Question

In the given figure, $O$ is the centre of circle, $\angle AEC={40}^o$, then find the value of $a+b+c$.

Answer 1

$\angle AEC={40}^{\circ }$

$\angle AEC=a$ angle in the same segment

$\angle AOC=2\times \angle AEC$

$\Rightarrow c=2\times {40}^{\circ }={80}^{\circ }$

In quadrilateral $ABCD$, opposite angle of a cyclic quadrilateral supplementary

$\angle ABC+\angle ADC={180}^{\circ }$

$\Rightarrow b+{40}^{\circ }={180}^{\circ }$

$\Rightarrow b={180}^{\circ }-{40}^{\circ }={140}^{\circ }$

And $P=a+b+c={40}^{\circ }+{140}^{\circ }+{80}^{\circ }={260}^{\circ }$