Question

$ABCD$ is a cyclic quadrilateral, in which $BC$ is parallel to $AD$, $\angle ADC={110}^{\circ }$ and $\angle BAC={50}^{\circ }$. What is the value of $\angle DAC$?

• A. ${40}^{\circ }$
• B. ${50}^{\circ }$
• C. ${60}^{\circ }$
• D. ${64}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

$ABCD$ is a cyclic quadrilateral.

Then, $\angle ABC+\angle ADC={180}^o$ ...[Opposite angles of a cyclic quadrilateral are supplementary]

$\Rightarrow$ $\angle ABC={180}^{\circ }-{110}^{\circ }={70}^{\circ }$.

In $△ABC$,

$\angle ABC+\angle ACB+\angle BAC={180}^o$ ...[Angle sum property]

$\Rightarrow$${70}^o+\angle ACB+{50}^o={180}^o$

$\Rightarrow$ $\angle ACB={180}^{\circ }-({50}^{\circ }+{70}^{\circ })={60}^{\circ }$.

Given, $BC$ and $AD$ are parallel,

$\Rightarrow \angle ACB=\angle DAC={60}^{\circ }$ ...[Alternate interior angles].

Hence, option $C$ is correct.