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.
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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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