Solution:
Given: A B CD is an inscribed polygon.
To Prove: ∠A and ∠C are supplementary angles.
Proof: Join AC and B D.
Angle in the same segment of a circle are equal.
∠ACB=∠ADB→→AB is a segment.
Also, ∠A B D=∠A CD→→AD is a Segment.
In Δ ABD
∠A+∠ABD+∠ADB=180°→→Angle sum property of triangle.
∠A+∠A CD+ ∠ACB=180°
∠A+∠C=180°
Hence proved, that is, ∠A and ∠C are supplementary angles.
The method Adopted by you
∠1=2 ∠A----(1)
and, ∠2=2 ∠C-------(2)
The theorem which has been used to prove 1 and 2, Angle subtended by an arc at the center is twice the angle subtended by it any point on the circle.→(Inscribed angle theorem)
Also, angle in a complete circle measures 360°.→→Chord arc theorem
∠1+∠2=360°→→Addition Property of Equality
2∠A+2∠C=360°→→[Using 1 and 2, Called Substitution Property]
Dividing both sides by 2→→Division Property of Equality
2∠A+2∠C=360°→→[Using 1 and 2]
∠A+∠C=180°
→→Correct work.
