# An Introduction to Circle Theorems ### 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

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𝑥 = 20 𝑜 ∠𝐴𝐵𝐶 = 58 𝑜 𝑥 = 24 𝑜 ∠𝐶𝐵𝐴 = 70 𝑜 ∠𝐶𝐷𝐴 = 110 𝑜 ∠𝑇𝑄𝑅 = 62 𝑜 𝑥 = 152 𝑜 , 𝑦 = 28 𝑜 ∠𝑅𝑂𝑄 = 106 𝑜 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