Question

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.

Answer 1

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.