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 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 8: Theorem 8

Class 9: Theorem 9

Class 10: Theorem 10

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Select the class required then clickmouse key to view class.

Class 7: Theorem 7 and the three deductions.(Two classes is advised)

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

cb

.

.Amount of spaceAngle

11. Three letters

a

bc

.

.

Using three letters, with the centre at the vertex. The angle is now referred to as :

abc or cba.

2. A number

cb

.

.1

a

Putting a number at the vertex of the angle. The angle is now referred to as 1.

3. A capital letter

b

.

.B

a

c

Putting a capital letter at the vertex of the angle.

The angle is now referred to as B.

Right angle

A quarter of a revolution is called a right angle.

Therefore a right angle is 90.

Straight angle

A half a revolution or two right angles makes a straight angle.

A straight angle is 180.

Measuring angles

We use the symbol to denote a right angle.

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 .

Angles at a point

Angles at a point add up to 360.

A+ B + C + D + E = 360

A B

A

B

D

E

C

Pairs of lines:

Consider the lines L and K :

.p

L

K

L intersects K at p written : L K = {p}

Intersecting

Parallel lines

L

K

L is parallel to K

Written: LK

Parallel lines never meet and are usually indicated by arrows.

Parallel lines always remain the same distance apart.

Perpendicular

L is perpendicular to K

Written: L K

The symbol is placed where two lines meet to show that they are perpendicular

L

K

Now work on practical examples in your maths book.

Parallel lines and Angles

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:

AB

C

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.

The line L intersects each of these lines.

L

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.

The angles marked with are also corresponding angles

.A

B

L

.

.

Remember: When a third line intersects two parallel lines the corresponding angles are equal.

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.

A

B

L

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

a

c

d

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

The following are different types of Quadrilaterals

..

...

.

Parallelogram

1. Opposite sides are parallel 2. Opposite sides are equal

3. Opposite angles 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 2. All sides are equal 3. All angles are right angles

4. Diagonals are equal and bisect each other

5. Diagonals intersect at right angles

6. Diagonals bisect each angle

..

....

..

Types of Triangles

Equilateral Triangle

.

.

.3 equal sides

3 equal angles

Isosceles Triangle

a b

2 sides equal

Base angles are equal a = b

(base angles are the angles opposite equal sides)

Scalene triangle

3 unequal sides 3 unequal angles

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.

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’.

a

b c

x

y z

abc xyz

Case 1

= Three sides of the other triangleThree sides of one triangle

SSS

Three sides

Case 2

Two sides and the included angle of one triangle

Two sides and the included angle of one triangle

=

SAS

(side, angle, side)

Case 3

One side and two angles of one triangle

Corresponding side and two angles of one triangle

=

ASA

(angle, side, angle)

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.

Given:

To prove :

Construction:

Proof: Straight angle

Straight angle

1=2

Label angle 3

1=2

Intersecting lines L and K, with vertically opposite angles 1 and 2.

1+3=180

2+3=180

Q.E.D.

L

K

1 2

1+3=3+2 .....Subtract 3 from both sides

3

Theorem: The measure of the three angles of a triangle sum to 180.

Given:

To Prove: 1+2+3=180

Construction:

Proof:

1=4 and 2=5 Alternate angles

1+2+3=4+5+3

But 4+5+3=180 Straight angle

1+2+3=180

The triangle abc with 1,2 and 3.

4 5

a

b c1 2

3

Q.E.D.

Draw a line through a, Parallel to

bc. Label angles 4 and 5.

Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure.

Given: A triangle with interior opposite angles 1 and 2 and the exterior angle 3.

To prove: 1+ 2= 3

Construction: Label angle 4

Proof: 1+ 2+ 4=180

3+ 4=180

Three angles in a triangle

1+ 2+ 4= 3+ 4

Straight angle

1+ 2= 3

a

b c3

1

2 4

Q.E.D.

a

b c1 2

Theorem: If to sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure.

Given: The triangle abc, with ab = ac and base angles 1 and 2.

To prove: 1 = 2Construction: Draw ad, the bisector of bac. Label angles 3 and 4. Proof:

ab = ac given

3 = 4 construction

ad = ad common

SAS

1 = 2 Corresponding angles

d

3 4

abd acd

Consider abd and acd:

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.

Given: Parallelogram abcd

a

b c

d

To prove:

Construction: Join a to c. Label angles 1,2,3 and 4.Proof:

1= 2 and 3= 4 Alternate angles

ac = ac common

ASA

ab = dcand ad = bc Corresponding sides

And abc = adc Corresponding angles

Similarly, bad = bcd

1

23

4

ab = dc , ad = bcabc = adc, bad = bcd

Consider abc and adc :

abc adc

Q.E.D.

Theorem:A diagonal bisects the area of a parallelogram.

a

b c

d

Given: Parallelogram abcd with diagonal [ac].

To prove: Area of abc = area of adc.

Proof:

ab = dc Opposite sides

ad = bc Opposite sides

ac = ac Common

SSS

Consider abc and adc:

abc adc

area abc = area adcQ.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.

Given: Circle, centre o, containing points a, b and c.To prove: boc = 2 bac

Construction: Join a to o and continue to d. Label angles 1,2,3,4 and 5.Proof:

d

a

b c

.o

1= 2 + 3 Exterior angleBut 2 = 3 1 = 2 2

Similarly, 5 = 2 4 1+ 5 = 2 2 + 2 4

1 + 5 = 2(2 + 4) i.e. boc = 2 bac

1

2

3

4

5

Consider aob:

Q.E.D.

Base angles in an isosceles

Deduction 1: All angles at the circumference on the same arc are equal in measure.

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.

a

bc

d

3

1 2

.o

Deduction 2: An angle subtended by a diameter at the circumference is a right angle.

To prove: bac = 90

Proof: 2 = 2 1 Angle at the centre is twice the angle on the circumference (both on the arc

bc) straight line. But 2 = 180 2 1 = 180 1 = 90 i.e. bac = 90

Q.E.D.

a

b co

2

1

.

Deduction 3: The sum of the opposite angles of a cyclic quadrilateral is 180.

To prove: bad + bcd = 180 3 = 2 1Proof: Angle at the centre is twice the angle on

the circumference. (both on minor arc bd)

4 = 2 2 Angle at the centre is twice the angle on the circumference. (Both on the major arc bd)

3 + 4 = 2 1 + 2 2But 3 + 4 = 360 Angles at a point

2 1 + 2 2 = 360 1 + 2 = 180 i.e. bad + bcd = 180

Q.E.D.

.o4

a

b

c

d3

2

1

Now work on practical examples from your maths books.

a

b

c Ld∟∟.

Theorem: A line through the centre of a circle perpendicular to a chord bisects the chord.

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.

Proof:1 = 2 = 90 Given

ca = cb Both radii cd = cd common

R H S Corresponding sides

Consider cda and cdb:

cda cdb ad = bd

1

2

Q.E.D.

Now work on practical examples from your maths books.

b

a

c

d

e f

2

1

2

3

1 3

Theorem: If two triangles are equiangular, the lengths of the corresponding sides are in proportion.

Given : Two triangles with equal angles.

To prove: |df|

|ac|=

|de|

|ab|

|ef|

|bc|=

Construction: On ab mark off ax equal in length to de. On ac mark off ay equal to df and label the angles 4 and 5.

Proof: 1 = 4

[xy] is parallel to [bc]

|ay|

|ac|=

|ax|

|ab|As xy is parallel to bc.

|df|

|ac|=

|de|

|ab|Similarly

|ef|

|bc|=

x y4 5

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.

Q.E.D.

cb

a

c

b

a

b

a c

b

a

c

1

2

3

45

To prove that angle 1 is 90º

Proof:3+ 4+ 5 = 180º ……Angles in a triangle

But 5 = 90º => 3+ 4 = 90º

=> 3+ 2 = 90º ……Since 2 = 4

Now 1+ 2+ 3 = 180º ……Straight line

=> 1 = 180º - ( 3+ 2 )

=> 1 = 180º - ( 90º ) ……Since 3+ 2 already proved to be 90º

=> 1 = 90º

Now work on practical examples from your maths books.