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Angles in Same Segment of a Circle

In the figure...

Question

In the figure, $∠ D B C = 58^{\circ} .$ BD is a diameter of the circle. Calculate :

(i) $∠ B D C$

(ii) $∠ B E C$

(iii) $∠ B A C$

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Solution

Given $BD$ is diameter

$\angle DBC = 58^o$

(i) $\angle DCB = 90^o$ (angle in semi circle)

In $\triangle BDC$

$58+\angle BDC+90=180^o$

$\angle BDC = 32^o$

(ii) $ABEC$ is a cyclic quadrilateral

$\angle BDC = \angle BAC = 32^o$ (angle in the same segment)

Therefore $\angle BEC + 32 = 180^o$

$\angle BEC = 148^o$

(iii) $\angle BAC = 32^o$ (angle in the same segment)

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