Circle Theorems Identify a tangent to a circle Find angles in circles

TangentsAngles in a semicircleCyclic quadrilateralMajor and minor segmentsAngles at the circumference/centre Alternate segments

(1) Tangents A tangent to a circle is a straight line

that touches the circle at one point.

The point where the tangent touches the circle is called the point of contact.

Not tangents

TangentsDrawing a radius to any tangent to a circle always produces a 900 angle

The angles in circles

For these triangles:

One side is the radius of the circle

One side is a tangent to the circle

Since these two lines meet at right angles- these are right angle triangles!

Right angle Right

angle

Finding angles

32º

d

d

Once you can spot the right angles triangles- you can find missing angles

Remember: Angles in a triangle add up to 1800

900

900

a = 580

320 + 900 + a = 1800

a

32 + 90 + 58 = 180 d + d + 900 = 1800

45 + 45 + 90 = 180

d = 450

Angles in a semi circle

Use the diameter of a circle as the base of a triangle

Any triangle you make using this base will form a right angled triangle

(2) Two tangents

Two tangents will create Isosceles Triangles

You need to identify:

Equal pairs of lines (length)

Equal pairs of angles

Two tangents.

40º

Use “Isosceles triangle” to show 700 angles

700

700

200

Use the right angle (tangent to radius) to calculate the 200

Use the isosceles triangle to show 200 angle

200

Since angles in a triangle add up to 1800 show the 1400

1400

(3) Cyclic Quadrilateral

A

B

C

D

Any 4 points on a circle joined to form a quadrilateral

In any cyclic Quadrilateral opposite corners sum to 1800

So:

A + C = 1800

D + B = 1800

E

FG

HSo:

E + G = 1800

F + H = 1800

Cyclic Quadrilateral Questions

70º

50º 40º

1300

What are the missing angles?

1100 900

1400

900

(4) Major and minor segments

Providing all the angles are in the same segments they will be equal.

Major segment

Minor segment

Any circle can be divided into two unequal parts. These are called segments

The larger part is the major segment.

The smaller part is the minor segment.

The line that divides the circle is called a chord.

You can use your chord as the base of a triangle

Or…

ab

c

d

a = b = c = d

(5) Angles at the centre/circumferenceFrom a chord you can create a triangle at the centre or at the circumference of the circle.

At the centre

At the circumference

The angle at the centre is twice the size of the angle at the circumference

Questions

130º

a

b

300

a = 650

b = 600

1100

2500

C = 1250c

Putting your rules together

40º

Angles at a tangent = 900

500

An isosceles triangle has two equal angles

400

Angles in a triangle add up to 1800

1000

Angle at the centre is twice that at the circumference

500

Angles round a point add up to 3600

2600

(6) Alternate segments

A

A

B

B

Angles in alternate segments are equal