Two shapes are congruent if they are exactly the same size and shape — one could be placed exactly on top of the other after moving, rotating or reflecting it. Two shapes are similar if they have the same shape but different sizes — one is an enlargement or reduction of the other, with matching angles and sides in the same ratio. For triangles, you don't need to check every side and angle to prove congruence or similarity — reliable shortcut conditions exist, such as SSS (all three sides match), SAS (two sides and the included angle match), and AA (two angles match, which is enough to prove similarity since the third angle is then automatically equal too).
Example
Two triangles with sides 3cm, 4cm, 5cm and 6cm, 8cm, 10cm aren't congruent (different sizes) but ARE similar — every side of the second is exactly double the first (a scale factor of 2), and matching angles are identical.
Key terms
Congruent:
Exactly the same size and shape.
Similar:
The same shape but a different size, with sides in the same ratio.
Scale factor:
The number you multiply by to enlarge or reduce a shape.
Questions
1. Congruent shapes are:
Exactly the same size and shape
The same shape but different sizes
Never related to each other
Always triangles only
2. Similar shapes have:
The same shape but different sizes, with sides in the same ratio
Identical sizes and shapes always
No relationship between their sides
Always different angles
3. SSS as a congruence condition means:
All three sides match
All three angles match
One side matches
No sides need to match
4. A scale factor tells you:
The number you multiply by to enlarge or reduce a shape
The colour of a shape
The number of sides a shape has
Whether a shape is a triangle
5. Triangles with sides 3,4,5 and 6,8,10 have a scale factor of:
2
3
1
0.5
6. AA (two matching angles) is enough to prove triangles are:
Similar
Congruent always
Unrelated
Different shapes entirely
7. Two congruent triangles could be made to match by:
Moving, rotating or reflecting one onto the other
Only by making one bigger
Only by changing its angles
They can never be made to match
8. Why is AA sufficient to prove similarity, without needing to check the third angle?
Since a triangle's angles always sum to 180°, the third angle is automatically determined
The third angle is always irrelevant to any triangle
AA never proves anything about triangles
Similarity requires checking all three angles individually
9. Two triangles have sides 5,12,13 and 10,24,26. What is the scale factor?
2
3
1.5
4
10. SAS as a congruence condition requires:
Two sides and the included angle between them to match
Any two random sides to match
Three angles to match
Only one side to match
11. If two triangles are similar with a scale factor of 3, and one side of the smaller triangle is 4cm, the matching side on the larger triangle is:
12cm
7cm
4cm
1.33cm
12. Why might a surveyor use similar triangles to measure the height of a tall tree without climbing it?
Comparing a known small triangle (like a stick and its shadow) to a proportional larger one lets you calculate an unknown height
Similar triangles have no practical real-world use
Height can only ever be measured directly
This method would give a completely random result
13. Two shapes that are congruent must also be:
Similar (with a scale factor of exactly 1)
Never similar
A different shape entirely
Impossible to compare
14. A model car is built at a scale of 1:20. If the real car is 4m long, how long is the model?
0.2m (20cm)
4m
80m
2m
15. Why is proving congruence or similarity using shortcut conditions (like SSS or AA) more efficient than checking every single side and angle?
These conditions guarantee the remaining sides and angles must also match, without checking each one
Shortcut conditions are actually less reliable than checking everything
Every side and angle must always be individually verified regardless
SSS and AA only work for very specific, rare triangles
16. Two triangles share two equal angles but have different side lengths. What can you conclude?
They are similar but not congruent
They must be both similar and congruent
They share no mathematical relationship
This is mathematically impossible
17. Why might architects use similar triangles when creating scale drawings of buildings?
Scale drawings preserve the proportions and angles of the real structure at a manageable size
Scale drawings have no mathematical relationship to the real building
Similar triangles are irrelevant to architecture
Scale drawings must always be the exact same size as the real building
18. A triangle is enlarged by a scale factor of 2.5. If the original area was 8cm², what happens to the area (note: area scales by the square of the scale factor)?
New area = 8 × 2.5² = 50cm²
New area = 8 × 2.5 = 20cm²
The area stays exactly the same
The area cannot be calculated
19. Why can two triangles with matching SSS (all three sides equal) never differ in shape?
Fixing all three side lengths leaves only one possible triangle shape (up to reflection/rotation)
Matching sides never guarantees anything about a triangle's shape
Triangles with the same sides can still have completely different angles
SSS only applies to squares, not triangles
20. Two triangles have identical angles (AAA) but one is clearly larger than the other. Why does this prove similarity, not congruence?
Matching angles alone guarantee the same shape, but not the same size
AAA always proves congruence, not just similarity
This situation is mathematically impossible
Angles have no bearing on either congruence or similarity
21. Understanding congruence and similarity mainly helps you to:
Prove relationships between shapes efficiently and apply scale reasoning to real problems
Assume all shapes with any matching feature are identical
Avoid ever comparing two shapes
Treat every triangle as unrelated to any other
Answer key (parent copy)
1. Exactly the same size and shape
2. The same shape but different sizes, with sides in the same ratio
3. All three sides match
4. The number you multiply by to enlarge or reduce a shape
5. 2
6. Similar
7. Moving, rotating or reflecting one onto the other
8. Since a triangle's angles always sum to 180°, the third angle is automatically determined
9. 2
10. Two sides and the included angle between them to match
11. 12cm
12. Comparing a known small triangle (like a stick and its shadow) to a proportional larger one lets you calculate an unknown height
13. Similar (with a scale factor of exactly 1)
14. 0.2m (20cm)
15. These conditions guarantee the remaining sides and angles must also match, without checking each one
16. They are similar but not congruent
17. Scale drawings preserve the proportions and angles of the real structure at a manageable size
18. New area = 8 × 2.5² = 50cm²
19. Fixing all three side lengths leaves only one possible triangle shape (up to reflection/rotation)
20. Matching angles alone guarantee the same shape, but not the same size
21. Prove relationships between shapes efficiently and apply scale reasoning to real problems