Geometry 6

Level 1

Parts of a circle

Why is this triangle isosceles?

Two sides are radii

Find the value of x

35 x

Find the value of x

60

x

Find the value of x

x

Find the value of x

x20

Reminder: Exterior angle equals the sum of the opposite interior angles.

Find the value of x.

70 50

x

Let’s investigate

We now have two isosceles triangles

In each case their base angles are equal.

The exterior angle is twice the opposite internal angle

The exterior angle is twice the opposite internal angle

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

at centre

2x

x

Find the value of x

x

50

Find the value of x

x

160

Find the value of x

x

90

Find the value of x

x

60

Another way of looking at this.

2x

x

Find the value of x.

240

x

Find the value of x.

110

x

Find the value of x.

135

x

Notice if AC is a diameter

2x

x

x

Find the value of x.

55x

Find the value of x.

80

x

Find the value of x.

140

x

Find the value of x.

50

x

Find the value of x.

74

x 62

Find the value of x.

74x

62

Find the value of x

x75

Find the value of y

y

75 75

Find the value of z

z

75 7575

Angles on the same arc are the same.

Angles on the same arc are the same.

Angles same arc

Why is it important?

Find A, B and C

If BC is a diameter…

x

x

2x

If BC is a diameter…

x

x

2x

y y

2y

If BC is a diameter…

x

x

2x

y y

2y

If BC is a diameter…

x

x

2x

y y

2y

in a semi-circle

If AC is a diameter, find the value of x.

50 x

If AC is a diameter, find the value of x.

50

x

If AC is a diameter, find the value of x

40

x

2x

x

360 - 2x

180 - x

x

180 - x

Cyclic Quadrilateral

• Opposite angles in a cyclic quadrilateral sum to 180 degrees.Web animation

Cyclic quadrilateral

Opposite angles add up to 180 in a cyclic quadrilateral

Find the value of x

x120

Find the value of angles A and B.

Find the value of angles A and B.

110

A B

PQ is a diameter. RPQ = 30.Find RSP.

Chord

If a diameter cuts a chord in half, it form a right angle.

Tangent

Tgt rad

A radius is always perpendicular to a tangent at the point of contact.

Find the values of x, y and z

72x

yz

Find the values of x, y and z

72x

yz

Find the values of x, y and z

72x

yz

Tgt. — rad

In semi-circle

’s ∆

Notice z

72x

yz

Notice z

72x

yz

z

Exterior angle at a tangent always equals the

opposite interior angle.

72

z

Find the values of angles x and y.

62

x

y

71

Find the angle ACB.

Find the value of x.

x

128

x

42

Find the value of x.

x = 52

128

x =69

42

Find the value of x

Find, in terms of x, ADB, AOD. ABD

ADB = 2 x, AOD = 180 - 6 x,

ABD = 90 - 3 x

If the angle PCQ = 72, what is the reflex angle POQ ?

252

Find the angle ABC

112

a) Calculate the angle between the tangent and the line AC.

b) Calculate the angle ABC.

Determine the angle QAC in terms of x. Calculate the value of x.

.

Determine the angle QAC. Calculate the value of x.

.

75

75

a) Work out the following angles:i) ACD ii) BCA iii) CDAb) Show that the triangle EBA is isosceles.

ABC and ADE are straight lines. The diameter of the larger circle is CE.

Express in terms of x, the following angles:a) ABD b) BDC c) BAD