DAY 131 INTERIOR ANGLES OF CYCLIC QUADRILATERAL INTRODUCTION

DAY 131 – INTERIOR ANGLES OF CYCLIC QUADRILATERAL

INTRODUCTION We already know that a quadrilateral is a closed plane figure with four straight sides. At times, we may have a quadrilateral circumscribed in a circle. In such case, the sides of a quadrilateral would be the chords of the circle. In this lesson, we will prove properties of a quadrilateral with its vertices lying on the circumference of a single circle.

VOCABULARY Cyclic quadrilateral This is a quadrilateral in which its all vertices lie on the circumference of one circle.

A quadrilateral in which its vertices lie on the circumference of one circle such that its sides are chords of that circle is said to be inscribed in that circle. Such a quadrilateral is called a cyclic or inscribed quadrilateral. The circle involved is called the circumscribed circle. Consider the cyclic quadrilateral JKLM below. K J L M

We will prove each of these properties. However, to prove these properties, we need to remind ourselves of the following concepts: 1. A perpendicular bisector of a chord passes through the center of the circle and if chord is perpendicular to the radius of a circle, then the radius is a perpendicular bisector to that chord. 2. The size of an inscribed angle is half the size of a central angle subtended by the same arc.

The perpendicular bisectors to the sides of a cyclic quadrilateral meet at one point. Proof We want to prove that the perpendicular bisectors of sides JK, KM, LM and MJ will meet at a single point. Each side of a quadrilateral is a chord to the circumscribed circle. Since the perpendicular bisector of a chord passes through the center of the circle, the perpendicular bisectors to each side will pass through the center of the circle. Therefore the perpendicular bisectors will meet at the center of the circumscribed circle.

K L J O M

K L J O M

B C A D

O K J L M

T U S W V

HOMEWORK T U S V

ANSWERS TO HOMEWORK 17

THE END