In the given figure, AE is the diameter of the circle. Write down the numerical value of $∠$ ABC + $∠$ CDE. Give reasons for your answer.

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Solution

$\angle AOC = \frac{180}{2} = 90^o$

$\angle AOC = 2 \angle AEC$ (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

$\Rightarrow \angle AEC = \frac{90}{2} = 45^o$

$ABCE$ is a cyclic quadrilateral

$\therefore \angle ABC + \angle AEC = 180 ^o$

$\Rightarrow \angle ABC = 180-45 = 135^o$

Similarly, $\angle CDE = 135^o$

$\therefore \angle ABC + \angle CDE = 135^o + 135^o = 270^o$

$\\$

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