GCSE Maths · Higher · Chapter 6: Angles

Angle facts, polygons and bearings

Grades 4–9About 80 minutesNo calculator90 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Two angles on a straight line are $12 7^{\circ}$ and $x$ . Work out $x$ .
[1 mark]
Answer

A2
Three angles meet at a point. Two of them are $9 5^{\circ}$ and $14 0^{\circ}$ . Work out the third angle.
[1 mark]
Answer
A3
Two angles of a triangle are $5 3^{\circ}$ and $7 4^{\circ}$ . Work out the third angle.
[1 mark]
Answer
A4
Write down the three-figure bearing of the direction due west.
[1 mark]
Answer
A5
An isosceles triangle has one angle of $4 0^{\circ}$ between its two equal sides. Work out the size of each of the other two angles.
[2 marks]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

The angles of a triangle are $x^{\circ}$ , $(2 x + 15)^{\circ}$ and $(3 x - 21)^{\circ}$ . Work out the size of the largest angle.

Hint The three angles add up to

18 0^{\circ}

. Solve for

x

first, then work out all three angles.

B1
The angles of a triangle are $(x + 20)^{\circ}$ , $2 x^{\circ}$ and $(4 x - 29)^{\circ}$ . Work out the size of the largest angle.
[3 marks]
Answer

Example 2

Work out the size of each interior angle of a regular polygon with 12 sides.

Hint Find one exterior angle first: the exterior angles of any polygon add up to

36 0^{\circ}

B2
Work out the size of each interior angle of a regular polygon with 18 sides.
[2 marks]
Answer

Example 3

Each interior angle of a regular polygon is $16 2^{\circ}$ . How many sides does the polygon have?

Hint The interior and exterior angles at a corner add up to

18 0^{\circ}

. Then use the

36 0^{\circ}

total of the exterior angles.

B3
Each interior angle of a regular polygon is $16 5^{\circ}$ . How many sides does the polygon have?
[2 marks]
Answer

Example 4

The bearing of $B$ from $A$ is $06 4^{\circ}$ . Work out the bearing of $A$ from $B$ .

Hint Draw a north line at each point. The two north lines are parallel, so the back bearing differs by

18 0^{\circ}

B4
The bearing of $Q$ from $P$ is $29 5^{\circ}$ . Work out the bearing of $P$ from $Q$ .
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The diagram shows two parallel lines and a straight line crossing them.
Not drawn accurately
(a)
Work out the size of angle $a$ . Give a reason for your answer.
[2 marks]
Answer
(b)
Work out the size of angle $b$ . Give a reason for your answer.
[2 marks]
Answer
Total for C1: 4 marks

C2
$B C D$ is a straight line. Work out the size of angle $y$ .
[2 marks]
Not drawn accurately
Answer
C3
The diagram shows a hexagon. Work out the value of $x$ .
[3 marks]
Not drawn accurately
Answer
C4
$A B C D$ is a parallelogram. Work out the size of the smallest angle of the parallelogram.
[3 marks]
Not drawn accurately
Answer
C5
$B$ is due east of $A$ . The bearing of $C$ from $A$ is $05 2^{\circ}$ . The bearing of $C$ from $B$ is $31 8^{\circ}$ .
Work out the size of angle $A C B$ .
[3 marks]
Not drawn accurately
Answer
C6
A map is drawn to a scale of $1 : 40000$ . Two farms are $6.3$ cm apart on the map. Work out the real distance between the farms in kilometres.
[2 marks]
Answer
C7
The interior angles of a polygon add up to $198 0^{\circ}$ . How many sides does the polygon have?
[2 marks]
Answer
C8
A regular pentagon and a regular hexagon are joined along one side, so that they do not overlap. At one end of the shared side, angle $x$ is the angle outside both shapes.
Work out the size of angle $x$ .
[2 marks]
Answer
C9
$A B C D$ is a kite with $A B = A D$ and $C B = C D$ .
Not drawn accurately
(a)
Work out the size of angle $A B C$ .
[2 marks]
Answer
(b)
The diagonal $A C$ is drawn. Work out the size of angle $B A C$ .
[1 mark]
Answer
Total for C9: 3 marks

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The diagram shows two parallel lines and a straight line crossing them.
Work out the value of $x$ . Give reasons for your working.
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$

D2
Priya says, "A regular polygon can have interior angles of $14 5^{\circ}$ ."
Explain why Priya is wrong.
[2 marks]
D3
Town $B$ is 12 km from town $A$ on a bearing of $07 0^{\circ}$ . Town $C$ is 12 km from town $A$ on a bearing of $15 0^{\circ}$ .
Not drawn accurately
(a)
Work out the size of angle $B A C$ .
[1 mark]
Answer
(b)
Work out the bearing of $C$ from $B$ .
[4 marks]
Answer
(c)
Hence write down the bearing of $B$ from $C$ .
[1 mark]
Answer
Total for D3: 6 marks
D4
$A B C D E F$ is a regular hexagon. $A B G H$ is a square. $F$ is joined to $H$ .
Work out the size of angle $A H F$ . You must show your working.
[5 marks]
Not drawn accurately
Answer
D5
Each exterior angle of a regular polygon is $x^{\circ}$ . Each interior angle is $4 x^{\circ}$ .
Work out the number of sides of the polygon.
[3 marks]
Answer
D6
On a map with a scale of $1 : 20000$ , two churches are $4.5$ cm apart. Ellie cycles from one church to the other at an average speed of 12 km/h.
How many minutes does the journey take?
[3 marks]
Answer
D7
$A B C$ is an isosceles triangle with $A B = A C$ . The line $D A E$ is parallel to $B C$ . Angle $D A B = 6 4^{\circ}$ .
Work out the size of angle $B A C$ . Give reasons for your working.
[4 marks]
Not drawn accurately
Answer
D8
The diagram shows triangle $A B C$ with the side $B C$ extended to $D$ .
Show that $d = a + b$ . Give a reason for each step.
[3 marks]
Not drawn accurately
D9
The angles of a quadrilateral are in the ratio $2 : 3 : 4 : 6$ .
(a)
Work out the size of the largest angle.
[2 marks]
Answer
(b)
Explain why the quadrilateral cannot have a pair of parallel sides.
[2 marks]
Total for D9: 4 marks
D10
The bearing of $X$ from $Y$ is three times the bearing of $Y$ from $X$ .
Work out the bearing of $Y$ from $X$ .
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
The diagram shows a five-pointed star drawn by joining the corners of a pentagon. The pentagon is not regular.
Work out $a + b + c + d + e$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Look for triangles inside the star. An exterior angle of a triangle equals the two opposite interior angles added.
Hint 2 · The first line
Each angle of the small pentagon in the middle has an exterior angle that equals two of the tip angles added together.
Hint 3 · The full method
The five exterior angles of the middle pentagon add up to $36 0^{\circ}$ , and each tip angle is counted twice in them. So $2 (a + b + c + d + e) = 360$ .
Answer

E2
Two identical regular polygons and a regular decagon fit together exactly around a point, with no gaps.
How many sides does each of the two identical polygons have?
[3 marks]
Hint 1 · What to try
Find the interior angle of a regular decagon.
Hint 2 · The first line
Each interior angle of the decagon is $14 4^{\circ}$ , so the two identical angles add up to $21 6^{\circ}$ .
Hint 3 · The full method
Each identical polygon has interior angle $10 8^{\circ}$ , so exterior angle $7 2^{\circ}$ , and $360 \div 72$ sides.
Answer
E3
A regular polygon has interior angles of at least $17 0^{\circ}$ , and each exterior angle is a whole number of degrees.
How many different polygons like this are there?
[3 marks]
Hint 1 · What to try
Turn the condition on the interior angle into a condition on the exterior angle.
Hint 2 · The first line
The exterior angle is at most $1 0^{\circ}$ , and it must divide exactly into $360$ .
Hint 3 · The full method
List the whole numbers from 1 to 10 that divide into 360: $1, 2, 3, 4, 5, 6, 8, 9, 10$ . Count them.
Answer
E4
Three boats are at $A$ , $B$ and $C$ . The bearing of $B$ from $A$ is $04 0^{\circ}$ . The bearing of $C$ from $B$ is $10 0^{\circ}$ . The bearing of $A$ from $C$ is $25 0^{\circ}$ .
Show that $B$ is the same distance from $A$ as it is from $C$ .
[4 marks]
Hint 1 · What to try
Draw a sketch with a north line at each boat. Work out each angle of triangle $A B C$ from the bearings.
Hint 2 · The first line
Angle $B A C = 070 - 040 = 3 0^{\circ}$ and angle $A B C = 220 - 100 = 12 0^{\circ}$ .
Hint 3 · The full method
So angle $B C A = 180 - 30 - 120 = 3 0^{\circ}$ . Two angles are equal, so the triangle is isosceles and $B A = B C$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use angle facts on lines, at a point, in triangles and across parallel lines, and give reasons.
I can work out interior and exterior angles of polygons, and use the properties of special quadrilaterals.
I can use bearings and scales to solve problems.

Answers and mark scheme

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