Which statement describes congruent shapes?
Congruent means identical.
Congruent shapes are exactly the same shape and size.
Congruence and Similarity
Congruent shapes are identical in size and shape, while similar shapes have the same shape but different sizes. This topic links closely to transformations, ratio and angle rules.
Overview
Congruent shapes are exactly the same size and shape.
Similar shapes have the same shape but different sizes.
Congruent = same shape + same size
Similar = same shape + proportional sides
Similar = same shape + proportional sides
In exam questions, you often need to decide whether shapes are congruent or similar, then use matching angles or scale factors to find missing lengths.
What you should understand after this topic
- Understand the difference between congruent and similar shapes
- Identify matching sides and angles
- Use scale factors with similar shapes
- Recognise how congruent triangles are proven
- Use similarity to find missing lengths
Key Definitions
Congruent
Exactly the same shape and exactly the same size.
Similar
Same shape, but size may be different.
Scale Factor
The multiplier that changes one shape into another similar shape.
Corresponding Sides
Sides in matching positions in two shapes.
Corresponding Angles
Angles in matching positions in two shapes.
Proportional
In the same ratio.
Key Rules
Congruent shapes
All corresponding sides equal and all corresponding angles equal.
Similar shapes
All corresponding angles equal and corresponding sides in the same ratio.
Congruent triangles
Use SSS, SAS, ASA or RHS.
Similar shapes scale factor
New length = old length × scale factor.
Congruence Triangle Criteria
SSS
Three sides equal.
SAS
Two sides and included angle equal.
ASA
Two angles and a side equal.
RHS
Right angle, hypotenuse and one side equal.
How to Solve
Congruent or similar?
Start by asking one key question: are the two shapes exactly the same size, or is one a scaled version of the other?
Congruent shapes have the same shape and the same size.
Similar shapes have the same shape but different sizes.
Exam tip: Decide whether the shapes are congruent or similar before doing any calculations.
Congruent shapes
Congruent shapes match exactly. One shape may be moved, reflected or rotated, but its size does not change.
Congruent shapes can be mapped onto each other exactly.
This links directly to transformations, especially translations, rotations and reflections.
Similar shapes
Similar shapes have equal corresponding angles, and all corresponding sides are in the same ratio.
Example: one triangle has sides 3, 4, 5.
Another triangle has sides 6, 8, 10.
Each side has been multiplied by 2, so the triangles are similar.
This uses ideas from ratio
How to check similarity
Exam tip: One incorrect ratio means the shapes are not similar.
- Identify corresponding sides.
- Compare the side ratios carefully.
- Check that the same scale factor works for every pair of sides.
- Confirm that corresponding angles match if required.
Using scale factor
If two shapes are similar, you can use the scale factor to find missing lengths.
\( \text{Scale factor} = \frac{\text{new length}}{\text{original length}} \)
Example: a side of 5 cm becomes 15 cm.
Scale factor \(= \frac{15}{5} = 3\).
All corresponding sides are multiplied by 3.
Scale factors are also used in transformations and problems involving transformations.
Congruent triangles
In triangle questions, congruence means you have enough information to prove two triangles are identical.
These conditions rely on angle facts and matching corresponding sides.
SSS
All three sides match.
SAS
Two sides and the included angle match.
ASA
Two angles and one side match.
RHS
Right-angled triangle with equal hypotenuse and one side.
Example Questions
Edexcel
Exam-style questions focusing on recognising congruent and similar shapes.
Edexcel
Two shapes are congruent if they are exactly the same shape and size.
Are the two triangles congruent?
Edexcel
Two squares have side lengths 4 cm and 8 cm.
Are the squares congruent or similar?
AQA
Exam-style questions focusing on scale factors and similar triangles.
AQA
Triangle A has side lengths 3 cm, 4 cm and 5 cm. Triangle B has side lengths 6 cm, 8 cm and 10 cm.
Are the triangles similar? Give a reason.
AQA
A side of 7 cm in one shape corresponds to a side of 21 cm in a similar shape.
Find the scale factor from the smaller shape to the larger shape.
OCR
Exam-style questions focusing on using similarity and congruence rules in reasoning.
OCR
A triangle has side lengths 5 cm, 8 cm and 10 cm. A similar triangle has shortest side 15 cm.
Find the longest side of the larger triangle.
OCR
Two right-angled triangles have equal hypotenuse and one equal shorter side.
Explain why the triangles are congruent.
OCR
The two rectangles are similar.
Find the missing height x.
Exam Checklist
Step 1
Decide whether the question is about congruence or similarity.
Step 2
Match corresponding sides carefully.
Step 3
Check whether all ratios match.
Step 4
Use the same scale factor throughout.
Most common exam mistakes
Wrong matching
Using sides that do not correspond.
Ratio mistake
Using different scale factors for different sides.
Definition mistake
Confusing similar with congruent.
Triangle rule mistake
Forgetting the correct congruence test such as SSS or RHS.
Common Mistakes
These are common mistakes students make when working with congruence and similarity in GCSE Maths.
Thinking similar shapes have equal sides
Incorrect
A student assumes similar shapes must have the same side lengths.
Correct
Similar shapes have the same shape but not the same size. Their sides are in proportion, not equal.
Using inconsistent scale factors
Incorrect
A student uses different scale factors for different sides.
Correct
All corresponding sides must be multiplied by the same scale factor. If the scale factor changes, the shapes are not similar.
Matching incorrect corresponding sides
Incorrect
A student compares sides that do not match in position.
Correct
Always match corresponding sides and angles correctly. Look for equal angles or parallel sides to identify the correct pairing.
Ignoring rotation or reflection
Incorrect
A student says shapes are not congruent because they look different.
Correct
Congruent shapes can be rotated or reflected. If they are the same size and shape, they are still congruent.
Confusing congruence and similarity rules
Incorrect
A student applies congruence rules when only similarity is required.
Correct
Congruence means exactly the same size and shape, while similarity allows different sizes. Make sure you apply the correct concept.
Try It Yourself
Practise identifying congruent and similar shapes using mathematical reasoning.
Foundation
Foundation Practice
Recognise congruent shapes and identify similarity using scale factors.
Which pair of triangles are congruent?
Compare size and shape.
The triangles are identical, so they are congruent.
A shape is enlarged with scale factor 3. A side was 4 cm. What is the new length?
Multiply by the scale factor.
4 × 3 = 12 cm.
Which statement describes similar shapes?
Think scaling.
Similar shapes have the same shape but different sizes.
Two similar triangles have scale factor 2. One side is 5 cm. Find the corresponding side.
Multiply by scale factor.
5 × 2 = 10 cm.
Which pair are NOT similar?
Check proportions.
The sides are not proportional, so they are not similar.
A shape is reduced by scale factor 0.5. A side is 12 cm. Find the new length.
Multiply by 0.5.
12 × 0.5 = 6 cm.
Congruent triangles must have:
Think identical.
All sides and angles must match.
A triangle is enlarged from 3 cm to 9 cm. Find the scale factor.
Divide new by original.
9 ÷ 3 = 3.
If two shapes are similar, which is always true?
Think shape.
Angles are always equal in similar shapes.
Higher
Higher Practice
Apply congruence rules (SSS, SAS, ASA) and similarity reasoning with ratios.
Which condition proves triangles are congruent?
Three sides equal.
SSS proves congruence.
Which rule proves triangles are congruent?
Side-Angle-Side.
SAS is a valid congruence rule.
Two similar triangles have sides 4 cm and 10 cm. Find the scale factor from small to large.
Divide larger by smaller.
10 ÷ 4 = 2.5.
Two similar triangles have scale factor 3. Area of smaller triangle is 5 cm². Find area of larger triangle.
Area scale factor = square of linear scale factor.
3² = 9, so 5 × 9 = 45 cm².
Which condition proves similarity?
Angles equal.
AAA proves similarity.
A triangle is enlarged by scale factor 4. Volume scale factor (for 3D extension) would be:
Cube the scale factor.
4³ = 64.
Two triangles have equal angles but different sizes. They are:
Angles define shape.
Same angles means similar.
A triangle has sides 3, 4, 5. Another has 6, 8, 10. Are they similar? Enter yes or no.
Check ratios.
All sides scale by 2, so they are similar.
Which is NOT a congruence rule?
AAA only proves similarity.
AAA does not prove congruence.
A triangle is reduced from 12 cm to 3 cm. Find the scale factor.
Divide new by original.
3 ÷ 12 = 0.25.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does congruent mean?
Shapes that are identical in size and shape.
What does similar mean?
Shapes with the same shape but different size.
What stays the same in similar shapes?
Angles stay the same, sides are proportional.