Congruence and Similarity –

Congruence and similarity are fundamental geometry concepts tested at both Foundation and Higher tiers across AQA, Edexcel, and OCR GCSE Maths papers. Two shapes are congruent if they are exactly the same shape and size. Two shapes are similar if they are the same shape but different sizes — one is an enlargement of the other. You need to identify congruent and similar shapes, prove triangle congruence using formal conditions, and use scale factors to find missing lengths, areas, and volumes. This guide covers everything, with worked examples and practice questions throughout. For related content on enlargement, see our enlargement and scale factor guide.

What Are Congruence and Similarity?

Congruence

Two shapes are congruent if they have the same shape and the same size. All corresponding sides are equal and all corresponding angles are equal. One shape can be mapped onto the other by a combination of reflections, rotations, and translations — but not by an enlargement.

Similarity

Two shapes are similar if they have the same shape but not necessarily the same size. All corresponding angles are equal, and all corresponding sides are in the same ratio. One shape is an enlargement of the other.

Conditions for Congruent Triangles

To prove two triangles are congruent, you must show one of these four conditions:

• SSS — three pairs of equal sides.
• SAS — two pairs of equal sides and the included angle between them is equal.
• ASA (or AAS) — two pairs of equal angles and a corresponding side is equal.
• RHS — both triangles have a right angle, equal hypotenuses, and one other pair of equal sides.

Scale Factors for Similar Shapes

Step-by-Step Method

Proving Congruence

1. Identify corresponding sides and angles in the two triangles.
2. List the equal pairs with reasons (e.g. "given", "common side", "vertically opposite angles").
3. State which congruence condition is satisfied (SSS, SAS, ASA, or RHS).
4. Conclude that the triangles are congruent.

Finding Missing Lengths in Similar Shapes

1. Identify two corresponding sides where both lengths are known.
2. Calculate the linear scale factor: divide the larger by the smaller (or new by original).
3. Multiply (or divide) the known side by the scale factor to find the missing length.

Finding Missing Areas or Volumes

1. Find the linear scale factor.
2. For area: square the linear scale factor and multiply.
3. For volume: cube the linear scale factor and multiply.

Worked Example 1 — Foundation Level

Question: Two similar rectangles have widths of 4 cm and 10 cm. The smaller rectangle has a length of 6 cm. Find the length of the larger rectangle.

Working:

Step 1 — Find the linear scale factor: Scale factor = 10 / 4 = 2.5

Step 2 — Multiply the known length by the scale factor: Length of larger rectangle = 6 × 2.5 = 15 cm

Answer: The length of the larger rectangle is 15 cm.

Worked Example 2 — Higher Level

Question: Two similar cylinders have heights of 5 cm and 15 cm. The smaller cylinder has a surface area of 80 cm². Find the surface area of the larger cylinder.

Working:

Step 1 — Find the linear scale factor: Scale factor = 15 / 5 = 3

Step 2 — Find the area scale factor: Area scale factor = 3² = 9

Step 3 — Multiply: Surface area of larger cylinder = 80 × 9 = 720 cm²

Answer: The surface area of the larger cylinder is 720 cm².

Common Mistakes

• Using the wrong congruence condition. SSA (two sides and a non-included angle) is not a valid condition for congruence. Only SSS, SAS, ASA/AAS, and RHS work.
• Confusing congruence with similarity. Congruent shapes are the same size; similar shapes are the same shape but can be different sizes. All congruent shapes are similar, but not all similar shapes are congruent.
• Using the linear scale factor for area. If the linear scale factor is 3, the area factor is 9, not 3. Always square for area and cube for volume.
• Matching the wrong corresponding sides. When finding the scale factor, make sure you are comparing sides that are in the same position on each shape. The longest side on one shape corresponds to the longest side on the other.
• Not simplifying the scale factor. A scale factor of 6/4 should be simplified to 3/2 or written as 1.5 for clarity.

Exam Tips

• For congruence proofs, list your pairs of equal elements clearly and number them. Then state the condition: "By SAS, triangle ABC is congruent to triangle DEF."
• Look for hidden similar triangles. When two triangles share an angle and have parallel sides, they are often similar. This is common in Higher tier questions involving parallel lines cutting a triangle.
• Write the scale factor as a fraction if dividing gives a recurring decimal — this avoids rounding errors.
• Always identify which shape is the "original" and which is the "new". This determines whether you multiply or divide by the scale factor.
• Volume and area questions often appear in context — water tanks, paint coverage, model buildings. Read carefully to identify whether the question asks for length, area, or volume.

Practice Questions

Question 1: Two similar triangles have corresponding sides of 6 cm and 9 cm. What is the linear scale factor?

Question 2: Two similar shapes have a linear scale factor of 4. The area of the smaller shape is 20 cm². Find the area of the larger shape.

Question 3: Two similar solids have volumes of 27 cm³ and 216 cm³. Find the linear scale factor.

Question 4: Triangle PQR has PQ = 5 cm, QR = 7 cm, and PR = 5 cm. Triangle XYZ has XY = 5 cm, YZ = 7 cm, and XZ = 5 cm. Are the triangles congruent? If so, state the condition.

Question 5: Two similar containers have heights 10 cm and 25 cm. The smaller container holds 400 ml of water. How much does the larger container hold?

Test your congruence and similarity skills with adaptive AI questions at GCSEMathsAI — get instant feedback and detailed solutions for every problem.

Related Topics

• Transformations: Enlargement — enlargement produces similar shapes.
• Transformations: Reflection, Rotation, Translation — these produce congruent shapes.
• Circle Theorems — proofs that use congruent triangles.
• Trigonometry — finding missing sides in similar right-angled triangles.

Summary

Congruent shapes are identical in shape and size; similar shapes are the same shape but different sizes. To prove triangles are congruent, use SSS, SAS, ASA, or RHS — making sure to list corresponding equal elements with reasons. For similar shapes, the linear scale factor connects corresponding lengths; the area scale factor is the square of the linear factor, and the volume scale factor is the cube. These relationships are tested extensively at both tiers, particularly in context-based Higher questions involving real-world objects. Clear, structured working and correct use of terminology will earn you full marks.