non-cyclic. Now, let us do a construction such that join CD' where D' is the point of intersection of side AD with the circle.

Now, ABCD' is cyclic

⇒ ∠3 + ∠4 = 180°

Now, it is given that the sum of pair opposite angles of a quadrilateral ABCD is 180°

Therefore, ∠2 + ∠4 = 180°

From above two equations we get

∠3 + ∠4 = ∠2 + ∠4

⇒ ∠3 = ∠2

Now, in triangle CDD', by external angle property

∠3 = ∠1 + ∠2

⇒ ∠1 = 0 , hence the side CD' and CD coincides

⇒ Point D lies on circle

Hence, our supposition is wrong quadrilateral ABCD is cyclic.

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