Question

Question 10
Write True or False and justify your answer :

In the figure, if AOB is a diameter and $\angle ADC={120}^{\circ },then\angle CAB={30}^{\circ }$.

Answer 1

True

Construction
Join CA and CB

Since ADCB is a cyclic quadrilateral,
$\angle ADC+\angle CBA={180}^{\circ }$ [sum of opposite angles of cyclic quadrilateral is ${180}^{\circ }$]
$\Rightarrow \angle CBA={180}^{\circ }-{120}^{\circ }={60}^{\circ }[\therefore \angle ADC={120}^{\circ }]$

$In\Delta CAB,\angle ABC+\angle ACB={180}^{\circ }$ [by angle sum property of a training]
$\angle CAB+{60}^{\circ }+{90}^{\circ }={180}^{\circ }$ [angle formed from diameter to the circle is ${90}^{\circ }i.e.,\angle ACB={90}^{\circ }$]
$\Rightarrow \angle CAB={180}^{\circ }-{150}^{\circ }={30}^{\circ }$