A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ∠ADC=130o, then, which of the following is the measure of ∠BAC?

Given, a quadrilateral

ABCD

is inscribed in a circle such that

AB

is a diameter and

∠ ADC = 130^{o}

.
Draw the diagram according to the given data,

In a cyclic quadrilateral sum of opposite sides are supplementary.
Then,

\Rightarrow ∠ ADC + ∠ ABC = 180^{o}

\Rightarrow ∠ ABC = 180^{o} - ∠ ADC

\Rightarrow ∠ ABC = 180^{o} - 130^{o}

\Rightarrow ∠ ABC = 50^{o}

So, in

ΔABC

, by angle sum property,

\Rightarrow ∠ BAC + ∠ ACB + ∠ ABC = 180 °

\Rightarrow ∠ BAC = 180^{o}

- (∠ ACB + ∠ ABC)

…(i)
The angle subtended by the diameter of a circle at the circumference is

90^{o}

\Rightarrow ∠ ACB = 90^{o}

Then using this in equation (i),

∠ BAC = 180^{o} - (50^{o} + 90^{o})

\Rightarrow ∠ BAC = 180^{o} - 140 °

\Rightarrow ∠ BAC = 40^{o}

Hence, the measure of

∠ BAC

40^{o}

.
Therefore, option 3 is correct.

A quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $∠ ADC = 130^{o}$ , then, which of the following is the measure of $∠ BAC$ ?