State true or false.

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

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True

False

answer is A.

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Detailed Solution

Consider ABCD are the points in a quadrilateral circumscribing a circle PQRS
Join OP, OQ, OR and OS
Angles at each section are represented by a, b, c, d, e, f, g, h respectively.
Question Image

In triangle

Δ D O S a n d Δ D O R

D O = D O (c o m m o n s i d e) OS = O R (r a d i u s) D R = DS (T h e o r e m 2)

Therefore,

Δ D O S ≅ Δ D O R

∠ D O S = ∠ D O R (by CPCT)

Here,

⇒ ∠ c = ∠ d

…. (1)
Similarly,

⇒ ∠ a = ∠ b

…. (2)

⇒ ∠ e = ∠ f

…. (3)

⇒ ∠ g = ∠ h

…. (4)
In a circle,

⇒ ∠ a + ∠ b + ∠ c + ∠ d + ∠ e + ∠ f + ∠ g + ∠ h = 360 °

Substitute the equations (1), (2), (3), (4) in the above equation, we get

⇒ 2 (∠ a + ∠ e + ∠ h + ∠ d) = 360 ° ⇒ (∠ a + ∠ h) + ∠ e + ∠ d = 360 ° 2 ⇒ ∠ A O B + ∠ D O C = 180 °

Similarly,

⇒ 2 (∠ b + ∠ f + ∠ g + ∠ c) = 360 ° ⇒ (∠ b + ∠ c) + ∠ g + ∠ f = 360 ° 2 ⇒ ∠ A O D + ∠ B O C = 180 °

.
Hence, it is true that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Therefore, option (1) is correct.

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