Angles and Construction

SOOFIA INTERNATIONAL SCHOOL

LOWER SECONDARY
MATHEMATICS
GEOMETRY AND MEASURE

LOWER SECONDARY MATHEMATICS 1

 Angles and Construction

Angles and Construction

1.1. A sum of 360

At the end of section, the learner should be able to:

 Understand that the sum of angles at a point is 360°.
 Show and use the fact that angles of all quadrilaterals adds up
to 360°

An angle is the amount of turn between two lines that meet each other. When two lines
meet at a point, an angle is formed. Angles are measured in degrees (°).
A complete turn is equal to 360°. Half of a turn which forms a straight line is equal to
180°.
There are different types of angles

Type of angle Description Diagram

Acute angle Less than 90°

0 ≤ 𝑥° < 90

Right angle Exactly 90°

Obtuse angle More than 90 ° and less

than 180°

90 ≤ 𝑥° < 180

LOWER SECONDARY MATHEMATICS 2

 Angles and Construction

Reflex angle More than 180 ° and less

than 360°

180 ≤ 𝑥° < 360

Angles on a straight line

Angles on a straight line always add up to 180°. A straight line is considered to be half
of a full turn or full rotation.

This is a straight angle

shown by half of a full turn.

Angles at a point
The sum of angles at a point is always 360°. This is a full rotation.

Complete angle

360°

Steps on how to find missing angles.

Step 1: Add all the given angles together.

Step 2: Subtract the value from the sum of all the angles.

Step 3: Form and solve an equation where necessary.

LOWER SECONDARY MATHEMATICS 3

 Angles and Construction

Worked example 1.1

1.
a) Work out the size of the angle that has a letter.

Add all the given angles

55° + 30° = 85°
Subtract from 180°
180° - 85° = 95°

d = 95°

b) Work out the size of the angle that has a letter

Add all the given angles

52° + 23° = 75°
Subtract from 180°
180° - 75° = 105°

b = 105°

LOWER SECONDARY MATHEMATICS 4

 Angles and Construction

2. Calculate the size of each angle that has a letter

a)

Add all the given angles

97° + 73° + 140° = 310°
Subtract from 360°
360° - 310° = 50°

e = 50°

b)

Add all the given angles

100° + 75° + 80° = 255°
Subtract from 360°
360° - 255° = 105°

𝒙 = 105°

LOWER SECONDARY MATHEMATICS 5

 Angles and Construction

End of section questions

1. Work out the size of angle 𝒂

2. Find the size of angle marked 𝒙

3. Find the size of the angle marked below.

LOWER SECONDARY MATHEMATICS 6

 Parallel, Perpendicular, Transversal and Intersecting lines.

1.2. Parallel, Perpendicular, Transversal and Intersecting lines.

At the end of section, the learner should be able to:

 Define parallel lines, perpendicular, transversal and
intersecting lines.
 Identify whether a pair of lines is parallel or perpendicular.
 Understand that we can denote parallel lines by a pair of
arrows.

Perpendicular lines are two lines that intersect at a right angle (90° angle).

90°

The two lines intersect at a right

angle. The angle formed between
these two lines is 90°

Parallel lines are straight lines that never intersect and stay the same distance apart.
Parallel lines are never ending, which means they continue in the same direction
forever.

These two lines do not

intersect and the distance
between them never changes

LOWER SECONDARY MATHEMATICS 7

 Parallel, Perpendicular, Transversal and Intersecting lines.

When you combine segments of parallel lines, you can form various polygons as seen
below.

Transversal lines and Parallel lines

A transversal is defined as a line that crosses two lines at two distinct points.

The blue line is a transversal line

A transversal intersection with two lines produces different types of angles in pairs,
such as vertically opposite angles, corresponding angles, alternate angles and
consecutive interior angles.

Intersecting lines
Intersect means to meet. If we have two lines that meet at a point, they are called
intersecting lines.
P Q
R
Lines P and Q intersect each other at
point R (point of intersection).

LOWER SECONDARY MATHEMATICS 8

 Parallel, Perpendicular, Transversal and Intersecting lines.

Two intersecting lines always form a plane and they create pairs of vertically opposite
angles. Remember, vertically opposite angles are always equal.

a=c
b=d

Vertically opposite angles

These are angles that are formed when two lines meet each other at a point. Vertically
opposite angles are always equal to each other.

∠ Q and ∠ U are ∠ S and ∠ T are

equal equal

∠ V and ∠ Z are ∠ X and ∠ Y are

equal equal

LOWER SECONDARY MATHEMATICS 9

 Parallel, Perpendicular, Transversal and Intersecting lines.

Corresponding angles

In the diagram, the following angles are

called corresponding angles:
1 and 5
3 and 7 Corresponding angles
2 and 6 are equal

4 and 8

∠ S and ∠ X are ∠ Q and ∠ V are

equal equal

∠ T and ∠ Y are ∠ U and ∠ Z are

equal equal

LOWER SECONDARY MATHEMATICS 10

 Parallel, Perpendicular, Transversal and Intersecting lines.

Alternate angles

In the diagram, the following angles are

called alternate angles:
Alternate interior angles
3 and 6 Alternate angles are
always equal
4 and 5
Alternate exterior angles
1 and 8
2 and 7

∠ T and ∠ X are ∠ V and ∠ U are

equal (interior) equal (interior)

∠ Q and ∠ Z are
∠ S and ∠ Y are
equal (exterior)
equal (exterior)

LOWER SECONDARY MATHEMATICS 11

 Parallel, Perpendicular, Transversal and Intersecting lines.

Consecutive interior angles

These are the pairs of angles that are formed when a transversal lines cuts through two
parallel lines. The consecutive interior angles are formed within the inner region of two
parallel lines.

Consecutive interior angles always

add up to 180°, only if the two lines
being crossed by a transversal line
are parallel to each other.
In the diagram, the following angles
are called consecutive interior angles:
3 and 5
4 and 6

g + h = 180° i + j = 180°

k + l = 180° m + n = 180°

LOWER SECONDARY MATHEMATICS 12

 Parallel, Perpendicular, Transversal and Intersecting lines.

Worked example 1.2

1. On the diagram, AB is parallel to CD.

d e
A B
f g

C h i D
j k

Which angle is vertically opposite to ∠ j? ∠i

Which angle is vertically opposite to ∠ f? ∠e

Which angle is corresponding to ∠ d? ∠h

Which angle is alternate to ∠ f? ∠i

Which angle is corresponding to ∠ k? ∠g

Which angle is alternate to ∠ g? ∠j

LOWER SECONDARY MATHEMATICS 13

 Parallel, Perpendicular, Transversal and Intersecting lines.

2. On the diagram, AB is parallel to CD.

A B

C D

Find the sizes of angles b, c, d, e and f.

a) ∠ d = 60° (vertically opposite to ∠ a)

b) ∠ b = 120°

180° - 60° = 120° (angles on a straight line add up to 180°)

c) ∠ c = 120° (vertically opposite to ∠ b)

d) ∠ e = 60° ( ∠ c and ∠ e are consecutive interior angles. They must add up

to 180°)

OR

(∠ a and ∠ e are corresponding angles. They are equal)

e) ∠ f = 120° (∠ c and ∠ f are alternate interior angles. They are equal)

LOWER SECONDARY MATHEMATICS 14

 Parallel, Perpendicular, Transversal and Intersecting lines.

End of section questions

1. AB and CD are parallel lines.

a) Find the value of the angle 𝑥 .

b) Find the value of the angle 𝑦.

2. AB and CD are parallel lines.

a) Name the letter of one angle that is equal to angle 110°.

b) Which letters represent alternate angles?

c) Which letters represent corresponding angles?

LOWER SECONDARY MATHEMATICS 15

 The exterior angle of a triangle

1.3. The exterior angle of a triangle

At the end of section, the learner should be able to:

 Know the definition of exterior angle of a triangle.
 Know that exterior angles of a triangle sum up to 360°.
 Use the fact that exterior angle of a triangle is equal to the
sum of the two interior opposite angles.

The concept of the exterior angle of a triangle is quite easy to remember. To get the
exterior angle, you can extend any one side of the triangle.

Angles 𝒙 and 𝒚 will be interior

angles opposite to angle d

Angle z is adjacent
to angle e

You need to understand the relation between the triangle’s exterior and interior angles.

In the triangle above, angle e will be the sum of angle 𝒙 and 𝒚.

The exterior angle of a triangle

will equal the sum of the interior
e=𝑥+𝑦 opposite angles.

LOWER SECONDARY MATHEMATICS 16

 The exterior angle of a triangle

Worked example 1.4

Calculate the sizes of the labelled angles.

a)

125° = 𝑥 + 55°

𝑥 = 125° - 55°= 70°

b)

𝑥 = 90° + 50°
= 140°

c)

120° = 𝑥 + 60°

𝑥 = 120° - 60°= 60°

LOWER SECONDARY MATHEMATICS 17

 The exterior angle of a triangle

End of section questions

1. PQ is a straight line

a) Find the angle marked 𝑥

b) Find the angle marked 𝑦

2. In the diagram, ABC is a triangle.

ACD is a straight line.

Work out the size of angle 𝒙

LOWER SECONDARY MATHEMATICS 18

 The exterior angle of a triangle

3. BCD is a straight line.

ABC is a triangle.

Show that triangle ABC is an isosceles triangle.

LOWER SECONDARY MATHEMATICS 19

 Interior and Exterior angles of polygons

Interior and Exterior angles of polygons

The interior angles of a polygon are angles found inside of a shape. The exterior angles
of a polygon are angles found outside of the shape and they are formed between any
side of the polygon and a line extended from the side next to it.

The sum of interior angles of a polygon = (n – 2) × 180°

n represents the number of sides

𝟑𝟔𝟎°
Each exterior angle of a polygon =
𝒏
The table below indicates the polygons together with the number of interior and exterior
angles and the respective sum of angles.

Polygon Sides Sum of interior Each interior Sum of exterior

angles angle of regular angles
polygon
Triangle 3 180° 60° 360°

Quadrilateral 4 360° 90° 360°

Pentagon 5 540° 108° 360°

Hexagon 6 720° 120° 360°

Heptagon 7 900° 128.57°… 360°

Octagon 8 1080° 135° 360°

(𝒏 − 𝟐) × 𝟏𝟖𝟎°
Any polygon n (n – 2) × 180° 𝒏 360°

LOWER SECONDARY MATHEMATICS 20

 Interior and Exterior angles of polygons

Worked example 1.5

1.

a) Work out the sum of the interior angles of the polygon above.

(n – 2) × 180°

= (8 – 2) × 180°

= 6 × 180°

= 1080°

b) Work out the size of each interior angle.

(𝒏 − 𝟐) × 𝟏𝟖𝟎°
𝒏
(𝟖−𝟐)×𝟏𝟖𝟎°
=
𝟖

𝟔 ×𝟏 𝟖𝟎°
=
𝟖

𝟏𝟎𝟖𝟎°
=
𝟖

= 135°

LOWER SECONDARY MATHEMATICS 21

 Interior and Exterior angles of polygons

2. Work out the size of the exterior angle of a regular hexagon.

A regular hexagon has 6 sides

𝟑𝟔𝟎° 𝟑𝟔𝟎°
= = 60°
𝒏 𝟔

End of section questions

1.

a) Work out the sum of the interior angles of the pentagon.

b) Work out the size of each interior angle.

2. The size of each exterior angle in a regular polygon is 40°. Work out how many
sides the polygon has.

3. A regular polygon has 10 sides. Work the sum of interior angles of this
polygon.

LOWER SECONDARY MATHEMATICS 22

 Interior and Exterior angles of polygons

LOWER SECONDARY MATHEMATICS 23