Question

ABCD is a kite. If $\angle DAB$ is two times $\angle ADC$, find the value of $\angle ABC$ in terms of $\angle ADC$.

• A. $\angle ABC={180}^{\circ }-4\angle ADC$
• B. $\angle ABC={360}^{\circ }+4\angle ADC$
• C. $\angle ABC={360}^{\circ }-4\angle ADC$
• D. $\angle ABC={180}^{\circ }+4\angle ADC$

Answer 1

The correct option is C $\angle ABC={360}^{\circ }-4\angle ADC$

$Let\angle ADC=x$
Then$\angle DAB=2x$
$\angle DAB=\angle DCB\text{(One pair of diagonally opposite angles are equal)}$
$\angle DAB+\angle ABC+\angle BCD+\angle CDA={360}^{\circ }$
$\text{(Angle sum property of a quadrilateral)}$
$2x+\angle ABC+2x+x={360}^{\circ }$
$4x+\angle ABC={360}^{\circ }$
$\angle ABC={360}^{\circ }-4x$
$\angle ABC={360}^{\circ }-4\angle ADC$