An Introduction to Circle Theorems

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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.

THE CIRCLE

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

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

π‘₯ = 20 π‘œ ∠𝐴𝐡𝐢 = 58 π‘œ π‘₯ = 24 π‘œ ∠𝐢𝐡𝐴 = 70 π‘œ ∠𝐢𝐷𝐴 = 110 π‘œ ∠𝑇𝑄𝑅 = 62 π‘œ π‘₯ = 152 π‘œ , 𝑦 = 28 π‘œ ∠𝑅𝑂𝑄 = 106 π‘œ π‘₯ = 30 π‘œ , 𝑦 = 60 π‘œ

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 − π‘₯ π‘₯ First, label two we know are right-angles.

Label 90 − π‘₯ .

Internal angles in a triangle: 𝑦 + 90 + 90 − π‘₯ = 180 𝑦 + 180 − π‘₯ = 180 𝑦 − π‘₯ = 0 𝑦 = 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 𝑦 that touches another vertex.

2𝑦 The triangle is isosceles. If one π‘₯ angle is 2𝑦 , the other two are… 180 − 2𝑦 = 90 − 𝑦 2 Lastly on a line, we get π‘₯ + 90 − 𝑦 + 90 = 180 .

Simplifying this, we get π‘₯ = 𝑦 .

PROPERTY 6

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

π‘₯ π‘₯

ANSWERS

1.

∠𝑂𝐢𝐴 = 12 π‘œ b. ∠𝐴𝑂𝐢 = 156 π‘œ c. ∠𝐴𝐢𝐡 = 38 π‘œ 2. ∠𝑇𝑄𝑅 = 62 π‘œ 3. ∠𝐴𝐡𝐢 = 118 π‘œ , ∠𝐡𝐴𝐢 = 42 π‘œ

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