In Fig. if AOB is a diameter and =120^ , then =30^ . - Vidyadip

Question

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

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Answer

True.Solution: Join CA and CB.

Since, ADCB is a cyclic quadrilateral. $\angle\text{ADC}+\angle\text{CBA}=180^\circ$ [sum of opposite angles of cyclic quadrilateral is 180°] $\Rightarrow\angle\text{CBA}=180^\circ-120^\circ=60^\circ\ \ [\therefore\angle\text{ADC}=120^\circ]$ In $\triangle\text{ACB,}\ \angle\text{CAB} + \angle\text{CBA} + \angle\text{ACB} = 180^\circ$ [by angle sum property of a triangle] $\angle\text{CAB}+60^\circ+90^\circ=180^\circ$ $\big[$triangle formed from diameter to the circle is 90° i.e., $\angle\text{ACB}=90^\circ\big]$ $\Rightarrow\angle\text{CAB}=180^\circ-150^\circ=30^\circ.$

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