120 o

120o 95

o

Angles round a pointAdd up to 360

o

115o

Two angles making astraight line add to 180

o

angles opposite each otherat a cross are equal.

34o 34

o

3 angles in a triangle ALWAYSadd up to 180

o.

50o

40o

65o

90o146o

146o

145o

RevisionRevisionAngle PropertiesAngle Properties

Two angles in an isoscelesare equal

ALL angles in an equilateral triangle are 60

o

d = 115oa

o

co

bo

go

fo

ho

eo

is corresponding to d and must be 115

o

is opposite to d and must be 115o

is must be 65o (straight line)

is alternate to c and must also be 65o

hbce

RevisionRevisionAngle PropertiesAngle Properties

* *

60º

60º

60º

Key Point for Angles in a Semi-circle

Angles in a Semi-CircleAngles in a Semi-Circle

A triangle ABC within a semicircle with base the length of the diameter

will ALWAYS be right angled at P on the circumference.

Remember - Angles in any triangle sum to 180o

B

A C

44

Hints

Example 1 : Sketch diagram and find all the missing angles.

70o

Look for right angle triangles43o

20o

47o

Remember ! Angles in any triangle

sum to 180o


Example 2 : Sketch the diagram.


C

A B

D

25o

60o

(a) Right down two right angle triangles

(a) Calculate all missing angles.

E

66

Angles in a Semi-CircleAngles in a Semi-CirclePythagoras Theorem

We have been interested in right angled triangles withina semi-circle. Since they are right angled we can usePythagoras Theorem to calculate lengths.

P

A B

4cm3cm

a cm

Example 1 : Calculate the value of a2 2 2c a b 2 2 24 3c 2 16 9c

25 5a cm

5cm

2 25a

77

Angles in a Semi-CircleAngles in a Semi-CirclePythagoras Theorem

Y

X Zcm

8cm

10 cm

Example 2 : Calculate the length of XY2 2 2a c b 2 2 210 8a 2 100 64a

36 6a cm

2 36a

6cm

A A tangent linetangent line is a line that is a line that touches a circle at touches a circle at only one point.only one point.

Which of theWhich of thelines are lines are tangent to tangent to the circle?the circle?


Tangent Line

The radius of the circle that touches the tangent The radius of the circle that touches the tangent line is called the line is called the point of contact radius.point of contact radius.

Special PropertySpecial Property

The point of contact radiusThe point of contact radiusis always perpendicular is always perpendicular

(right-angled)(right-angled)to the tangent line.to the tangent line.


Tangent Line

Q.Q.Sketch the diagram and find the size of the marked Sketch the diagram and find the size of the marked angle in this diagramangle in this diagram

A

B

8

10

C


Tangent Line

46º

aº

Angles in triangle = 180º

So a = 180 – (90 + 46) = 44º


Tangent Line

AB

C

36º

bº

find the missing angles b, c, d and e

Angles in triangle = 180º

So b = 180 – (90 + 36) = 54º

A

B

Cdº

eº

30º


Tangent Line

•Since isosceles triangle, angle CAD MUST = angle CDA, Therefore angle CAD = 30º

D

•Angles in triangle = 180º So d = 180 – (30 + 30) = 120º

•Straight line at point C so angle BCA = 180 – 120 = 60º

•Angles in triangle = 180º So e = 180 – (90 + 60) = 30º

120 o

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