Answer:
Option D is correct.
Step-by-step explanation:
Given C is the center of the circle.
we have to prove opposite angles of a quadrilateral inscribed in a circle are supplementary.
By the theorem, the angle subtended by an arc at the center is double the angle subtended at any point on the circumference of a circle.
i.e \(\angle ECG=2\angle EDG\\\\\angle EDG=\frac{1}{2}\angle ECG\thinspace \thinspace \thinspace i.e \thinspace \thinspace \thinspace \angle EDG=\frac{1}{2}\angle1\\\\\angle EFG=\frac{1}{2}\angle ECG\thinspace \thinspace \thinspace i.e \thinspace \thinspace \thinspace \angle EFG=\frac{1}{2}\angle2\\\\As, \angle1+\angle2=360^{\circ}\\\\\angle EDG + \angle EFG=\frac{1}{2}\angle1+\frac{1}{2}\angle2=\frac{1}{2}(\angle1+\angle2)=\frac{1}{2}\times 360=180^{\circ}\)
Hence, the fact which is used to prove the above is
\(\angle EDG=\frac{1}{2}\angle ECG\)
i.e option D is used.
