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[tex]\angle C=\frac{1}{2}\times arc (BED)[/tex][tex]2\angle C=arc (BED)[/tex] .... (1)[tex]\angle E=\frac{1}{2}\times arc (BCD)[/tex][tex]2\angle E=arc (BCD)[/tex] ..... (2)The complete central angles of a circle is 360°.[tex]arc (BED)+arc (BCD)=360^{\circ}[/tex]Using (1) and (2), we get[tex]2\angle C+2\angle E=360^{\circ}[/tex][tex]2(\angle C+\angle E)=360^{\circ}[/tex]Divide both sides by 2.[tex]\angle C+\angle E=180^{\circ}[/tex]The sum of angle C and E is 180°. So, C and E are supplementary angles.Hence proved.