Construction: Extend the line DO and BO at the points l and H on the circle. Also, join AC.
We know that, in a circle, the angle subtended by an arc at the centre is twice the angle subtended by it in the remaining part of the circle.
∴∠1=2∠6 ...(i) and
∠3=2∠7 ...(ii) In
ΔAOC,OC=OA [both are the radius of circle]
∠OCA=∠4 [angles opposite to equal sides are equal]
Also,
∠AOC+∠OCA+∠4=180∘ [by angle sum property of triangle]
⇒∠AOC+∠4+∠4=180∘ ⇒∠AOC=180∘−2∠4 ...(iii) Now, in
ΔAEC,∠AEC+∠ECA+∠CAE=180∘ [by angle sum propertyof a triangle]
⇒∠AEC=180∘−(∠ECA+∠CAE) ⇒∠AEC=180∘−[(∠ECO+∠OCA)+∠CAO+∠OAE] =180∘−(∠6+∠4+∠4+∠5) [in
ΔOCD,∠6=∠ECO, because, angles opposite to equal sides are equal]
=180∘−(2∠4+∠5+∠6) =180∘−(180∘−∠AOC+∠7+∠6) [From Eq. (iii) and in
ΔAOB,∠5=∠7, as angles opposite to equal sides are equal]
=∠AOC−∠32−∠12 [From Eqs. (i) and (ii)]
=∠AOC−∠12−∠22−∠32+∠22 [adding and subtracting ∠22] =∠AOC−12(∠1+∠2+∠3)+∠82 [
∵∠2=∠8 (vertically opposite angles)]
=∠AOC−∠AOC2+∠DOB2 ⇒∠AEC=12(∠AOC+∠DOB) =12 [Angle sutended by arc CXA at the centre + angle subtended by arc DYB at the centre.]