Congruence and similarity

Congruent shapes are exactly the same shape and size. Similar shapes are the same shape but one is an enlargement of the other, so all their lengths are in the same ratio and their angles are equal.

Key facts

  • Congruent triangles: SSS, SAS, ASA (or AAS), RHS.
  • Similar: corresponding angles equal and sides in the same ratio.
  • Length scale factor k gives area factor k² and volume factor k³.

Worked examples

Example 1

Two similar triangles have sides 4 cm and 10 cm that match. A side of 6 cm on the small one matches x on the big one. Find x.

  1. 10 ÷ 4 = 2.5
  2. x = 6 × 2.5 = 15 cm

Answer: 15 cm

Example 2

Two similar jugs have heights 10 cm and 20 cm. The small one holds 300 ml. How much does the big one hold?

  1. Length factor: 2.
  2. Volume factor: 2³ = 8.
  3. 300 × 8 = 2400 ml

Answer: 2400 ml