Congruence & similarity

Two shapes are congruent if they are identical — the same size and shape, just possibly rotated or flipped. Two shapes are similar if they have the same shape but different sizes — all corresponding angles are equal, and corresponding sides are in the same ratio (scale factor). Proving triangles similar (using tests like AA — two matching angles) lets you find unknown side lengths using that ratio.

Key terms

Congruent:

Identical in shape and size.

Similar:

Same shape, different size, with equal angles and proportional sides.

Scale factor:

The ratio between corresponding sides of similar shapes.

Questions

1. 1. Congruent shapes are:

   • Identical in size and shape
   • Only the same shape, different size
   • Always different colours
   • Never the same in any way

2. 2. Similar shapes have:

   • The same shape but different sizes
   • Nothing in common
   • Identical sizes always
   • Only different angles, same sides

3. 3. In similar shapes, corresponding angles are:

   • Equal
   • Always different
   • Irrelevant
   • Always right angles

4. 4. In similar shapes, corresponding sides are:

   • In the same ratio (proportional)
   • Always identical in length
   • Unrelated to each other
   • Always doubled

5. 5. A scale factor describes:

   • The ratio between corresponding sides of similar shapes
   • The number of sides a shape has
   • The colour of a shape
   • The angles only

6. 6. Two congruent triangles have:

   • Identical angles and identical side lengths
   • Identical angles but different side lengths
   • Different angles and different sides
   • Nothing in common at all

7. 7. A photograph enlarged to a poster is an example of:

   • A similar shape (same proportions, bigger size)
   • A congruent shape
   • An unrelated shape
   • Neither similar nor congruent

8. 8. A shape with a scale factor of 2 applied means each side becomes:

   • Twice as long
   • Half as long
   • The same length
   • Four times as long

9. 9. A flagpole casts a 5m shadow while a 3m tall fence post casts a 2m shadow at the same time. Using similar triangles, the flagpole's height is:

   • 7.5m
   • 5m
   • 3m
   • 10m

10. 10. Two triangles with two equal corresponding angles (AA) are:

    • Similar
    • Always congruent, not just similar
    • Never related
    • Always a right-angle triangle

11. 11. A model car built at a scale of 1:20 means the real car is how many times bigger than the model?

    • 20
    • 1
    • 0.05
    • 200

12. 12. If two similar triangles have a scale factor of 3, and a side in the smaller triangle is 4cm, the corresponding side in the larger triangle is:

    • 12cm
    • 4cm
    • 7cm
    • 1.33cm

13. 13. A shape and its mirror-image reflection (same size) are:

    • Congruent
    • Never related in any way
    • Always similar but not congruent
    • Impossible to compare

14. 14. If two shapes are congruent, their scale factor is:

    • 1 (identical size)
    • Always greater than 1
    • Always less than 1
    • Undefined

15. 15. A tree casts a 12m shadow while a 1.5m student casts a 2m shadow at the same time. Using similar triangles, the tree's height is:

    • 9m
    • 12m
    • 16m
    • 1.5m

16. 16. Two similar rectangles have a scale factor of 2. If the smaller rectangle has an area of 6m², what is the larger rectangle's area?

    • 24m² (area scales by the square of the scale factor)
    • 12m²
    • 8m²
    • 6m²

17. 17. A map has a scale of 1:50,000. A distance of 4cm on the map represents an actual distance of:

    • 2km
    • 200m
    • 20km
    • 0.2km

18. 18. Why is the AA (angle-angle) test sufficient to prove triangle similarity, without needing to check side lengths?

    • If two angles match, the third angle must also match (angles sum to 180°), forcing the same shape
    • Angles have no connection to a triangle's overall shape
    • Any two triangles with any matching angle are automatically similar
    • Similarity always requires checking every single side and angle

19. 19. A architect's scale drawing at 1:100 shows a wall as 8cm long. The actual wall length is:

    • 8m
    • 80m
    • 0.8m
    • 800m

20. 20. Two similar solids have a scale factor of 3. If the smaller solid has a volume of 10cm³, the larger solid's volume is:

    • 270cm³ (volume scales by the cube of the scale factor)
    • 30cm³
    • 90cm³
    • 10cm³

21. 21. Why can two triangles have three equal angles (AAA) but not be congruent?

    • Matching angles only guarantees similarity, not identical size, since one could be a scaled-up copy of the other
    • AAA always guarantees congruence with no exceptions
    • Angles have no bearing on whether triangles are congruent or similar
    • Congruence and similarity always mean exactly the same thing