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

c

b

.

.

Amount of space

Angle

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

c

b

.

.

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.

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

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.

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

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’.

Now work on practical examples from your maths books.

a

b

c

L

d

∟

∟

.

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

y

4

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.

c

b

a

c

b

a

b

a

c

b

a

c

1

2

3

4

5

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.