Question

The figure below shows Quadrilateral CDBE inscribed in a circle with center A.

Answer 1

The paragraph proof with missing statement proves that its opposite angles are supplementary. Which statement can be used to fill in the blank space?

Given that CDBE is a quadrilateral inscribed in a circle with center A, ∡DCE and ∡DBE are inscribed angles. Since the measure of an inscribed angle is one-half the measure of its intercepted arc, __________________________________
and ∡DBE is half of arc DCE. Since arc DBE and arc DCE add up to the whole circle, or 360 degrees, the total of ∡DCE and ∡DBE must be half of 360, or 180 degrees. Therefore, they are supplementary. By the definition of a quadrilateral, all interior angles must add to 360. Therefore, the other two angles must also be supplementary.

Answer 2

∡DCE is half of arc DBE

∡EDC is half of arc ECD

∡CBE is half of arc DCE

∡DBE is half of arc DCE