IGCSE Maths · Area 4 · Geometry

Angles

Every angle question in 0580 is marked on two things: the size of the angle and the reason you give for it. This topic collects the angle facts you need - at a point, on a straight line, vertically opposite, inside triangles and quadrilaterals, between parallel lines, and inside polygons - and shows the wording Cambridge accepts as a reason. The reason is worth marks of its own, so learn it with the fact.

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Syllabus coverage

  • Calculate unknown angles and give simple explanations, using that the sum of the angles at a point is 360 degrees and at a point on a straight line is 180 degrees
  • Use that vertically opposite angles are equal
  • Use that the angle sum of a triangle is 180 degrees and the angle sum of a quadrilateral is 360 degrees
  • Use the three-letter notation for angles, for example angle ABC, and use correct geometrical terminology when giving reasons
  • Calculate unknown angles within parallel lines and give geometric explanations: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180 degrees (supplementary)
  • Know and use the angle properties of regular polygons, including interior and exterior angles and the angle sum
  • Extended only: the same angle properties applied to irregular polygons

Part of the IGCSE Maths syllabus 0580.

cdabghefc = 68° (given)d = 112°: 180 - 68 = 112a = 68° vertically oppositeg = 68° correspondinge = 68° alternateABCDFEAB parallel to CD (matching arrows)EF is the transversalOne given angle of 68° fixes all eight: four angles of 68° (a, c, e, g) and four of 112° (b, d, f, h).
One angle of 68° fixing all eight angles at two parallel lines, with the reason written beside each.

Angle language, and the notation that earns marks

An angle measures the turn between two lines that meet, and it is measured in degrees. A right angle is 90°, an acute angle is less than 90°, an obtuse angle is between 90° and 180°, and a reflex angle is more than 180°. Half a turn is a straight line and a full turn is 360°.

Cambridge requires the three-letter notation for angles. Angle ABC is the angle at the vertex B between the lines BA and BC, so the middle letter is always the vertex, which is where the angle is. In triangle ABC, angle ABC is at B, angle BCA is at C and angle CAB is at A. If the diagram names an angle with one letter, such as x, then x is that angle and you can use it in a calculation straight away.

A protractor measures and draws angles. Put the centre of the protractor on the vertex with the 0 line along one arm, then read the scale that starts from 0 on that arm - a protractor has two scales, and reading the wrong one is out by as much as 180°. Measure to the nearest degree, and when you are asked to draw an angle, mark the arc and write the size beside it.

  • The middle letter of angle ABC is the vertex, B
  • Acute is less than 90°, obtuse is between 90° and 180°, reflex is between 180° and 360°
  • Angles on one side of a straight line add to 180°; angles around a point add to 360°
Type of angleSizeExample
AcuteLess than 90°63°
RightExactly 90°the corner of a square
ObtuseBetween 90° and 180°117°
StraightExactly 180°the angles on a straight line
ReflexBetween 180° and 360°253°
Read the letters in the question. 'Work out angle ABC' is asking about the angle at B, not at A, and picking the wrong vertex loses a whole question for no reason.

At a point, on a straight line, and vertically opposite

Three facts do most of the work in this topic. Angles at a point add to 360°, because they close a full turn. Angles on a straight line add to 180°, because a straight line is half a turn. Vertically opposite angles are equal: when two straight lines cross, the angles opposite each other are the same size, and the two angles next to each other add to 180° because they sit on a straight line.

So if three angles meet at a point and two of them are 137° and 118°, the third is 360−137−118=105∘. If one angle on a straight line is 47°, the other is 180−47=133∘. If two lines cross and one angle is 68°, the angle opposite it is also 68°, and each of the other two is 180−68=112∘.

These three facts are also how you check an answer you are unsure about. Three angles at a point that add to 350° are wrong, however careful the working looked, because the full turn has to close.

  • Angles at a point: 137+118+105=360∘
  • Angles on a straight line: 47+133=180∘
  • Vertically opposite angles are equal, so two crossing lines make two pairs of equal angles
Write the reason with the number every time, even when the question does not ask for one. 'x = 133, angles on a straight line' is a complete answer; a bare '133' leaves the examiner to guess which fact you used.

Angles in a triangle

The angle sum of a triangle is 180°. Every triangle, whatever its shape, has three angles adding to 180°, so in a triangle whose angles include 58° and 74°, the third angle is 180−58−74=48∘.

The names of the triangles carry angle facts with them. An equilateral triangle has three equal angles of 60°. An isosceles triangle has two equal sides and, opposite those sides, two equal angles, so if the apex angle is 40° the two base angles are each (180−40)÷2=70∘. A right-angled triangle has one angle of 90°, so the other two add to 90°. A scalene triangle has three different angles and no shortcut at all.

One more fact follows from the two above and saves time: the exterior angle of a triangle, formed by producing one side, is equal to the sum of the two opposite interior angles. Cambridge does not list it as a reason on its own, so if you use it, quote the two facts it comes from - the exterior angle and its interior angle are on a straight line, and the three interior angles of the triangle add to 180°.

  • Angle sum of a triangle = 180°
  • Equilateral: 60°, 60°, 60°. Isosceles: the two base angles are equal
  • Right-angled: the two acute angles add to 90°
  • Exterior angle of a triangle = the sum of the two opposite interior angles
In an isosceles question the equal angles are opposite the equal sides, so find the marked equal sides first. Solving for the wrong pair of angles is the commonest error in this topic.

Quadrilaterals, and the angle sum of any polygon

A quadrilateral is any four-sided shape and its angle sum is 360°. The quickest way to see why is to draw a diagonal: it splits the quadrilateral into two triangles, and 180+180=360. So a quadrilateral with angles of 95°, 110° and 82° has a fourth angle of 360−(95+110+82)=360−287=73∘.

The same diagonal idea works for every polygon. A polygon with n sides can be cut into n−2 triangles from one vertex, so its angle sum is (n−2)×180∘. A pentagon is 3×180=540∘, a hexagon 720°, an octagon 1080° and a decagon 1440°.

The facts about particular quadrilaterals are worth knowing because they turn a two-step question into a one-step one: the angles of a square and of a rectangle are all 90°; the opposite angles of a parallelogram and of a rhombus are equal, and angles next to each other add to 180°; a kite has one pair of equal opposite angles.

  • Quadrilateral: 360°
  • Pentagon 540°, hexagon 720°, octagon 1080°, decagon 1440°
  • Angle sum of an n-sided polygon = (n−2)×180∘
A right angle marked on a diagram is a number you have been given: 90°. Read the diagram for the angles that are marked rather than described, and count how many angles the shape has before you subtract.

Interior and exterior angles of polygons

At each vertex of a polygon the interior angle is the angle inside the shape, and the exterior angle is the angle between one side and the next side produced - the turn you make if you walk round the shape. The interior angle and the exterior angle at the same vertex are on a straight line, so they add to 180°.

The exterior angles are the easy ones. Walking all the way round a polygon turns you through exactly one full turn, so the exterior angles of any polygon add to 360°, whether the polygon is regular or irregular. In a regular polygon all the exterior angles are equal, so each one is 360÷n, and each interior angle is 180−360÷n, which is the same as (n−2)×180÷n.

That gives you a quick answer to two standard questions. For the interior angle of a regular polygon, work out the exterior angle first and subtract it from 180. For the number of sides, given the interior angle, find the exterior angle and divide it into 360: a regular polygon with interior angle 144° has exterior angle 36°, so it has 360÷36=10 sides.

  • Exterior angles of any polygon add to 360°
  • Regular polygon: exterior angle =360÷n, interior angle =180−360÷n
  • Interior angle + exterior angle at a vertex = 180°
  • Given the interior angle, the number of sides is 360÷ the exterior angle
Regular polygonSidesExterior angleInterior angleAngle sum
Pentagon572°108°540°
Hexagon660°120°720°
Octagon845°135°1080°
Nonagon940°140°1260°
Decagon1036°144°1440°
12-sided polygon1230°150°1800°
Check the pair of numbers against each other: interior + exterior must be 180°. If you have 135° and 45° for an octagon you are right; if you have 135° and 55° you have made an arithmetic slip.

Parallel lines: corresponding, alternate and co-interior

When a straight line - called a transversal - crosses two parallel lines it makes eight angles, but only three facts are needed. Corresponding angles are equal: they are in matching positions at the two crossings, both above and to the right, say, making an F shape. Alternate angles are equal: they are on opposite sides of the transversal and between the two parallel lines, making a Z shape. Co-interior angles add to 180°: they are between the two parallel lines and on the same side of the transversal, making a C shape. Co-interior angles are also called supplementary or allied angles.

The three facts fit together, so one given angle gives you all eight. If the angle marked c is 68°, then the corresponding angle is 68°, the alternate angle is 68°, the angle vertically opposite c is 68°, and every angle that makes a C shape with one of those is 180−68=112∘.

These facts only work when the lines really are parallel, and a diagram shows that with arrows on the two lines. If the lines are not marked parallel, do not assume it, however parallel they look on the page.

  • Corresponding angles: equal, matching positions at the two crossings, an F shape
  • Alternate angles: equal, opposite sides of the transversal, a Z shape
  • Co-interior angles: add to 180°, same side of the transversal, a C shape
  • One angle of 68° gives four angles of 68° and four of 112°
Match the reason to the pair of angles. 'Alternate angles are equal' written next to a pair of co-interior angles scores nothing, even when the number happens to come out right.

Writing the reason so that it scores

Cambridge says candidates are expected to use the correct geometrical terminology when giving reasons for answers, and in this topic the reason carries marks of its own. A reason is a short named property: angles on a straight line; angles at a point; vertically opposite angles; angles in a triangle; angles in a quadrilateral; corresponding angles; alternate angles; co-interior angles; exterior angle of a regular polygon; angle sum of a polygon.

Two habits make reasons reliable. First, say which angles you mean: 'x and 68° are co-interior angles, so x+68=180' is a reason, while 'because of angles' is not. Second, give a reason for every step rather than only the first, because a two-step question usually carries a mark on each step.

Look at the shape the two angles make when you are not sure which property applies. A straight line is 180°, a full turn is 360°, an F shape is corresponding angles, a Z shape is alternate angles and a C shape is co-interior angles.

  • Name the property, then use it: 'co-interior angles add to 180°, so x=180−68=112∘'
  • One reason for each step: a two-step question is usually two marks
  • Never write 'it is obvious' or 'you can see it': name the property instead
  • Show the subtraction as working so the method mark is visible
If a question says 'give a reason for your answer', the reason is a mark in its own right. An answer of 112 with no reason can score one less than the same answer with 'co-interior angles sum to 180°' written beside it.

Key terms you must be able to define

Angle
The measure of the turn between two lines that meet, measured in degrees.
Vertex
The point where the two arms of an angle meet. In angle ABC the vertex is B, the middle letter.
Acute angle
An angle less than 90°.
Obtuse angle
An angle greater than 90° and less than 180°.
Reflex angle
An angle greater than 180° and less than 360°.
Right angle
An angle of exactly 90°, marked on a diagram with a small square.
Angles at a point
The angles that meet at one point and go all the way round it. They add to 360°.
Angles on a straight line
Angles that sit next to each other on one side of a straight line. They add to 180°.
Vertically opposite angles
The pair of angles opposite each other where two straight lines cross. They are equal.
Angle sum of a triangle
The three angles of any triangle add to 180°.
Isosceles triangle
A triangle with two equal sides, and opposite those sides two equal base angles.
Equilateral triangle
A triangle with three equal sides and three equal angles, each 60°.
Angle sum of a quadrilateral
The four angles of any quadrilateral add to 360°, because a diagonal splits it into two triangles.
Polygon
A closed shape with straight sides. A regular polygon has all its sides and all its angles equal.
Interior angle
The angle inside a polygon at one of its vertices.
Exterior angle
The angle between one side of a polygon and the next side produced. The exterior angles of any polygon add to 360°.
Transversal
A straight line that crosses two other lines, which is what creates corresponding, alternate and co-interior angle pairs.
Corresponding angles
Angles in matching positions at the two crossings of a transversal with two parallel lines. They are equal.
Alternate angles
Angles on opposite sides of a transversal and between the two parallel lines, making a Z shape. They are equal.
Co-interior angles
Angles between two parallel lines on the same side of a transversal, making a C shape. They add to 180° and are also called supplementary or allied angles.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

Angles at a point and on a straight line

Three angles meet at a point and two of them are 137° and 118°. The third is 360 - 137 - 118 = 105°, because angles at a point add to 360°. If instead the 105° sits on a straight line with another angle, that other angle is 180 - 105 = 75°.

An isosceles triangle

In triangle ABC, AB = AC and angle BAC = 40°. The triangle is isosceles, so angle ABC = angle ACB, and the two of them together take up 180 - 40 = 140°. Each is 140 / 2 = 70°. Two reasons, two marks: isosceles base angles are equal, and the angles of a triangle add to 180°.

Angles between parallel lines

Two parallel lines are crossed by a transversal and one angle is 73°. The angle in the matching position at the other crossing is 73° (corresponding angles are equal); the angle on the opposite side of the transversal between the lines is 73° (alternate angles are equal); and the angle in a C shape with it is 180 - 73 = 107° (co-interior angles add to 180°).

The interior angle of a regular polygon

Work out each interior angle of a regular octagon. The exterior angle is 360 / 8 = 45°, because the exterior angles of any polygon add to 360° and a regular polygon shares them equally. The interior angle is 180 - 45 = 135°. The check is the angle sum: 8 x 135 = 1080 = (8 - 2) x 180.

A quadrilateral with an unknown angle

The angles of a quadrilateral are 95°, 110°, 82° and x°. The angle sum of a quadrilateral is 360°, so x = 360 - (95 + 110 + 82) = 360 - 287 = 73°. Adding the three known angles first keeps the subtraction to a single step and makes the arithmetic easy to check.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What do the angles on a straight line add to?
180°. A straight line is half a turn.
Q2What do the angles at a point add to?
360°. They close a full turn.
Q3What is true about vertically opposite angles?
They are equal. When two straight lines cross, the angles opposite each other are the same size.
Q4What is the angle sum of a triangle?
180°. Every triangle, whatever its shape, has angles adding to 180°.
Q5What is the angle sum of a quadrilateral?
360°, because a diagonal splits it into two triangles.
Q6Which angles of an isosceles triangle are equal?
The two angles opposite the equal sides - the base angles. In a triangle with apex angle 40° they are 70° each.
Q7What is the angle sum of an n-sided polygon?
(n−2)×180∘. A pentagon is 540°, a hexagon 720°, an octagon 1080°.
Q8What do the exterior angles of any polygon add to?
360°, whether the polygon is regular or irregular, because a walk round the shape is one full turn.
Q9How do you find each exterior angle of a regular polygon?
Divide 360° by the number of sides. A regular octagon has exterior angles of 360 / 8 = 45°.
Q10How do you find each interior angle of a regular polygon?
Subtract the exterior angle from 180°, so 180 - 360/n. For an octagon, 180 - 45 = 135°.
Q11A regular polygon has interior angles of 144°. How many sides does it have?
10. The exterior angle is 180 - 144 = 36°, and 360 / 36 = 10.
Q12What are corresponding angles?
Angles in matching positions at the two crossings of a transversal with parallel lines. They are equal, and they make an F shape.
Q13What are alternate angles?
Angles on opposite sides of the transversal and between the parallel lines, making a Z shape. They are equal.
Q14What do co-interior angles add to?
180°. They are between the parallel lines on the same side of the transversal, making a C shape.
Q15In angle ABC, which letter is the vertex?
B, the middle letter. The angle is between the lines BA and BC.
0 of 15 marked as known

Exam-style questions

7 questions, 17 marks. Write your answer, then reveal the mark scheme and tick each point you actually made. Answering honestly here is what makes the rest of the site useful.

Work outAO12 marks
The angles 47° and x° are on a straight line. Work out the value of x. Give a reason for your answer.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • A1x = 133, with the reason 'angles on a straight line' namednot checked yet
Examiner tip: Two marks, and the reason is one of them. Write 'angles on a straight line add to 180°' beside your answer - a bare 133 can lose the mark even though the number is right.
Work outAO12 marks
Three angles meet at a point. Two of them are 137° and 118°. Work out the size of the third angle, giving a reason for your answer.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • A1105°, with the reason 'angles at a point' namednot checked yet
Examiner tip: Subtract both given angles in one step and show it. The reason is 'angles at a point add to 360°', which is a different fact from the straight-line one.
Work outAO23 marks
In triangle ABC, AB = AC and angle BAC = 40°. Work out the size of angle ABC. Give a reason for each step of your working.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • M1
  • A1angle ABC = 140÷2=70∘not checked yet
Examiner tip: Two facts, two marks, then the answer. Both reasons must be named: 'isosceles base angles are equal' and 'the angles of a triangle add to 180°'. Writing only the number loses the two method marks.
Work outAO23 marks
AB and CD are parallel lines. A straight line crosses both of them. The angle marked 73° and the angle marked x are between the two parallel lines and on the same side of the crossing line. Work out the value of x, giving a reason for your answer.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • M1
  • A1x = 107°, with the reason 'co-interior angles add to 180°' namednot checked yet
Examiner tip: The reason is a mark of its own here. Name the pair as co-interior, not alternate - alternate angles are on opposite sides of the transversal, and calling them alternate scores nothing even with the right number.
Work outAO13 marks
Work out the size of each interior angle of a regular octagon.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • M1
  • A1135°not checked yet
Examiner tip: Start from the exterior angle: 360 divided by the number of sides is one short step, and the interior angle is what is left of 180°. Quote the property as well as the number - 'exterior angles of a polygon sum to 360°'.
Work outAO23 marks
The angles of a quadrilateral are 95°, 110°, 82° and x°. Work out the value of x, giving a reason for your answer.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • M1
  • A1x = 73°, with the reason 'the angles of a quadrilateral sum to 360°' namednot checked yet
Examiner tip: Add the three given angles before subtracting, and say why the total is 360° - a quadrilateral is two triangles. Using 180° here is the commonest error in the question.
StateAO11 mark
Two parallel lines are crossed by a transversal. Two angles are in matching positions at the two crossings, one above and to the right of each crossing. State the name given to this pair of angles.
Write a short answer in words.

Mark scheme — tick each point you actually made

  • B1corresponding angles, which are equal because they are in matching positions at the two crossingsnot checked yet
Examiner tip: Matching positions at the two crossings make an F shape, and that is the description of corresponding angles. Alternate angles are on opposite sides of the transversal, in a Z shape, and co-interior angles are on the same side and add to 180°.

Exam technique for this topic

  • Every angle answer needs its reason. Write the property in words after the number - 'x = 133, angles on a straight line' - because the reason is a mark in this topic and an unnamed number is not.
  • Say which pair of angles you are talking about before you use a property: 'x and 68° are co-interior, so they add to 180°'. Naming the pair is what separates a full-mark reason from a guess.
  • Check that the lines really are parallel before using corresponding, alternate or co-interior angles. Parallel lines are marked with arrows, and without them the three facts do not apply.
  • In a polygon question, decide first whether it is about interior or exterior angles. 'How many sides?' is an exterior-angle question: divide the exterior angle into 360.
  • Show the subtraction as working. '180 - 58 - 74 = 48' earns the method mark even if the arithmetic slips, while a bare 48 does not.

Why students lose marks here

A student wrotex = 133, because the two angles look the same size.
Why it loses marksA reason has to name a geometrical property. 'They look the same' is a description of the diagram, not a reason, and the reason is where the mark is.
The answer that scoresx = 133, because the angles on a straight line add to 180° and 180 - 47 = 133.
A student wroteThe two angles between the parallel lines are alternate, so they are equal.
Why it loses marksAlternate angles are on opposite sides of the transversal. Two angles on the same side of the transversal, between the parallel lines, are co-interior and add to 180° instead of being equal.
The answer that scoresThey are co-interior angles, so x = 180 - 68 = 112°.
A student wroteEach interior angle of a regular hexagon is 360 / 6 = 60°.
Why it loses marks360 divided by the number of sides gives the exterior angle, not the interior angle. The interior angle is what is left of the straight line: 180 - 60.
The answer that scoresExterior angle = 360 / 6 = 60°, so interior angle = 180 - 60 = 120°, and the angle sum is 6 x 120 = 720 = (6 - 2) x 180.
A student wroteThe angles of the quadrilateral add to 180°, so the fourth angle is 180 - 105 - 87 - 98.
Why it loses marks180° is the triangle sum. A quadrilateral is two triangles, because a diagonal splits it in two, so its angles add to 360°.
The answer that scores360 - (105 + 87 + 98) = 360 - 290 = 70°.
A student wroteThe base angles of the isosceles triangle are 70° each, so the third angle is also 70°.
Why it loses marksOnly the two angles opposite the equal sides are equal. The third angle is whatever is left of 180°, which here is 180 - 70 - 70 = 40°.
The answer that scoresThe two base angles are 70° each and the apex angle is 40°, because 70 + 70 + 40 = 180°.

Where this fits in IGCSE Maths

This page is one topic of Cambridge IGCSE Maths 0580. IGCSE Maths is the course home: every area and every topic in Cambridge's own order, the papers you sit and what each one tests, and the exam-practice tools. The full topic list, with each topic linked to its revision page, is on the IGCSE Maths syllabus page.