Question

Is my work correct?

Given: ABCD is an inscribed polygon.

Prove: ∠A and​ ∠C ​ are supplementary angles.

Answer 1

A two-column proof to prove that angles A and​ C ​ are supplementary angles should be completed as follows;

Statement Reason______________

ABCD is an inscribed polygon Given

mBCD = 2(m∠A) Inscribed Angle Theorem

mDAB = 2(m∠C) Inscribed Angle Theorem

mBCD + mDAB = 360° The sum of arcs that make a circle is 360°

2(m∠A) + 2(m∠C) = 360° Substitution Property

m∠A + m∠C = 180° Division Property of Equality

∠A and∠C are supplementary angles Defintion of supplementary angles.

In Mathematics and Euclidean Geometry, the inscribed angle theorem states that the measure of an inscribed angle is one-half the measure of the intercepted arc in a circle or the inscribed angle of a circle is equal to half of the central angle of a circle.

Generally speaking, a supplementary angle refers to two angles or arc whose sum is equal to 180 degrees.

Based on the defintion of supplementary angles, we can logically deduce that angle A and angle C are supplementary angles.

Answer 2

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.