# 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.

𝑥 = 20 𝑜

2.

∠𝐴𝐵𝐶 = 58 𝑜

3.

𝑥 = 24 𝑜

4.

∠𝐶𝐵𝐴 = 70 𝑜 ∠𝐶𝐷𝐴 = 110 𝑜

5.

∠𝑇𝑄𝑅 = 62 𝑜

6.

𝑥 = 152 𝑜 , 𝑦 = 28 𝑜

7.

∠𝑅𝑂𝑄 = 106 𝑜

8.

𝑥 = 30 𝑜 , 𝑦 = 60 𝑜

### PROPERTY 6

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

𝑦

90 − 𝑥 𝑥

First, label two we know are right-angles.

Label 90 − 𝑥 .

Internal angles in a triangle: 𝑦 + 90 + 90 − 𝑥 = 180 𝑦 + 180 − 𝑥 = 180 𝑦 − 𝑥 = 0 𝑦 = 𝑥

### 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 𝑦 that touches another vertex.

2𝑦

The triangle is isosceles. If one 𝑥 angle is

2𝑦

, the other two are…

180 − 2𝑦

= 90 − 𝑦

2

Lastly on a line, we get 𝑥 + 90 − 𝑦 + 90 = 180 .

Simplifying this, we get 𝑥 = 𝑦 .

### PROPERTY 6

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

𝑥 𝑥

1.

∠𝑂𝐶𝐴 = 12 𝑜 b.

∠𝐴𝑂𝐶 = 156 𝑜 c. ∠𝐴𝐶𝐵 = 38 𝑜

2.

∠𝑇𝑄𝑅 = 62 𝑜

3.

∠𝐴𝐵𝐶 = 118 𝑜 , ∠𝐵𝐴𝐶 = 42 𝑜