Question

Given:quadrillateral ABCD inscribed in a circle

Answer 1

Answer:

Step-by-step explanation:

Given: quadrilateral ABCD inscribed in a circle

To Prove:

1. ∠A and ∠C are supplementary.

2. ∠B and ∠D are supplementary.

Construction : Join AC and BD.

Proof: As, angle in same segment of circle are equal.Considering AB, BC, CD and DA as Segments, which are inside the circle,

∠1=∠2-----(1)

∠3=∠4-----(2)

∠5=∠6-------(3)

∠7=∠8------(4)

Also, sum of angles of quadrilateral is 360°.

⇒∠A+∠B+∠C+∠D=360°

→→∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8=360°→→→using 1,2,3,and 4

→→→2∠1+2∠4+2∠6+2∠8=360°

→→→→2( ∠1 +∠6) +2(∠4+∠8)=360°⇒Dividing both sides by 2,

→→→∠B + ∠D=180°as, ∠1 +∠6=∠B , ∠4+∠8=∠B------(A)

As, ∠A+∠B+∠C+∠D=360°

∠A+∠C+180°=360°

∠A+∠C=360°-180°------Using A

∠A+∠C=180°

Hence proved.

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