Geometry

What you'll learn

Geometry forms a substantial component of the Pearson Edexcel International IGCSE Mathematics specification, accounting for approximately 20% of examination marks. This revision guide covers all testable geometry content including angle properties, polygon properties, circle theorems, and geometric constructions. You will develop skills in spatial reasoning, formal proof, and applying geometric principles to both abstract problems and real-world contexts.

Key terms and definitions

Congruent — shapes that are identical in size and shape; corresponding sides and angles are equal

Similar — shapes that have the same shape but different sizes; corresponding angles are equal and corresponding sides are in the same ratio

Supplementary angles — two angles that sum to 180°

Complementary angles — two angles that sum to 90°

Subtended angle — an angle formed at a point by two lines drawn from the endpoints of an arc or line segment

Interior angle — an angle formed inside a polygon between two adjacent sides

Exterior angle — an angle formed between one side of a polygon and the extension of an adjacent side

Chord — a straight line joining two points on the circumference of a circle

Core concepts

Angle properties

Angles form the foundation of geometric reasoning. Master these fundamental relationships:

Angles at a point sum to 360°

Angles on a straight line sum to 180°

Vertically opposite angles are equal when two straight lines intersect

Parallel lines and transversals create specific angle relationships:

• Corresponding angles are equal (F-pattern)
• Alternate angles are equal (Z-pattern)
• Co-interior angles sum to 180° (C-pattern)

When solving problems involving parallel lines, mark known angles systematically and identify which angle relationship applies. Always provide geometric reasons in your working.

Triangle properties

Triangles contain several critical properties tested extensively:

Angle sum: The interior angles of any triangle sum to 180°

Exterior angle theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles

Triangle inequality: The sum of any two sides must exceed the third side

Types of triangles by sides:

• Equilateral: all sides equal, all angles 60°
• Isosceles: two sides equal, two angles equal
• Scalene: all sides different

Types of triangles by angles:

• Acute: all angles less than 90°
• Right: one angle exactly 90°
• Obtuse: one angle greater than 90°

For isosceles triangles, remember that equal sides are opposite equal angles. This property is frequently tested in multi-step problems.

Polygon properties

Polygons require you to apply formulae and understand internal structure:

Sum of interior angles = (n - 2) × 180° where n is the number of sides

Each interior angle of a regular polygon = (n - 2) × 180° ÷ n

Sum of exterior angles = 360° for any polygon

Each exterior angle of a regular polygon = 360° ÷ n

Common polygons tested include:

• Pentagon (5 sides)
• Hexagon (6 sides)
• Octagon (8 sides)
• Decagon (10 sides)

For regular polygons, you may need to find the number of sides given one interior or exterior angle. Rearrange the formulae algebraically.

Circle theorems

Circle theorems represent one of the most challenging geometry topics. Eight key theorems are testable:

1. Angle at the centre is twice the angle at the circumference (both subtended by the same arc)

2. Angle in a semicircle equals 90° (where the diameter forms the base)

3. Angles in the same segment are equal (subtended by the same chord)

4. Opposite angles in a cyclic quadrilateral sum to 180° (a quadrilateral with all vertices on the circumference)

5. The angle between a tangent and radius is 90° (at the point of contact)

6. Two tangents from an external point are equal in length

7. The alternate segment theorem: the angle between a tangent and chord equals the angle in the alternate segment

8. Perpendicular from the centre to a chord bisects the chord

When answering circle theorem questions, you must state which theorem you have used. Learn the exact names. Sketching additional construction lines (radii or chords) often reveals which theorem applies.

Congruence and similarity

Understanding when shapes are congruent or similar is essential:

Congruence conditions for triangles:

• SSS (three sides equal)
• SAS (two sides and included angle equal)
• ASA (two angles and corresponding side equal)
• RHS (right angle, hypotenuse, and one side equal)

Note that AAA proves similarity, not congruence, and SSA is not a valid congruence condition.

Similarity: If two shapes are similar with linear scale factor k:

• Corresponding lengths are in ratio k:1
• Areas are in ratio k²:1
• Volumes are in ratio k³:1

For similar triangles, calculate the scale factor by dividing corresponding sides. Apply this to find unknown lengths. For area and volume problems, remember to square or cube the linear scale factor respectively.

Constructions and loci

You must perform accurate constructions using only a ruler and compass:

Standard constructions:

• Perpendicular bisector of a line segment (creates the locus of points equidistant from two points)
• Bisector of an angle (creates the locus of points equidistant from two lines)
• Perpendicular to a line from a point
• 60° angle (using equilateral triangle properties)

Leave all construction arcs visible. Use a sharp pencil and maintain consistent compass width.

Common loci:

• Locus equidistant from a point: circle
• Locus equidistant from two points: perpendicular bisector
• Locus equidistant from a line: parallel lines on either side
• Locus equidistant from two lines: angle bisector

Loci questions often combine multiple conditions. Shade or label the required region clearly.

Worked examples

Example 1: Parallel lines and angles

In the diagram, AB is parallel to CD. Angle BAE = 68° and angle CDE = 115°. Find angle AED.

Solution:

Step 1: Extend line AE to meet CD (or identify the transversal)

Step 2: Find angle AEC using co-interior angles: Angle AEC + angle ECD = 180° (co-interior angles, AB || CD) The angle adjacent to 115° = 180° - 115° = 65°

Step 3: Use alternate angles: Angle AED = angle BAE (alternate angles, AB || CD... if configured differently, adjust accordingly)

Alternatively, using triangle properties: In triangle AED: 68° + angle AED + (180° - 115°) = 180° 68° + angle AED + 65° = 180° Angle AED = 180° - 133° = 47° (2 marks)

Note: The exact method depends on diagram configuration; always identify the angle relationship used.

Example 2: Circle theorem

A, B, C, and D lie on a circle. AC is a diameter. Angle BAC = 35°. Find angle BDC.

Solution:

Step 1: Find angle ABC Angle ABC = 90° (angle in a semicircle) (1 mark)

Step 2: Find angle ACB In triangle ABC: 35° + 90° + angle ACB = 180° (angle sum of triangle) Angle ACB = 55° (1 mark)

Step 3: Find angle BDC Angle BDC = angle BAC = 35° (angles in the same segment subtended by chord BC) (1 mark)

Total: 3 marks

Example 3: Similar shapes

Two similar rectangles have areas of 48 cm² and 108 cm². The length of the smaller rectangle is 8 cm. Find the length of the larger rectangle.

Solution:

Step 1: Find the area scale factor Area scale factor = 108 ÷ 48 = 2.25 or 9/4 (1 mark)

Step 2: Find the linear scale factor Linear scale factor = √(area scale factor) = √2.25 = 1.5 or 3/2 (1 mark)

Step 3: Calculate the length Length of larger rectangle = 8 × 1.5 = 12 cm (1 mark)

Total: 3 marks

Common mistakes and how to avoid them

• Confusing corresponding and alternate angles: Draw clear F and Z patterns on diagrams. Corresponding angles form an F-shape; alternate angles form a Z-shape

• Forgetting to square or cube the scale factor: Always check whether the question asks about length (use k), area (use k²), or volume (use k³). Write down which you're using

• Not stating circle theorems: Writing "angle in semicircle" or "angles in same segment" is essential for method marks. The calculation alone is insufficient

• Mixing interior and exterior angles: Remember that interior + exterior = 180° at each vertex. Check whether the formula requires interior or exterior angles

• Incomplete geometric reasoning: Every angle statement needs a reason: "vertically opposite," "isosceles triangle," "co-interior angles sum to 180°." This demonstrates understanding and earns method marks

• Poor construction technique: Use a sharp pencil, keep compass point fixed when drawing arcs, and maintain the same compass radius for paired arcs. Leave construction lines visible

Exam technique for "Geometry"

• Command word "calculate" requires working and a numerical answer. Show each step clearly with geometric reasons. Marks are awarded for method and accuracy separately

• Mark allocation indicates steps: A 3-mark angle question typically requires three pieces of reasoning or three calculations. If you reach the answer in one step, you've likely missed intermediate angles

• State theorems explicitly: Write "angle at centre = 2 × angle at circumference" or use standard abbreviated forms. Examiners cannot infer which theorem you've applied from calculations alone

• Check angle sum to 360° or 180°: As a final verification, add angles around a point or on a line. This catches arithmetic errors before you submit

Quick revision summary

Geometry questions require precise knowledge of angle relationships, polygon formulae, and circle theorems. Master parallel line angle patterns (F, Z, C), triangle and polygon angle sums, and all eight circle theorems with exact names. For similarity and congruence, identify the scale factor and apply it correctly to length, area, or volume. Always show geometric reasoning explicitly—state which property or theorem justifies each step. Accurate constructions with visible arcs demonstrate technique. Practice multi-step problems combining several concepts, as these reflect authentic examination questions.