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Similar or congruent?

By exploring the minimum conditions required for triangles to be congruent or similar, students will develop a better understanding of the nuances of the tests. A series of activities for exploring congruence is provided in another part of the resource.

When proving results involving similarity and congruence, some students may still find it challenging to decide which test to use.

  • Problems involving equality of lengths usually involve congruence.
  • Problems involving proportions involve similarity.

In selecting a congruence test, students may make these errors:

  • if there is a right angle, using the RHS test and ignoring the possibility of SAS
  • when using SAS, not checking that the angle is included
  • when using AAS, not checking that the side is matching

In selecting a similarity test, students may make these errors:

  • incorrectly matching sides and angles
  • overlooking the SAS test
  • incorrectly referring to equal sides when proportion is required

There is no need to prove that three angles in a triangle are equal as only two are required. Although not an error, this common practice does waste time.

Exposure to a variety of problems which cover these situations will help your student avoid these pitfalls.

What is wrong with this proof?

When assessing similarity and congruence proofs, an awareness of some common sources of error will help you quickly recognise when they have been made.

Starting a congruence or similarity proof

Before writing a congruence or similarity proof students need to choose the appropriate test. This is best done through a process of elimination.

Complete the congruence proof

This cloze activity provides frameworks of varying support for developing congruence proofs.

Curriculum links

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