# An Introduction to Circle Theorems

### PART 2

Slideshow 47, Mathematics

Mr Richard Sasaki, Room 307

OBJECTIVES

• Review circle properties

• Learn some properties regarding angles and circles

### THE CIRCLE

Let’s learn and recall some basic circle property names.

### CIRCLE PROPERTIES

So far we know…

A tangent is always 90 o to its radius.

a

An angle at the edge is half the angle at the centre.

2a a b

For a cyclic quadrilateral, opposite angles add up to 180 o .

### PROPERTY 4

For a triangle with the diameter of the circle as an edge, the opposite angle touching the circle’s edge is a right-angle.

180 o

You should have showed this before on the worksheet!

We can see this as a quadrilateral with an 180 o angle.

### PROPERTY 5

In circles, angles in the same segment are equal to one another.

2a a a

We know the central angle is twice the angle at the edge.

The position at the edge makes no difference.

So the angles at the edges are equal.

### PROPERTY 5

In circles, angles in the same segment are equal to one another.

a a

Be careful, nothing here is congruent! They are similar though.

a a

1.

π₯ = 20 π

2.

∠π΄π΅πΆ = 58 π

3.

π₯ = 24 π

4.

∠πΆπ΅π΄ = 70 π ∠πΆπ·π΄ = 110 π

5.

∠πππ = 62 π

6.

π₯ = 152 π , π¦ = 28 π

7.

∠πππ = 106 π

8.

π₯ = 30 π , π¦ = 60 π

### PROPERTY 6

The last we’ll learn. An angle between the tangent and a chord is equal to the angle in the alternate segment.

π¦

90 − π₯ π₯

First, label two we know are right-angles.

Label 90 − π₯ .

Internal angles in a triangle: π¦ + 90 + 90 − π₯ = 180 π¦ + 180 − π₯ = 180 π¦ − π₯ = 0 π¦ = π₯

### PROPERTY 6

Actually, for this property to work, the chord doesn’t need to pass through the origin.

First add two radii. One that touches the tangent, the other π¦ that touches another vertex.

2π¦

The triangle is isosceles. If one π₯ angle is

2π¦

, the other two are…

180 − 2π¦

= 90 − π¦

2

Lastly on a line, we get π₯ + 90 − π¦ + 90 = 180 .

Simplifying this, we get π₯ = π¦ .

### PROPERTY 6

An angle between the tangent and a chord is equal to the angle in the alternate segment.

π₯ π₯

1.

∠ππΆπ΄ = 12 π b.

∠π΄ππΆ = 156 π c. ∠π΄πΆπ΅ = 38 π

2.

∠πππ = 62 π

3.

∠π΄π΅πΆ = 118 π , ∠π΅π΄πΆ = 42 π