Math, asked by Sony5145, 1 year ago

Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle.Also ∠A+∠C=180°, then prove that the vertex D also lie on the same circle.

Answers

Answered by MaheswariS
9

Answer:

Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle.Also ∠A+∠C=180°, then prove that the vertex D also lie on the same circle.

Given:

ABCD is a cyclic quadrilateral such that ∠A+∠C=180°.........(1)

we know that,

Sum of all angles of a quadrilateral =360°

That is

∠A+∠B+∠C+∠D=360°

==>(∠A+∠C)+(∠B+∠D)=360°

==>180°+(∠B+∠D)=360°

==> ∠B+∠D=360°-180°

==> ∠B+∠D=180°.............(2)

From (1) and (2), we get

Opposite angles of a cyclic quadrilateral ABCD are supplementary

Hence, ABCD is a cyclic quadrilateral

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