The opposite angle of a quadrilateral in a circle are supplementary is to find the intercepted arc of opposite angles of a cyclic quadrilateral.
We have given,
Rahim is constructing proof to show that the opposite angles of a quadrilateral inscribed in a circle are supplementary.
What is the cyclic quadrilateral?
The cyclic quadrilateral is a quadrilateral that lies on a circle.
SupposeABCD is a cyclic quadrilateral.
The First step to show that the opposite angle of a quadrilateral ABCD in a circle is supplementary is to find the intercepted arc of opposite ∠A and ∠C of a cyclic quadrilateral.
An intercepted are of ∠A is Arc(BCD) and intercepted arc of ∠C is Arc(DAB).
arc(BCD)=2A...................(1)
arc(DAB)=2C...................(2)
We know that
arcBCD)+arc(DAB)=360..............(3)
Put values of Arc(BCD) and Arc(DAB) in equation (3)
[tex]2A+2C=360[/tex].............(4)
[tex]2(\∠A+\∠ C)={360}[/tex]
[tex]\∠A+\∠ C=\frac{360}{2}[/tex]
∠A+∠C=180
Hence,
The opposite angles A and angle C of a cyclic quadrilateral are supplementary.
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