Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Angles, parallel lines and polygons. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Angle facts on straight lines, around a point, in triangles and quadrilaterals, on parallel lines and in polygons. Questions ask you to work out angles and give reasons; harder ones chain several facts together or use algebra. This topic comes up on both tiers, on calculator and non-calculator papers.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
Use angles on a line and around a pointAngles on a straight line add up to 180° and angles around a point add up to 360°.
3
Find missing angles in triangles and quadrilateralsAngles in a triangle add up to 180°, angles in a quadrilateral add up to 360°, and the base angles of an isosceles triangle are equal.
3
Know the properties of special quadrilateralsKnow the sides, angles, diagonals and symmetry of squares, rectangles, parallelograms, rhombuses, kites and trapezia.
4
Use alternate, corresponding and co-interior anglesOn parallel lines, alternate angles and corresponding angles are equal, and co-interior angles add up to 180°.
4
Find interior and exterior angles of polygonsSum of interior angles = (n − 2) × 180°, and the exterior angles of any polygon add up to 360°.
5
Give a geometric reason for every stepWrite the angle fact in words each time you use it, e.g. 'alternate angles are equal'.
5
Form and solve equations from angle factsWhen angles are given in terms of x, use an angle fact to write an equation, solve it, then find the angle.
Notes
Basic angle facts
Angles on a straight line add up to 180°. Angles around a point add up to 360°.
Vertically opposite angles (opposite each other where two straight lines cross) are equal.
Angles in a triangle add up to 180°. Angles in a quadrilateral add up to 360°.
An isosceles triangle has two equal sides, and the two base angles (opposite the equal sides) are equal. Each angle in an equilateral triangle is 60°.
The exterior angle of a triangle equals the sum of the two interior opposite angles.
Parallel lines
Parallel lines are marked with matching arrows. A line that crosses them is called a transversal.
Alternate angles are equal. They are between the parallel lines, on opposite sides of the transversal (a Z shape).
Corresponding angles are equal. They are in the same position at each crossing (an F shape).
Co-interior angles add up to 180°. They are between the parallel lines, on the same side of the transversal (a C shape).
Special quadrilaterals
Square: 4 equal sides, 4 right angles; diagonals are equal and cross at 90°.
Rectangle: opposite sides equal, 4 right angles; diagonals are equal and bisect each other.
Parallelogram: opposite sides equal and parallel, opposite angles equal; diagonals bisect each other.
Rhombus: 4 equal sides, opposite sides parallel, opposite angles equal; diagonals bisect each other at 90°.
Kite: two pairs of equal adjacent sides, one pair of equal opposite angles; diagonals cross at 90°.
Trapezium: one pair of parallel sides.
Lines of symmetry: square 4, rectangle 2, rhombus 2, kite 1, isosceles trapezium 1, parallelogram 0. Order of rotational symmetry: square 4; rectangle, rhombus and parallelogram 2.
Polygons
From one vertex, an n-sided polygon splits into n − 2 triangles, so the sum of the interior angles is (n − 2) × 180°.
The exterior angles of any polygon add up to 360°. At each vertex, interior angle + exterior angle = 180°.
In a regular polygon all sides and angles are equal, so each exterior angle = 360° ÷ n, and n = 360° ÷ exterior angle.
Work out the size of an angle. Give reasons for your answer.
3 to 4 marks4
Mark each angle on the diagram as soon as you find it.
Work towards the angle you want one fact at a time.
After each step, write the reason in full words, e.g. 'corresponding angles are equal'.
Check the answer fits the diagram (acute or obtuse).
Example. PQ and RS are parallel horizontal lines, with PQ above RS. T is a point on PQ, with P to the left of T. U and V are points on RS, with U to the left of V. TU = TV. Angle PTU = 52°. Work out the size of angle UTV. Give a reason for each stage of your working.
Show the model answer
Angle TUV = 52° because alternate angles are equal (M1) Angle TVU = 52° because base angles of an isosceles triangle are equal Angle UTV = 180 − 52 − 52 (M1) = 76° (A1) Reasons given: alternate angles are equal; base angles of an isosceles triangle are equal; angles in a triangle add up to 180° (C1)
Interior and exterior angles of polygons
2 to 4 marks4
For a regular polygon, exterior angle = 360° ÷ n and interior angle = 180° − exterior angle.
To find the number of sides, work out 360° ÷ exterior angle. It must be a whole number.
For the total of the interior angles, use (n − 2) × 180°.
Example. Each interior angle of a regular polygon is 156°. (a) Work out the number of sides of the polygon. (2 marks) (b) Work out the sum of the interior angles of the polygon. (2 marks)
Choose the angle fact that links the angles, e.g. angles in a quadrilateral add up to 360°.
Write the equation, collect like terms and solve for x.
Substitute x back in to find the angle the question asks for.
Example. The angles of a quadrilateral are x°, 2x°, (x + 30)° and (3x + 15)°. Work out the size of the largest angle.
Show the model answer
x + 2x + x + 30 + 3x + 15 = 360 (M1) 7x + 45 = 360, so 7x = 315 (M1) x = 45 (A1) Largest angle = 3 × 45 + 15 = 150° (A1)
Shortcuts and memory tricks
Z for alternate, F for corresponding, C for co-interior helps you find the angles, but never write 'Z angles' as a reason.
Polygons: exterior angles are the easy way in. Start with 360 ÷ n, then take it from 180°.
Quick check: 360 ÷ exterior angle must be a whole number, or the polygon cannot be regular.
Count triangles for the angle sum: a hexagon splits into 4 triangles, so its angles add up to 4 × 180° = 720°.
Where marks are lost
Writing 'Z angles' or 'F angles' instead of 'alternate angles' or 'corresponding angles'; these do not score as reasons.
Using 360 ÷ n as the interior angle of a regular polygon; it gives the exterior angle.
Assuming lines are parallel or sides are equal because they look it; use only what is marked or stated.
Measuring with a protractor when the diagram is labelled 'not accurately drawn'.
Giving the value of x when the question asks for an angle.
Exam technique
'Give reasons' means a reason for every step. A correct angle without full reasons loses the communication mark.
Write the angles you find on the diagram; they can earn method marks.
Angle ABC is the angle at B, between the lines BA and BC.
Write the full fact, e.g. 'angles on a straight line add up to 180°', not just '180°' or 'straight line'.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
\(ABC\) is a straight line. Diagram NOT accurately drawn
[object Object]
(a) Work out the size of the angle marked \(x\).[2]
(b) Give a reason for your answer.[1]
Show the answer and mark scheme
(a)Answer: \(81^\circ\) °
M1 for 180 − 57 − 42 or 180 − 99
A1 for 81
Worked solution: \(x = 180 - 57 - 42 = 81\)
(b)Answer: Angles on a straight line add up to 180°
C1 for angles on a straight line add up to 180°
Question 2Medium4 marks
(a) Explain why a regular polygon cannot have an interior angle of 100°.[2]
(b) A regular polygon has interior angles of 156°. How many sides does it have?[2]
Show the answer and mark scheme
(a)Answer: The exterior angle would be 80°, and \(360 \div 80 = 4.5\), which is not a whole number of sides.
M1 for exterior angle \(180 - 100 = 80\)
C1 for \(360 \div 80 = 4.5\), which is not a whole number, so there is no such polygon
Worked solution: Exterior angle \(= 180 - 100 = 80^\circ\). The exterior angles of a polygon add up to 360°, so \(n = 360 \div 80 = 4.5\). A polygon must have a whole number of sides.
(b)Answer: 15
M1 for \(360 \div (180 - 156)\)
A1 for 15
Worked solution: Exterior angle \(= 24^\circ\), so \(n = 360 \div 24 = 15\).