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Class 1: Angles
Class 2: Parallel lines and angles
Class 3: Quadrilaterals and types of triangles.
Class 4: Congruent triangles.
Class 5: Theorems 1- 4
Class 6: Theorems 5 & 6
Class 7: Theorem 7 and the three deductions.(Two classes is advised)
Class 8: Theorem 8
Class 9: Theorem 9
Class 10: Theorem 10
Select the class required then click
mouse key to view class.
Angles
An angle is formed when two lines meet. The size of the angle measures
the amount of space between the lines. In the diagram the lines ba and bc
are called the ‘arms’ of the angle, and the point ‘b’ at which they meet is
called the ‘vertex’ of the angle. An angle is denoted by the symbol .An
angle can be named in one of the three ways:
.
a
b
Angle
.
Amount of space
c
1. Three letters
.
a
b
.
c
Using three letters, with the centre at the
vertex. The angle is now referred to as :
abc or cba.
2. A number
.
a
b
1
.
c
Putting a number at the vertex of the angle. The
angle is now referred to as 1.
3. A capital letter
.
a
b
B
.
c
Putting a capital letter at the vertex of the angle.
The angle is now referred to as B.
Measuring angles
Right angle
A quarter of a revolution is called a right angle.
Therefore a right angle is 90.
We use the symbol
to denote a right angle.
Straight angle
A half a revolution or two right angles makes a
straight angle.
A straight angle is 180.
Acute, Obtuse and reflex Angles
Any angle that is less than 90 is called an
acute angle.
An angle that is greater than 90 but
less than 180 is called an obtuse
angle.
An angle greater than 180 is called a
reflex angle.
Angles on a straight line
Angles on a straight line add up to 180.
A + B = 180 .
A
Angles at a point
Angles at a point add up to 360.
B
A+ B + C + D + E = 360
C
A
D
E
B
Pairs of lines:
Intersecting
Consider the lines L and K :
L
.
p
K
L intersects K at p
written : L  K = {p}
Parallel lines
L
K
L is parallel to K
Written: LK
Parallel lines never meet and are usually indicated by arrows.
Parallel lines always remain the same distance apart.
Perpendicular
L
K
L is perpendicular to K
Written: L K
The symbol
is placed where two lines meet to show that they are
perpendicular
Now work on practical examples
in your maths book.
Parallel lines and Angles
C
1.Vertically opposite angles
When two straight lines cross, four
angles are formed. The two angles that
are opposite each other are called
vertically opposite angles. Thus a and b
are vertically opposite angles. So also
are the angles c and d.
From the above diagram:
A
B
D
A+ B = 180 …….. Straight angle
B + C = 180 ……... Straight angle
A + C = B + C ……… Now subtract c from both
sides
A=B
2. Corresponding Angles
The diagram below shows a line L and four other parallel lines intersecting it.
L
The line L intersects each of these lines.
All the highlighted angles are in corresponding positions.
These angles are known as corresponding angles.
If you measure these angles you will find that they are all equal.
In the given diagram the line L intersects two
parallel lines A and B. The highlighted angles
are equal because they are corresponding
angles.
.
L
.
The angles marked with are also
corresponding angles
.
Remember: When a third line intersects two parallel lines the
corresponding angles are equal.
A
B
3. Alternate angles
The diagram shows a line L intersecting two
parallel lines A and B.
The highlighted angles are between the parallel
lines and on alternate sides of the line L. These
shaded angles are called alternate angles and are
equal in size. Remember the Z shape.
L
A
B
Now work on practical examples
from your maths books.
Quadrilaterals
A quadrilateral is a four sided figure.
The four angles of a quadrilateral sum to 360.
b
c
a  + b  + c  + d  = 360
(This is because a quadrilateral can be divided up
into two triangles.)
Note: Opposite angles in a cyclic quadrilateral sum to 180.
a  + c  = 180
b  + d  = 180
a
d
The following are different types of
Quadrilaterals
Parallelogram
1. Opposite sides are parallel
3. Opposite angles are equal
.
..
..
.
2. Opposite sides are equal
4. Diagonals bisect each other
Rhombus
1. Opposite sides are parallel
2. All sides are equal
3. Opposite angles are equal
..
.
..
4. Diagonals bisect each other
5. Diagonal intersects at right
angles
.
6. Diagonals bisect opposite
angles
.. .
. ..
Rectangle
1. Opposite sides are parallel
2. Opposite sides are equal
3. All angles are right angles
4. Diagonals are equal and bisect each
other
Square
1. Opposite sides are parallel
4. Diagonals are equal and
bisect each other
2. All sides are equal
5. Diagonals intersect at
right angles
3. All angles are right angles
6. Diagonals bisect each
angle
..
..
..
..
Types of Triangles
Isosceles Triangle
Equilateral Triangle
.
.
.
3 equal sides
3 equal angles
Scalene triangle
3 unequal sides
3 unequal angles
a
b
2 sides equal
Base angles are equal
a = b
(base angles are the angles
opposite equal sides)
Now work on practical examples
from your maths books.
Congruent triangles
Congruent means identical. Two triangles are said to be congruent if they have
equal lengths of sides, equal angles, and equal areas. If placed on top of each other
they would cover each other exactly.
a
b
abc
c
xyz
y
x
z
The symbol for congruence is
. For two triangles to be congruent (identical), the
three sides and three angles of one triangle must be equal to the three sides and three
angles of the other triangle. The following are the ‘ tests for congruency’.
Case 1
Three sides of one triangle
=
SSS
Three sides
Three sides of the other triangle
Case 2
Two sides and the included angle of
one triangle
=
SAS
(side, angle, side)
Two sides and the included angle of
one triangle
Case 3
One side and two angles of
one triangle
=
ASA
(angle, side, angle)
Corresponding side and two
angles of one triangle
Case 4
A right angle, the hypotenuse and
the other side of one triangle
=
A right angle, the hypotenuse and
the other side of one triangle
RHS
(Right angle, hypotenuse, side)
Now do practical examples on
congruent triangles in your maths
book.
Theorem: Vertically opposite angles are equal in measure.
L
Given: Intersecting lines L and K, with vertically
opposite angles 1 and 2.
3
1 2
To prove :
1=2
Construction: Label angle 3
K
Proof: 1+3=180
Straight angle
2+3=180
Straight angle


1+3=3+2 .....Subtract 3 from both sides
1=2
Q.E.D.
Theorem: The measure of the three angles of a triangle sum to 180.
Given:
4
a
3
The triangle abc with 1,2 and 3.
1+2+3=180
To Prove:
5
Construction: Draw a line through a, Parallel to
bc. Label angles 4 and 5.
1
b
2
Proof: 1=4
c
and
2=5
Alternate angles
1+2+3=4+5+3
But 4+5+3=180
Straight angle
 1+2+3=180
Q.E.D.
Theorem: An exterior angle of a triangle equals the sum of the two interior opposite
angles in measure.
a
1
b
Given:
2
3
c
A triangle with interior opposite angles 1 and 2 and the exterior angle 3.
To prove:
1+ 2= 3
Construction:
Label angle 4
Proof:
4
1+ 2+ 4=180
3+ 4=180
Three angles in a triangle
Straight angle
 1+ 2+ 4= 3+ 4
 1+ 2= 3
Q.E.D.
Theorem: If to sides of a triangle are equal in measure, then the angles
opposite these sides are equal in measure.
a
3 4
b
Given:
To prove:
Construction:
Proof:
2
1
d
c
The triangle abc, with ab = ac and base angles 1 and 2.
1 = 2
Draw ad, the bisector of bac. Label angles 3 and 4.
Consider
abd and
acd:
given
ab = ac
construction
3 = 4
common
ad = ad
SAS

abd
acd
Corresponding angles
 1 = 2
Q.E.D.
Now work on practical examples
from your maths books.
Theorem: Opposite sides and opposite angles of a parallelgram are respectively
equal in measure.
a
d
4
1
3
b
2
c
Parallelogram abcd
Given:
ab = dc , ad = bc
abc = adc, bad = bcd
Construction: Join a to c. Label angles 1,2,3 and 4.
Consider
abc and
adc :
Proof:
1= 2 and 3= 4
To prove:

ac = ac
abc
adc
 ab = dcand ad = bc
And abc = adc
Similarly, bad = bcd
Alternate angles
common
ASA
Corresponding sides
Corresponding angles
Q.E.D.
Theorem:A diagonal bisects the area of a parallelogram.
a
d
b
Given:
To prove:
Proof:
c
Parallelogram abcd with diagonal [ac].
Area of abc = area of adc.
Consider

 area
abc and
ab = dc
ad = bc
ac = ac
abc
abc = area
adc:
Opposite sides
Opposite sides
Common
SSS
adc
adc
Q.E.D.
Now work on practical examples
from your maths books.
Theorem: The measure of the angle at the centre of the circle is twice the measure
of the angle at the circumference, standing on the same arc.
a
24
.
o
3
15
c
b
d
Circle, centre o, containing points a, b and c.
Given:
boc = 2 bac
To prove:
Construction: Join a to o and continue to d. Label angles 1,2,3,4 and 5.
Proof:
Consider
aob:
Exterior angle
1= 2 + 3
But
2 = 3
Base angles in an isosceles

1 = 2 2
Similarly, 5 = 2 4

1+ 5 = 2 2 + 2 4

1 + 5 = 2(2 + 4)
Q.E.D.
i.e.
boc = 2 bac
Deduction 1: All angles at the circumference on the same arc are equal in
measure.
d
a
2
1
.
o
3
c
b
To prove:
bac = bdc
Proof:
3 = 2 1
Angle at the centre is twice the angle on the
circumference (both on the arc bc)
3 = 2 2
Angle at the centre is twice the angle on
the circumference (both on arc bc)
 2 1 = 2 2
 1 = 2
i.e. bac = bdc
Q.E.D.
Deduction 2: An angle subtended by a diameter at the circumference is a right
angle.
a
1
b
.
o
c
2
To prove:
bac = 90
Proof:
2 = 2 1
But


i.e.
2 = 180
2 1 = 180
1 = 90
bac = 90
Angle at the centre is twice the angle on
the circumference (both on the arc bc)
straight line.
Q.E.D.
Deduction 3: The sum of the opposite angles of a cyclic quadrilateral is 180.
a
1
.
4o
3
d
b
2
c
To prove:
Proof:
bad + bcd = 180
3 = 2 1
4 = 2 2
Angle at the centre is twice the angle on
the circumference. (both on minor arc bd)
Angle at the centre is twice the angle on the
circumference. (Both on the major arc bd)

3 + 4 = 2 1 + 2 2
Angles at a point
But
3 + 4 = 360
 2 1 + 2 2 = 360
i.e. bad + bcd = 180
 1 + 2 = 180
Q.E.D.
Now work on practical examples
from your maths books.
Theorem: A line through the centre of a circle perpendicular to a chord bisects the
chord.
c
1 d
2
∟
.
∟
a
L
b
Given:
Circle, centre c, a line L containing c, chord [ab], such
that L ab and L ab = d.
To prove:
ad = bd
Construction: Label right angles 1 and 2.
Consider
cda and
cdb:
Proof:
1 = 2 = 90
ca = cb
cd = cd

cda
cdb
ad = bd

Given
Both radii
common
RHS
Corresponding sides
Q.E.D.
Now work on practical examples
from your maths books.
Theorem: If two triangles are equiangular, the lengths of the corresponding sides are in
proportion.
Two triangles with equal angles.
Given :
|ab|
|de|
To prove:
|ac|
|df|
=
=
|bc|
|ef|
a
On ab mark off ax equal in length to de.
On ac mark off ay equal to df and label
d the angles 4 and 5.
2
2
Construction:
Proof:
1 = 4
[xy] is parallel to [bc]
x 4
5 y
e 1
3 f
|ab|
|ax|
b 1
3 c
|ab|
|de|
=
=
|ac|
|ay|
|ac|
|df|
As xy is parallel to bc.
Similarly =
|bc|
|ef|
Q.E.D.
Now work on practical examples
from your maths books.
Theorem: In a right-angled triangle, the square of the length of the side
opposite to the right angle is equal to the sum of the squares of the other two
sides.
To prove that angle 1 is 90º
Proof:
b
a 2
But 5 = 90º => 3+ 4 = 90º
c
c
1
b
3
b
a
4
a
=> 3+ 2 = 90º ……Since 2 = 4
Now 1+ 2+ 3 = 180º ……Straight line
c
c
5
3+ 4+ 5 = 180º ……Angles in a triangle
a
b
=> 1 = 180º - ( 3+ 2 )
=> 1 = 180º - ( 90º ) ……Since 3+ 2 already
proved to be 90º
=> 1 = 90º
Q.E.D.
Now work on practical examples
from your maths books.