1. Measuring angles
An angle is a measure of turn and is usually measured in degrees.
A full turn measures 360°.
360°
Links:
S1 Line and angles – calculating angles
S6 Construction and loci – drawing angles

2. Measuring angles
An angle is a measure of turn and is usually measured in degrees.
A half turn measures 180°.
This is a straight line.
180°
Links:
S1 Line and angles – calculating angles
S6 Construction and loci – drawing angles

3. Measuring angles
An angle is a measure of turn and is usually measured in degrees.
A quarter turn measures 90°.
It is called a right angle.
90°
We label a right angle with a small square.
Links:
S1 Line and angles – calculating angles
S6 Construction and loci – drawing angles

4. Measuring angles
An angle is a measure of turn and is usually measured in degrees.
A three-quarter turn measures 270°.
270°
Links:
S1 Line and angles – calculating angles
S6 Construction and loci – drawing angles

5. Acute, obtuse and reflex angles
All angles are acute, obtuse or reflex.
An acute angle is between 0º and 90º.
An obtuse angle is between 90º and 180º.
An reflex angle is between 180º and 360º.
Pupils should be able to classify angles according to whether they are acute, right angle, obtuse, straight lines or reflex.

6. Acute, obtuse and reflex angles
Look at each interior angle in this shape. Is it acute, obtuse or reflex? B D C A I
Establish that A is acute, B is obtuse, C is reflex, D is acute, E is acute, F is reflex, G is acute, H is obtuse, and I is reflex. F E H G

7. Using a protractor We measure angles with a protractor.
Discuss the two scales on the protractor. The outside scale goes from 0° to 180° and the inner scale goes from 180° to 0°.
Notice that the protractor has two scales.
Before you measure an angle decide whether it is acute or obtuse.

8. Labelling angles When two lines meet at a point an angle is formed. AC
An angle is a measure of the rotation of one of the line segments relative to the other.
Pupils often find the naming of angles difficult, particularly when there is more than one angle at a point.
At key stage 3 this confusion is often avoided by using single lower case letters to name angles.
We label points using capital letters.
The angle can then be described as ABC
or ABC
or B.
Sometimes instead an angle is labelled with a lower case letter.

9. 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.

10. 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°.

11. 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.

12. 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.

13. Complementary angles
When two angles add up to 90° they are called complementary angles. a b
Ask pupils to give examples of pairs of complementary angles. For example, 32° and 58º.
Give pupils an acute angle and ask them to calculate the complement to this angle.
a + b = 90°
Angle a and angle b are complementary angles.

14. Supplementary angles
When two angles add up to 180° they are called supplementary angles. b a
Ask pupils to give examples of pairs of supplementary angles. For example, 113° and 67º.
Give pupils an angle and ask them to calculate the supplement to this angle.
a + b = 180°
Angle a and angle b are supplementary angles.

15. Corresponding angles
There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h
Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines. g g
d = h because
Corresponding angles are equal

16. Corresponding angles
There are four pairs of corresponding angles, or F-angles. a a b b d d c c e e f f h h g g
a = e because
Corresponding angles are equal

17. Corresponding angles
There are four pairs of corresponding angles, or F-angles. a b d c c e f h g g
c = g because
Corresponding angles are equal

18. Corresponding angles
There are four pairs of corresponding angles, or F-angles. a b b d c e f f h g
b = f because
Corresponding angles are equal

19. Alternate angles There are two pairs of alternate angles, or Z-angles.b d d c e f f h g
d = f because
Alternate angles are equal

20. Alternate angles There are two pairs of alternate angles, or Z-angles.b d c c e e f h g
c = e because
Alternate angles are equal

21. 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.

22. Exterior angles in polygons
When we extend the sides of a polygon outside the shape
exterior angles are formed. e d f

23. Sum of the interior angles in a quadrilateral
What is the sum of the interior angles in a quadrilateral? d c f a e b
We can work this out by dividing the quadrilateral into two triangles.
a + b + c = 180°
and
d + e + f = 180°
So,
(a + b + c) + (d + e + f ) = 360°
The sum of the interior angles in a quadrilateral is 360°.

24. Sum of interior angles in a polygon
We already know that the sum of the interior angles in any triangle is 180°. c
a + b + c = 180 ° a b
We have just shown that the sum of the interior angles in any quadrilateral is 360°. a b c d
Pupils should be able to understand a proof that the that the exterior angle is equal to the sum of the two interior opposite angles.
Framework reference p183
a + b + c + d = 360 °
Do you know the sum of the interior angles for any other polygons?

25. Sum of the interior angles in a pentagon
What is the sum of the interior angles in a pentagon? c d a f g b e h i
We can work this out by using lines from one vertex to divide the pentagon into three triangles .
a + b + c = 180°
and
d + e + f = 180°
and
g + h + i = 180°
So,
(a + b + c) + (d + e + f ) + (g + h + i) = 560°
The sum of the interior angles in a pentagon is 560°.

26. Sum of the interior angles in a polygon
We’ve seen that a quadrilateral can be divided into two triangles …
… and a pentagon can be divided into three triangles.
A hexagon can be divided into four triangles.
How many triangles can a hexagon be divided into?

27. Sum of the interior angles in a polygon
The number of triangles that a polygon can be divided into is always two less than the number of sides.
We can say that:
A polygon with n sides can be divided into (n – 2) triangles.
The sum of the interior angles in a triangle is 180°.
So,
The sum of the interior angles in an n-sided polygon is (n – 2) × 180°.

28. Interior angles in regular polygons
A regular polygon has equal sides and equal angles.
We can work out the size of the interior angles in a regular polygon as follows:
Name of regular polygon
Sum of the interior angles
Size of each interior angle
Equilateral triangle
180°
180° ÷ 3 =
60°
Square
2 × 180° = 360°
360° ÷ 4 =
90°
Ask pupils to complete the table for regular polygons with up to 10 sides.
Regular pentagon
3 × 180° = 540°
540° ÷ 5 =
108°
Regular hexagon
4 × 180° = 720°
720° ÷ 6 =
120°

29. Interior and exterior angles in an equilateral triangle
Every interior angle measures 60°.
60°
120°
Every exterior angle measures 120°.
The sum of the interior angles is 3 × 60° = 180°.
120°
60°
60°
120°
The sum of the exterior angles is 3 × 120° = 360°.

30. Interior and exterior angles in a square
Every interior angle measures 90°.
90°
90°
90°
90°
Every exterior angle measures 90°.
The sum of the interior angles is 4 × 90° = 360°.
90°
90°
90°
The sum of the exterior angles is 4 × 90° = 360°.
90°

31. Interior and exterior angles in a regular pentagon
Every interior angle measures 108°.
108°
72°
72°
Every exterior angle measures 72°.
108°
108°
72°
The sum of the interior angles is 5 × 108° = 540°.
108°
108°
72°
72°
The sum of the exterior angles is 5 × 72° = 360°.

32. Interior and exterior angles in a regular hexagon
Every interior angle measures 120°.
60°
120°
120°
60°
Every exterior angle measures 60°.
60°
120°
120°
60°
The sum of the interior angles is 6 × 120° = 720°.
120°
120°
60°
60°
The sum of the exterior angles is 6 × 60° = 360°.

33. The sum of exterior angles in a polygon
For any polygon, the sum of the interior and exterior angles at each vertex is 180°.
For n vertices, the sum of n interior and n exterior angles is n × 180° or 180n°.
The sum of the interior angles is (n – 2) × 180°.
We can write this algebraically as 180(n – 2)° =
180n° – 360°.

34. The sum of exterior angles in a polygon
If the sum of both the interior and the exterior angles is 180n°
and the sum of the interior angles is 180n° – 360°,
the sum of the exterior angles is the difference between these two.
The sum of the exterior angles = 180n° – (180n° – 360°)
= 180n° – 180n° + 360°
Discuss this algebraic proof that the sum of the exterior angles in a polygon is always 360°.
= 360°
The sum of the exterior angles in a polygon is 360°.

35. Bearings Bearings are a measure of direction taken from North.
If you were travelling North you would be travelling on a bearing of 000°.
If you were travelling from the point P in the direction shown by the arrow then you would be travelling on a bearing of 000°.
If you were travelling from the point P in the direction shown by the arrow then you would be travelling on a bearing of 075°. N
Bearings are always measured clockwise from North and are written as three figures.
75° P

36. Compass points 000° N 315° 045° NW NE 270° W E 090° SW SE 225° 135° S
Revise the basic compass points. These can be remembered using a mnemonic such as ‘Naughty Elephants Squirt Water’.
Introduce the points NE, SE, SW and NW, pointing out that N or S always comes before E and W.
Ask pupils to tell you how many degrees there are between each compass point (45°).
Ask questions such as,
A ship is sailing due southwest. What bearing is it sailing on?
Reveal the directions of the compass using bearings in orange.
You may like to mention that in addition to these points we can also have NNE (north by north east), ENE (east by north east), ESE (east by south east), SSE (south by south east), SSW (south by south west), WSW (west by south west), WNW (west by north west) and NNW (north by north west). As an extension exercise ask pupils to draw these compass points using a ruler and a protractor and give the bearing of each one. There will be 22.5º between each compass point. SW SE
225°
135° S
180°

37. Bearings The bearing from point A to point B is 105º.
What is the bearing from point B to point A? N
The angle from B to A is
105º + 180º = N
285º
105º A
This is called a reciprocal bearing or back bearing.
105º ?
Explain that the north lines are parallel and so we can work out he corresponding angle of 105°. The angle from B to A is therefore 105° + 180° = 285°.
If a given bearing is less than 180º, we find the reciprocal or back bearing by adding 180°. If a given bearing is more than 180º, we find the reciprocal or back bearing by subtracting 180°. B
180°