State whether the given statement is True or False.

Statement: If the sum of a pair of opposite angles of a quadrilateral is $180^{0}$ , then that quadrilateral is cyclic.

Question

State whether the given statement is True or False.Statement: If the sum of a pair of opposite angles of a quadrilateral is 1800, then that quadrilateral is cyclic.

Accepted Answer

Let us assume a cyclic quadrilateral ABCD as shown in the below figure .

https://www.vedantu.com/question-sets/2e609bba-9833-4945-9980-a0b8b23f96e92152188780974804627.png

Here we need to prove the sum of pair of opposite angles of cyclic quadrilateral is 1800,
from the above figure we can write
∠BAD + ∠BCD =

180^{0}

or ∠ABC + ∠ADC =

180^{0}

From the diagram we can write the relation between the internal angles as given below
From the Chord AB, ∠5 = ∠8 ............ (i) Angles in same segment are equal.
Similarly,
From the Chord BC, ∠1 = ∠6 ............(ii)
From the Chord CD, ∠2 = ∠4............(iii)
From the Chord AD, ∠7 = ∠3............(iv)
We know that the sum of angles in a quadrilateral is

360^{0}

, hence we can write
∠BAD + ∠ABD + ∠BCD + ∠ADC =

360^{0}

From the diagram we can write ∠BAD = ∠1 + ∠2, ∠ABD = ∠3 + ∠4, ∠BCD = ∠7 + ∠8, ∠ADB = ∠5 + ∠6 ∠BAD + ∠ABD + ∠BCD + ∠ADC =

360^{0}

⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 =

360^{0}

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

360^{0}

From equations (i), (ii), (iii), (iv), substituting the values of ∠3, ∠4, ∠5, ∠6 in the above equation, then we will get
(∠1 + ∠2 + ∠7 + ∠8) + (∠3 + ∠4 + ∠5 + ∠6) =

360^{0}

⇒ (∠1 + ∠2 + ∠7 + ∠8) + (∠7 + ∠2 + ∠8 + ∠1) =

360^{0}

⇒(∠1 + ∠2 + ∠7 + ∠8) + (∠1 + ∠2 + ∠7 + ∠8)=

360^{0}

2(∠1 + ∠2 + ∠7 + ∠8) =

360^{0}

(∠1 + ∠2 + ∠7 + ∠8) =

180^{0}

(∠1 + ∠2) + (∠7 + ∠8) =

180^{0}

From the diagram the value of ∠1 + ∠2 = ∠BAD, ∠7 + ∠8 = ∠BCD, then
(∠1 + ∠2) + (∠7 + ∠8) =

180^{0}

⇒ ∠BAD + ∠BCD =

180^{0}

Similarly,
∠ABC + ∠ADC =

180^{0}

From this we can say that in a cyclic quadrilateral the sum of pair of opposite angles is equal to

180^{0}

or If the sum of pair of opposite angles in a quadrilateral is equal to

180^{0}

, then the quadrilateral is cyclic quadrilateral.
So, option (1) is correct.

State whether the given statement is True or False.

Statement: If the sum of a pair of opposite angles of a quadrilateral is $180^{0}$ , then that quadrilateral is cyclic.

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