Geometrical reasoning

Revise Geometrical reasoning for Mathematics A 4MA1 (O Level) — revision notes, 6 practice questions and instant AI marking.

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Geometrical reasoning

The big idea: A geometry answer is a short argument: each angle you find needs the fact that justifies it. The number earns one mark; the reasons earn the rest.

Summary — What This Chapter Covers

Foundation: give informal reasons for calculations based on lines, triangles and polygons (including chords and tangents).
Higher: give reasons using standard geometrical statements in any context involving lines, polygons and circles.
One step, one fact, one reason. Name the angle you are finding each time.
A self-check: if two routes give the same angle, your chain is consistent.

1. Reasons for Lines, Triangles and Polygons

Foundation & Higher

Standard wording

Write each step as "angle XYZ = … (reason)". These are the phrases mark schemes accept:

Fact	Write
180° on a line	angles on a straight line add up to 180°
360° at a point	angles at a point add up to 360°
Crossing lines	vertically opposite angles are equal
Parallel lines	alternate angles are equal / corresponding angles are equal / allied (co-interior) angles add up to 180°
Triangle	angles in a triangle add up to 180° / exterior angle of a triangle equals the sum of the interior opposite angles
Isosceles	base angles of an isosceles triangle are equal
Polygon	angles in a quadrilateral add up to 360° / exterior angles of a polygon add up to 360°
Tangent	a tangent is perpendicular to the radius

Worked Example

ABC is a triangle. BCD is a straight line. Angle ABC = 52° and angle ACD = 118°. Work out the size of angle BAC. Give reasons for your answer.

Practice Question 1

PQ and RS are parallel lines. A line crosses PQ at X and RS at Y. Angle PXY = 74°. Find angle XYS, giving a reason.

2. Reasons with Circles

Higher tier

Chaining circle theorems

At Higher tier any context can carry reasons, including the circle theorems. Mark the centre, draw radii (they make isosceles triangles), and look for tangents (right angles).

Worked Example

A, B and C lie on a circle, centre O. TA is the tangent at A, and C lies on the opposite side of AB from T. Angle TAB = 58°. Find (a) angle ACB, (b) angle AOB, (c) angle OAB. Give reasons.

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Past papers that test Geometrical reasoning

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4MA1 Jun 2022 · Paper 1F Edexcel · mark scheme Open → 4MA1 Jun 2022 · Paper 1H Edexcel · mark scheme Open → 4MA1 Nov 2021 · Paper 2F Edexcel · mark scheme Open →

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Geometrical reasoning