ABCD is a cyclic quadrilateral in which BC is parallel to AD, ang… - Vidyadip

Question

ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110° and angle BAC = 50°. Find angle DAC and angle DCA.

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Answer

$A B C D$ is a cyclic quadrilateral in which $A D \| B C$
$
\begin{aligned}
& \angle ADC =110^{\circ}, \angle BAC =50^{\circ} \\
& \angle B +\angle D =180^{\circ}
\end{aligned}
$
(Sum of opposite angles of a quadrilateral)
$
\begin{aligned}
& \Rightarrow \angle B +110^{\circ}=180^{\circ} \\
& \Rightarrow \angle B =70^{\circ}
\end{aligned}
$
Now in $\triangle ABC$,
$
\begin{aligned}
& \angle BAC +\angle ABC +\angle ACB =180^{\circ} \\
& \Rightarrow 50^{\circ}+70^{\circ}+\angle ACB =180^{\circ} \\
& \Rightarrow \angle ACB =180^{\circ}-120^{\circ}=60^{\circ} \\
& \because AD I BC \\
& \therefore \angle DAC =\angle ACB =60^{\circ} \text { (alternate angles) }
\end{aligned}
$
Now in $\triangle A D C$,
$
\begin{aligned}
& \angle DAC +\angle ADC +\angle DCA =180^{\circ} \\
& \Rightarrow 60^{\circ}+110^{\circ}+\angle DCA =180^{\circ} \\
& \Rightarrow \angle DCA =180^{\circ}-170^{\circ}=10^{\circ}
\end{aligned}
$

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