In figure, A, B, C, D are four points on the circle. AC and BD in… - Vidyadip

Question

In figure, $A, B, C, D$ are four points on the circle. $AC$ and $BD$ intersect at a point $E$ such that $\angle BEC = 130^\circ $ and $\angle ECD = 20^\circ $ Find $\angle BAC.$

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Answer

Given: $\angle BEC = 130^\circ $ and $\angle ECD = 20^\circ $
$ \angle DEC = 180^\circ - \angle BEC = 180^\circ - 130^\circ = 50^\circ [$Linear pair$]$
Now in $\triangle DEC,$
$ \angle DEC + \angle DCE + \angle EDC = 180^\circ [$Angle sum property$]$
$\Rightarrow 50^\circ + 20^\circ + \angle EDC = 180^\circ \Rightarrow \angle EDC = 110^\circ $
$\Rightarrow \angle BAC = \angle EDC = 110^\circ [$Angles in same segment$]$

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