Home > Geometric reasoning > Misunderstandings > Similar or congruent? > Starting a congruence or similarity proof

Starting a congruence or similarity proof

As an initial strategy, add any information contained in the text of the problem to the diagram.

Then identify the different triangles by highlighting them in different colours. Re-draw the triangles at the side of the main diagram if visualisation is difficult. The naming of corresponding vertices, angles and sides in the two triangles is critical.

Download and watch the Looking beyond the lines slide show presentation.

List the four tests. Then systematically eliminate the inappropriate tests by looking for:

  • a right angle
  • side relationships
  • angle relationships.

Once the correct test has been selected, write a structured proof.

  • Introduce the two triangles.
    In subsequent lines, information relating to the first triangle should be written first.
  • State the facts in the order required by the test.
    This alerts students to look for the matching side or the included angle and also clearly supports the test.
  • Conclude with a statement of congruence or similarity, writing the vertices in matching order.
    This is a requirement of the congruence (\(\cong\)) and similarity (\(\sim\)) symbols. It also helps in the subsequent matching of sides, angles or proportions

The facts which were previously unknown are now confirmed true and can be used to prove further results.

Curriculum links

Year 10: Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes.