Question

Is my work correct?

Answer 1

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.

Answer 2

Check the picture below.

so, notice, the intercepted arc by the inscribed angle A is the red arc there, namely arcBCD, and the one intercepted by angle C is the arcBAD in blue there.

now, both arcs take up the whole circle, namely the whole 360°, and we also know that arcBCD = 2A, whilst arcBAD = 2C.

$\bf \stackrel{2(\measuredangle A)}{\widehat{BCD}}+\stackrel{2(\measuredangle C)}{\widehat{BAD}}=360\implies 2A+2C=360\implies 2(A+C)=360 \\\\\\ A+C=\cfrac{360}{2}\implies A+C=180$