1. KS3 Mathematics S1 Lines and Angles
The aim of this unit is to teach pupils to:
Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions and derived properties
Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
S1 Lines and Angles

2. S1.2 Parallel and perpendicular lines
Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1.3 Calculating angles
S1.4 Angles in polygons

3. Lines in a plane What can you say about these pairs of lines?
When we discuss lines in geometry, they are assumed to be infinitely long. That means that two lines in the same plane (that is in the same flat two-dimensional surface) will either intersect at some point or be parallel.
These lines cross, or intersect.
These lines do not intersect.
They are parallel.

4. Lines in a plane A flat two-dimensional surface is called a plane.
Any two straight lines in a plane either intersect once …
This is called the point of intersection.
Ask pupils how we could draw two infinitely long lines that will never meet. The answer would be to draw them in different planes. We can imagine, for example, one plane made by one wall in the room and another plane made by the opposite wall. If we drew a line on one wall and a line on the other, they would never meet, even if the walls extended to infinity.

5. Lines in a plane … or they are parallel.
We use arrow heads to show that lines are parallel.
Parallel lines will never meet. They stay an equal distance apart.
Pupils should be able to identify parallel and perpendicular lines in 2-D and 3-D shapes and in the environment.
For example: rail tracks, double yellow lines, door frame or ruled lines on a page.
This means that they are always equidistant.
Where do you see parallel lines in everyday life?

6. What is special about the angles at the point of intersection here?
Perpendicular lines
What is special about the angles at the point of intersection here? a
a = b = c = d b d
Each angle is 90. We show this with a small square in each corner. c
Pupils should be able to explain that perpendicular lines intersect at right angles.
Lines that intersect at right angles are called perpendicular lines.

7. Types of angle a a a a Acute angle 0º < a < 90º Right angle
Obtuse angle
90º < a < 180º a
Reflex angle 180º < a < 360º a
Pupils should be able to classify angles according to whether they are acute, right-angled, obtuse or reflex.

8. Angles on a straight line and at a pointb
Angles on a line add up to 180 c
Angles at a point add up to 360 a b c d
a + b + c = 180°
a + b + c + d = 360
because there are 180° in a half turn.
This slide is a reminder of rules established at KS3.
because there are 360 in a full turn.

9. Vertically opposite angles
When two lines intersect, two pairs of vertically opposite angles are formed. a b c d
a = c
and
b = d
Vertically opposite angles are equal.

10. Intersecting lines
Use this activity to demonstrate that vertically opposite angles are always equal.

11. Angles on a straight line
Use this activity to demonstrate that the angles on a straight line always add up to 180°.
Hide one of the angles and ask pupils to work out its value. Add another angle to make the problem more difficult.

12. Angles on a straight line
Angles on a line add up to 180. a b
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
a + b = 180°
because there are 180° in a half turn.

13. Angles around a point
Move the points to change the values of the angles. Show that these will always add up to 360º.
Hide one of the angles, move the points and ask pupils to calculate the size of the missing angle.

14. Angles around a point Angles around a point add up to 360. b a c d
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
a + b + c + d = 360
because there are 360 in a full turn.

15. Calculating angles around a point
Use geometrical reasoning to find the size of the labelled angles.
69°
68° d
167° a
43° c
103°
43° b
Point out that that there are two intersecting lines in the second diagram.
Click to reveal the solutions.
137°

16. The angles in a triangle add up to 180°.b c
For any triangle,
a + b + c = 180°
The angles in a triangle add up to 180°.

17. Angles in a triangle
Change the triangle by moving the vertex. Pressing the play button will divide the triangle into three pieces. Pressing play again will rearrange the pieces so that the three vertices come together to form a straight line. Conclude that the angles in a triangle always add up to 180º.
Pupil can replicate this result by taking a triangle cut out of a piece of paper, tearing off each of the corners and rearranging them to make a straight line.

18. Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles.
64° b
116°
33° a
31°
326°
82°
Ask pupils to calculate the size of the missing angles before revealing them.
49°
43°
25° d
88° c
28°
233°

19. Angles in an isosceles triangle
In an isosceles triangle, two of the sides are equal.
We indicate the equal sides by drawing dashes on them.
The two angles at the bottom on the equal sides are called base angles.
The two base angles are also equal.
If we are told one angle in an isosceles triangle we can work out the other two.

20. Angles in an isosceles triangle
For example,
88° a
46°
46° a
Find the size of the other two angles.
The two unknown angles are equal so call them both a.
We can use the fact that the angles in a triangle add up to 180° to write an equation.
As an alternative to using algebra we could use the following argument.
The three angles add up to 180º, so the two unknown angles must add up to 180º – 88º, that’s 92º.
The two angles are the same size, so each must measure half of 92º or 46º.
88° + a + a = 180°
88° + 2a = 180°
2a = 92°
a = 46°

21. Interior and exterior angles in a triangle
Drag the vertices of the triangle to show that the exterior angle is equal to the sum of the opposite interior angles.
Hide angles by clicking on them and ask pupils to calculate their sizes.

22. Corresponding, alternate and interior angles
Corresponding angles are equal a b
a = b
Alternate angles are equal a b
a = b
Interior angles add up to 180° a b
a + b = 180°
Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines.
Look for an F-shape
Look for a Z-shape
Look for a C- or U-shape