Question

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of a circle.

Answer 1

To Prove : ∠AOB + ∠COD = 180°;∠BOC + ∠DOA = 180°

Given: ABCD is circumscribing the circle.

Proof:

Let ABCD be a quadrilateral circumscribing a circle centred at O such that it touches the circle at point P, Q, R, S.

join the vertices of the quadrilateral ABCD to the centre of the circle.

Consider ΔOAP and ΔOAS:

AP = AS (Tangents from the same point)

OP = OS (Radii of the same circle)

OA = OA (Common side)

ΔOAP ≅ ΔOAS (SSS congruence criterion)

And thus, ∠ POA = ∠ AOS

∠1 = ∠ 8 Similarly,

∠2 = ∠ 3

∠4 = ∠ 5

∠6 = ∠ 7

∠1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 360°

(∠ 1 + ∠ 8) + (∠ 2 + ∠ 3) + (∠ 4 + ∠ 5) + (∠ 6 + ∠ 7) = 360°

2∠ 1 + 2∠ 2 + 2∠ 5 + 2∠ 6 = 360°

2(∠ 1 + ∠ 2) + 2(∠ 5 + ∠ 6) = 360°

(∠ 1 + ∠ 2) + (∠ 5 + ∠ 6) = 180°

∠ AOB + ∠ COD = 180°

Similarly, we can prove that ∠ BOC + ∠ DOA = 180°

Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

[Proved] Q.E.D

Answer 2

Hey Mate :D

Your Answer :---

Let ABCD be a quadrilateral circumscribing a circle centered at O such that it touches the circle at point P, Q, R, S.

Let us join the vertices of the quadrilateral ABCD to the center of the circle.

[ plz see attached file also :) ]

Consider ∆OAP and ∆OAS,

AP = AS (Tangents from the same point)

OP = OS (Radii of the same circle)

OA = OA (Common side)

∆OAP ≅ ∆OAS (SSS congruence criterion)

Therefore, A ↔ A, P ↔ S, O ↔ O

And thus,

∠POA = ∠AOS ∠1 = ∠8 Similarly, ∠2 = ∠3 ∠4 = ∠5 ∠6 = ∠7

∠1 + ∠2 + ∠ 3 + ∠4 + ∠5 + ∠ 6 + ∠7 + ∠ 8 = 360º

( ∠ 1 + ∠ 8) + ( ∠ 2 + ∠ 3) + ( ∠4 + ∠ 5) + ( ∠6 + ∠ 7) = 360º

2 ∠1 + 2 ∠2 + 2∠ 5 + 2∠ 6 = 360º

2( ∠1 + ∠ 2) + 2( ∠5 + ∠ 6) = 360º

( ∠1 + ∠2 ) + ( ∠5 + ∠ 6) = 180º

∠AOB + ∠COD = 180º

Similarly, we can prove that

∠BOC + ∠DOA = 180º

Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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