In the given figure, AB, BC, CD and DA are tangents to the circle with centre O forming a quadrilateral ABCD. Show that

Question

In the given figure, AB, BC, CD and DA are tangents to the circle with centre O forming a quadrilateral ABCD. Show that $∠ A O B + ∠ C O D = 180 °$

Smart Achievers · Accepted Answer

Given A quad. ABCD circumscribes a circle with centre O.

To Prove: $∠ A O B + ∠ C O D = 180 °$

and $∠ A O D + ∠ B O C = 180 °$

Join OP, OQ, OR and OS.

We know that the tangent drawn from an external point of a circle subtends equal angles at the centre.

$\Rightarrow ∠ 1 = ∠ 2, ∠ 3 = ∠ 4, ∠ 5 = ∠ 6, ∠ 7 = ∠ 8,$

and $∠ 1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 360 ° [∠ S at a Point]$

$\Rightarrow 2 (∠ 2 + ∠ 3) + 2 (∠ 6 + ∠ 7) = 360 °$

$2 (∠ 1 + ∠ 8) + 2 (∠ 4 + ∠ 5) = 360 °$

$\Rightarrow ∠ 2 + ∠ 3 + ∠ 6 + ∠ 7 = 180 °$

$∠ 1 + ∠ 8 + ∠ 4 + ∠ 5 = 180 °$

$\Rightarrow ∠ A O B + ∠ C O D = 180 °$

$∠ A O D + ∠ B O C = 180 °$

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