If ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and if ∠ADC=140o, then which of the following is the measure of ∠BAC?

Given that

ABCD

is a cyclic quadrilateral with

AB

as the diameter of the circle.
Also,

∠ ADC = 140^{o} .

Constructing a figure with the given data,

In a cyclic quadrilateral sum of opposite angles are supplementary.

\Rightarrow ∠ ADC + ∠ ABC = 180^{o}

\Rightarrow ∠ ABC = 180^{o} - 140^{o}

\Rightarrow ∠ ABC = 40^{o}

We also know that, the angle subtended by a diameter at the circumference of the circle, is

90^{o}

.
Thus,

\Rightarrow ∠ ACB = 90^{o}

∴

ΔABC,

by angle sum property,

∠ CAB + ∠ ACB + ∠ ABC = 180^{o}

\Rightarrow ∠ CAB = 180^{o} - (∠ ACB + ∠ ABC)

\Rightarrow ∠ CAB = 180^{o} - (90^{o} + 40^{o})

\Rightarrow ∠ CAB = 50^{o}

Hence, the measure of

∠ CAB

50^{o}

.
Therefore, option 2 is correct.

If $ABCD$ is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and if $∠ ADC = 140^{o}$ , then which of the following is the measure of $∠ BAC$ ?