Q:

# Quadrilateral BCDE is inscribed inside a circle as shown below. Write a proof showing that angles C and E are supplementary. Circle A is shown with an inscribed quadrilateral labeled BCDE.

Accepted Solution

A:
Answer:The sum of angle C and E is 180°. So, C and E are supplementary angles.Step-by-step explanation:Given information: Quadrilateral BCDE is inscribed inside a circle.To prove : angles C and E are supplementary, i.e., ∠C+∠E=180°.Proof:BCDE is a cyclic quadrilateral.According to central angles theorem, the inscribed angle on a circle is half of its central angle.By using Central angle theorem$$\angle C=\frac{1}{2}\times arc (BED)$$$$2\angle C=arc (BED)$$                 .... (1)$$\angle E=\frac{1}{2}\times arc (BCD)$$$$2\angle E=arc (BCD)$$              ..... (2)The complete central angles of a circle is 360°.$$arc (BED)+arc (BCD)=360^{\circ}$$Using (1) and (2), we get$$2\angle C+2\angle E=360^{\circ}$$$$2(\angle C+\angle E)=360^{\circ}$$Divide both sides by 2.$$\angle C+\angle E=180^{\circ}$$The sum of angle C and E is 180°. So, C and E are supplementary angles.Hence proved.