Properties of Circle 

 

 

 

Contents

 

Symmetrical Properties 

 

Angle Properties

 

Cyclic Quadrilaterals

 

Problems on angle properties

 

Tangents from External Point

 

Alternate Angle Segment

 

Quiz

 

 

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Problems on Angle Properties

The following examples involve the angle properties of circles.

Example 1

In the figure, CAD and CBE are straight lines. If CA is the diameter of the circle ABC, explain why ÐADE is a right angle.

x1 = 90° (rt.Ð in a semicircle)

x1 = x2 (ext. Ð of a cyclic quad.)
x2 = 90°
ÐADE = 90°

Example 2
 
In the figure, PAQ and RAS are straight lines. Show that ÐPXR = ÐQYS.
 
x1 = x2 (ext. Ð of a cyclic quad.)
x2 = x3 (ext.Ð of a cyclic quad.)
ÐPXR = ÐQYR

 
Example 3
 
ABCD is a cyclic quadrilateral and BC = CD. Show that AC bisects ÐBAD.
 
x1 = x2 (isos.△) 
x1 = x4 ( Ðs in same segment)
x2 = x3 ( Ðs in same segment)
x3 = x4
 
Hence, AC bisects ÐBAD.