What you'll learn
This topic examines how shapes relate to one another through congruence (identical size and shape) and similarity (same shape, different size). You'll prove triangles are congruent using four key conditions, calculate unknown lengths and angles in similar figures, and apply scale factors to solve real-world problems. These concepts appear regularly in CIE IGCSE Mathematics Paper 2 and Extended Paper 4, often combined with circle theorems, trigonometry, or transformations.
Key terms and definitions
Congruent shapes — figures that are identical in both shape and size; corresponding sides and angles are equal, though the shapes may be rotated, reflected, or translated.
Similar shapes — figures with the same shape but different sizes; corresponding angles are equal and corresponding sides are in the same ratio.
Scale factor — the multiplier that relates corresponding lengths in similar figures; if shape B is an enlargement of shape A with scale factor k, then every length in B = k × corresponding length in A.
Corresponding sides — sides that occupy the same relative position in similar or congruent figures, typically identified by matching angles at their endpoints.
SSS, SAS, ASA, RHS — the four conditions that prove triangle congruence (Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Right angle-Hypotenuse-Side).
Ratio of areas — when the linear scale factor between similar shapes is k, the ratio of their areas is k².
Ratio of volumes — when the linear scale factor between similar solids is k, the ratio of their volumes is k³.
Core concepts
Conditions for congruent triangles
Two triangles are congruent if they satisfy one of four conditions. The CIE IGCSE syllabus requires you to identify and apply each:
SSS (Side-Side-Side): All three sides of one triangle equal the three sides of another triangle. The order matters — you must match corresponding sides correctly.
Example: Triangle ABC with sides 5 cm, 7 cm, 9 cm is congruent to triangle DEF with sides 5 cm, 7 cm, 9 cm (when matched appropriately).
SAS (Side-Angle-Side): Two sides and the included angle (the angle between those two sides) of one triangle equal two sides and the included angle of another.
Critical point: The angle must be between the two sides. Two sides and a non-included angle do NOT guarantee congruence.
ASA (Angle-Side-Angle): Two angles and the corresponding side (the side between those angles or opposite to a specific angle) of one triangle equal two angles and the corresponding side of another.
Note: Since angles in a triangle sum to 180°, if two angles match, the third automatically matches. ASA is sometimes expressed as AAS (Angle-Angle-Side).
RHS (Right angle-Hypotenuse-Side): In right-angled triangles only, if the hypotenuse and one other side of one triangle equal the hypotenuse and one side of another, the triangles are congruent.
This condition applies exclusively to right triangles and is particularly useful in problems involving perpendicular lines or Pythagoras' theorem.
Proving congruence in exam questions
CIE examiners expect a structured proof:
- Identify the triangles clearly by naming vertices in corresponding order
- State three facts about equal sides or angles, marking them on diagrams when possible
- Name the condition (SSS, SAS, ASA, or RHS)
- Write the conclusion: "Therefore triangle ABC ≅ triangle DEF" (using the congruence symbol ≅)
Common diagram scenarios:
- Triangles sharing a common side (that side is equal to itself)
- Parallel lines creating equal alternate or corresponding angles
- Isosceles triangles with equal base angles
- Perpendicular bisectors creating right angles and equal segments
Conditions for similar figures
Two figures are similar when:
- All corresponding angles are equal
- All corresponding sides are in the same ratio (the scale factor)
For triangles specifically, you need only prove one of these conditions:
- AAA (Angle-Angle-Angle): If two triangles have equal corresponding angles, they are similar (proving two angles equal is sufficient, since the third follows automatically)
- SSS (proportional): If all three pairs of corresponding sides are in the same ratio
- SAS (proportional): If two pairs of sides are in the same ratio and the included angles are equal
Calculating with similar shapes
When two shapes are similar with linear scale factor k:
Finding the scale factor: Divide any length in the enlarged shape by the corresponding length in the original shape.
Scale factor k = length in image ÷ length in object
Finding unknown lengths: Multiply known lengths by k (when enlarging) or divide by k (when reducing).
Areas of similar shapes: If the linear scale factor is k, then:
- Area of image = k² × area of object
Example: Two similar rectangles have lengths 4 cm and 10 cm. The linear scale factor is 10 ÷ 4 = 2.5. If the smaller rectangle has area 12 cm², the larger has area 12 × 2.5² = 12 × 6.25 = 75 cm².
Volumes of similar solids: If the linear scale factor is k, then:
- Volume of image = k³ × volume of object
This relationship extends to capacity, mass (if density is constant), and surface area (which scales as k²).
Similar triangles in geometric problems
Similar triangles frequently appear in problems involving:
Parallel lines cutting transversals: When a line parallel to one side of a triangle cuts the other two sides, it creates a smaller triangle similar to the original.
If DE is parallel to BC in triangle ABC, then triangle ADE is similar to triangle ABC. This leads to the important relationship: AD/AB = AE/AC = DE/BC
Shadow problems and indirect measurement: An object and its shadow form a right triangle. At the same time of day, another object and its shadow form a similar triangle, allowing height calculations.
Nested triangles: Intersecting lines often create multiple similar triangles within one diagram. Identifying all pairs is crucial for complex problems.
Applications in coordinate geometry
Congruence and similarity connect to transformations:
- Translations, rotations, and reflections produce congruent shapes (no size change)
- Enlargements produce similar shapes (unless scale factor = ±1, which gives congruence)
When proving shapes are congruent or similar on coordinate grids, calculate side lengths using the distance formula: distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Then compare these lengths and use angle properties to establish the required condition.
Worked examples
Example 1: Proving triangle congruence
Question: In the diagram, AB = AD and BC = DC. Prove that triangle ABC is congruent to triangle ADC.
Solution: Consider triangles ABC and ADC.
AB = AD (given) BC = DC (given) AC = AC (common side, shared by both triangles)
All three sides of triangle ABC equal the three sides of triangle ADC.
Therefore triangle ABC ≅ triangle ADC (SSS condition)
[2 marks: 1 for identifying three equal elements, 1 for stating SSS]
Example 2: Finding lengths in similar triangles
Question: Triangle PQR is similar to triangle XYZ. PQ = 6 cm, QR = 8 cm, PR = 10 cm, and XY = 9 cm. Calculate the length of YZ and the perimeter of triangle XYZ.
Solution: The triangles are similar, so corresponding sides are in the same ratio.
Scale factor = XY ÷ PQ = 9 ÷ 6 = 1.5
YZ corresponds to QR: YZ = 1.5 × 8 = 12 cm
XZ corresponds to PR: XZ = 1.5 × 10 = 15 cm
Perimeter of triangle XYZ = 9 + 12 + 15 = 36 cm
[4 marks: 1 for correct scale factor, 1 for YZ, 1 for XZ, 1 for perimeter]
Example 3: Areas of similar shapes
Question: Two similar cylinders have heights 5 cm and 12 cm. The surface area of the smaller cylinder is 80 cm². Calculate: (a) the surface area of the larger cylinder (b) Given that the larger cylinder has volume 864 cm³, find the volume of the smaller cylinder.
Solution: (a) Linear scale factor k = 12 ÷ 5 = 2.4
Area scale factor = k² = 2.4² = 5.76
Surface area of larger cylinder = 80 × 5.76 = 460.8 cm²
(b) Volume scale factor = k³ = 2.4³ = 13.824
If larger volume = 864 cm³: 864 = smaller volume × 13.824 Smaller volume = 864 ÷ 13.824 = 62.5 cm³
[5 marks: 1 for linear scale factor, 2 for area calculation, 2 for volume calculation]
Common mistakes and how to avoid them
Mistake: Confusing congruence and similarity — stating triangles are congruent when they're only similar, or vice versa. Correction: Congruent means identical in every measurement. Similar means same shape, proportional sides. Check whether actual measurements are equal or merely proportional.
Mistake: Using two sides and a non-included angle to prove congruence (thinking "SSA" is valid). Correction: SSA is NOT a congruence condition. Only SAS works, where the angle must be between the two sides. Always check the angle's position relative to the given sides.
Mistake: Applying the linear scale factor to areas or volumes directly. Correction: Square the scale factor for areas (k²) and cube it for volumes (k³). If the height doubles, the area multiplies by 4 and volume by 8.
Mistake: Incorrectly matching corresponding sides in similar shapes, leading to wrong scale factors. Correction: Use angles to identify corresponding sides — sides opposite equal angles correspond. Label diagrams carefully and write ratios explicitly (e.g., AB/XY = BC/YZ).
Mistake: Forgetting to justify why a side or angle is equal in congruence proofs (e.g., assuming vertically opposite angles without stating it). Correction: Every fact needs a reason: "given", "common side", "vertically opposite angles", "alternate angles (AB || CD)", etc. CIE mark schemes require explicit justification.
Mistake: Calculating only one dimension when asked for perimeter or total measurements in similar shapes. Correction: Apply the scale factor to every relevant length, then sum to find perimeter. Read the question carefully to identify what's being asked.
Exam technique for congruence and similarity questions
Command words and mark allocation: "Prove" or "Show that" questions require full working and justification (typically 2-3 marks for congruence proofs). "Calculate" or "Find" questions need numerical answers with working (1-2 marks per value). "Explain" requires reasons in words, not just calculations.
Diagram annotation: When diagrams are given, mark equal sides with identical tick marks and equal angles with matching arcs. This visual organization prevents errors and communicates your reasoning to examiners, potentially earning method marks even if the final answer is incorrect.
Structured congruence proofs: Follow the four-step format rigidly: identify triangles → state three facts with reasons → name condition → write conclusion. Missing any step typically loses marks. Write triangle vertices in corresponding order (e.g., ABC ≅ DEF, not ABC ≅ EFD if that's incorrect).
Scale factor clarity: Always show the division that gives your scale factor (e.g., "scale factor = 15 ÷ 6 = 2.5"). Then explicitly show multiplication by k, k², or k³ as appropriate. This method earns full marks even if you make an arithmetic error, whereas jumping to an answer loses method marks when wrong.
Quick revision summary
Congruent shapes are identical; prove triangle congruence using SSS (three sides), SAS (two sides and included angle), ASA (two angles and corresponding side), or RHS (right angle, hypotenuse, one side). Similar shapes have equal angles and proportional sides; linear scale factor k relates lengths, k² relates areas, k³ relates volumes. Calculate k by dividing corresponding lengths. In proofs, justify every equality with reasons. In calculations, show scale factor working explicitly. Practice identifying corresponding sides using angle matching. Common errors include confusing linear and area/volume scale factors and using invalid conditions like SSA.