Question 5
In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that $∠ B E C = 130^{\circ}$ and $∠ E C D = 20^{\circ}$ . Find $∠ B A C$ .

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Solution

Given,
$∠ B E C = 130^{\circ}$
$∠ E C D = 20^{\circ}$
$∠ B E C + ∠ D E C = 180^{\circ}$ [Linear pair]
$130^{\circ} + ∠ D E C = 180^{\circ}$
$∠ D E C = 180^{\circ} - 130^{\circ} = 50^{\circ}$

Now, in $Δ D E C$
$∠ D E C + ∠ E C D + ∠ C D E = 180^{\circ}$ [Angle sum property of a triangle]
$50^{\circ} + 20^{\circ} + ∠ C D E = 180^{\circ}$
$∠ C D E = 180^{\circ} - 70^{\circ} = 110^{\circ}$
$∠ C D E = ∠ B A C$ [Angles in the same segment]
$∴ ∠ B A C = 110^{\circ}$

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Question 5 In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC=130∘ and ∠ECD=20∘. Find ∠BAC.

Question 5
In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that $∠ B E C = 130^{\circ}$ and $∠ E C D = 20^{\circ}$ . Find $∠ B A C$ .