Question

Quadrilateral BCDE is inscribed inside a circle as shown below. Write a proof showing that angles C and E are supplementary.

Answer 1

Each of those angles are Inscribed angles. A useful fact is that the measure of the intercepted arc of an inscribed angle is twice the measure of the angle.

The intercepted arc of ∠C is arc BED. The intercepted arc of ∠E is arc BCD.

m(arc BED) + m(arc BCD) = 2( m∠C + m∠E)

The two arcs combined make u p the entire circle, so the sum of their measures is 360.

360 = 2(m∠C + m∠E)
180 = m∠C + m∠E

The angles are supplementary.