A cyclic quadrilateral ABCD is drawn in a circle with centre O. A and C are joined to O. If $\angle A B C=2 p$ and $\angle A D C=3 p$, what is the measure (in degrees) of the $\angle A O C$ reflex?

216

\(\angle\)ABC + \(\angle\)ADC = \({180}^\circ\)

⇒ 2p + 3p = \({180}^\circ\)

⇒ 5p = \({180}^\circ\)

⇒ p = \({180}^\circ\)/5 = \({36}^\circ\)

So, \(\angle\)ABC = 2 x \({36}^\circ\) = \({72}^\circ\) and \(\angle\)ADC = 3 x \({36}^\circ\) = \({108}^\circ\)

Then \(\angle\)AOC = 2 x \({72}^\circ\) =  \({144}^\circ\)  [as, \(\angle\)ABC = \({72}^\circ\)] (Concept used)

Another side of \(\angle\)AOC = \({360}^\circ\) - \({144}^\circ\) = \({216}^\circ\)

Therefore, \(\angle\)AOC = \({216}^\circ\)