We have studied that a quadrilateral is a 4 – sided polygon with 4 angles and 4 vertices. For more details, you can consult the article “Quadrilaterals” in the “Polygon” section.

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In geometry exams, examiners make the questions complex by inscribing a figure inside another figure and ask you to find the missing angle, length, or area. One example from the previous article shows how an inscribed triangle inside a circle makes two chords and follows certain theorems.

This article will discuss what a quadrilateral inscribed in a circle is and the inscribed quadrilateral theorem.

What is a Quadrilateral Inscribed in a Circle?

In geometry, a quadrilateral inscribed in a circle, also known as a cyclic quadrilateral or chordal quadrilateral, is a quadrilateral with four vertices on the circumference of a circle. In a quadrilateral inscribed circle, the four sides of the quadrilateral are the chords of the circle.

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Solution

Sum of opposite angles = 180 o

(y + 2) o + (y – 2) o = 180 o

Simplify.

y + 2 + y – 2 =180 o

2y = 180 o

Divide by 2 on both sides to get,

y = 90 o

On substitution,

(y + 2) o ⇒ 92 o

(y – 2) o ⇒ 88 o

Similarly,

(3x – 2) o = (7x + 2) o

3x – 2 + 7x + 2 = 180 o

10x =180 o

Divide by 10 on both sides,

x = 18 o

Substitute.

(3x – 2) o ⇒ 52 o

(7x + 2) o ⇒ 128o

Practice Questions

1. All polygons can be inscribed in a circle.

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A. Yes

B. No

2. Inscribed quadrilaterals are also called _____

A. Trapped quadrilaterals

B. Cyclic quadrilaterals

C. Tangential quadrilaterals

D. None of these.

3. A quadrilateral is inscribed in a circle if and only if the opposite angles are ______