Question

In a given figure, Point O is the centre of the circle. show that ANGLEAOC=angleAFC+angleAEC

Answer 1

Let Angle AOC be x, Angle AFC be y and And angle AEC = z.
Therefore we need to prove x = y + z

Now we have,

x/2 = Angle ABC = Angle ADC (angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at the remaining part of the circle) (Angles in the same segment are equal).

In EBFD,

Angle EBF + Angle EDF + Angle BFD + Angle BED = 360°

(180°- x/2) + (180°-x/2) + y + z = 360° (Angle BFD = Angle AFC because the are vertically opposite)

360° - x + y + z = 360°

y + z = x

Proved.