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$ABCD$ is a cyclic quadrilateral $AE$ is drawn parallel to $CD$ and $BA$ is produced. If $∠ ABC = 92^{\circ}$ and $∠ FAE = 20^{\circ},$ then which of the following is the measure of $∠ B CD$ ?

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88^{\circ}

108^{\circ}

115^{\circ}

72^{\circ}

answer is B.

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Detailed Solution

Given,

ABCD

is a cyclic quadrilateral.

∠ ABC = 92^{\circ}

and

∠ FAE = 20^{\circ} .

Also, in a cyclic quadrilateral

ABCD,

sum of opposite angles is

180^{\circ}

\Rightarrow ∠ ADC + ∠ ABC = 180^{\circ}

\Rightarrow ∠ ADC + 92^{\circ}

180^{\circ}

\Rightarrow ∠ ADC = 180^{\circ} - 92^{\circ}

\Rightarrow ∠ ADC = 88^{\circ}

∠ EAD = ∠ ADC = 88^{\circ}

(Alternate interior angles of parallel lines

CD

and

AE

)

\Rightarrow ∠ FAD = ∠ EAD + ∠ FAE

\Rightarrow ∠ FAD = 20^{\circ}

88^{\circ}

\Rightarrow ∠ FAD = 108^{\circ}

\Rightarrow ∠ BCD = ∠ FAD = 108^{\circ}

(Exterior angle of a cyclic quadrilateral)
Hence, the measure of

∠ BCD

108^{\circ}

.
Therefore, option 2 is correct.

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