The idea, step by step
Enough information fixes a triangle
SSS uses three corresponding equal sides. SAS uses two corresponding equal sides and the equal angle between those sides. ASA uses two equal angles and their included side; AAS is also sufficient because the third angle is fixed by the angle sum.
RHS applies only to right-angled triangles: the right angle, equal hypotenuses and one corresponding equal shorter side establish congruence. The hypotenuse is opposite the right angle, not whichever side appears horizontal.
| Test | Information needed |
|---|---|
| SSS | Three corresponding sides |
| SAS | Two sides and their included angle |
| ASA or AAS | Two angles and a corresponding side |
| RHS | Right angle, hypotenuse and a corresponding shorter side |
Know what is not enough
Three equal angles establish the same shape but do not fix size. Triangles with sides 3, 4, 5 and 6, 8, 10 have equal angles but are not congruent.
Two sides and a non-included angle are not a general congruence test. The missing side can sometimes occupy two different positions. Check whether the angle really lies between the supplied sides before claiming SAS.
A common trap
Saying SAS whenever two sides and any angle are given.
Instead: For SAS, the angle must be the angle between the two known sides.
See the thinking
Recognise an included angle
In triangles ABC and DEF, AB = DE = 4 cm, AC = DF = 7 cm, and angles A and D both equal 50°. Prove congruence.
- At A, the given angle lies between AB and AC. At D, it lies between DE and DF.
- The two side pairs and their included angle match.
- Therefore by SAS.
Use a right angle
Two right-angled triangles each have hypotenuse 10 cm and a corresponding shorter side of 6 cm. Must they be congruent?
- Both triangles have a right angle.
- Their hypotenuses and one matching shorter side are equal.
- RHS therefore fixes the same triangle size and shape.
Check your understanding
Both triangles have side lengths 5 cm, 6 cm and 7 cm. Which congruence test applies?
Show answer
SSS.
All three corresponding side lengths are equal.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Triangles have corresponding side lengths 4, 6, 8 cm and 4, 6, 8 cm. State the congruence test.1 mark
- Two triangles each have angles 40° and 60°, and the side between these angles is 7 cm in both. State the test.2 marks
- Two right triangles have hypotenuse 13 cm and a corresponding leg of 5 cm. State the test.2 marks
- Triangles each have angles 35°, 65° and 80°. Is congruence guaranteed? Explain.2 marks
- AB = DE, AC = DF and angles A and D are equal. State a test and identify why it applies.3 marks
- A student uses SSA as a general congruence test. Explain why this is unsafe.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Triangles have corresponding side lengths 4, 6, 8 cm and 4, 6, 8 cm. State the congruence test.1 mark
All three side pairs match.
SSS. - Two triangles each have angles 40° and 60°, and the side between these angles is 7 cm in both. State the test.2 marks
Two angles and their included side match.
ASA. - Two right triangles have hypotenuse 13 cm and a corresponding leg of 5 cm. State the test.2 marks
Right angle, hypotenuse and one matching shorter side are equal.
RHS. - Triangles each have angles 35°, 65° and 80°. Is congruence guaranteed? Explain.2 marks
AAA gives similarity; one triangle could be an enlargement of the other.
No. Equal angles do not fix the triangle’s size. - AB = DE, AC = DF and angles A and D are equal. State a test and identify why it applies.3 marks
AB and AC meet at A.
DE and DF meet at D.
SAS, because A and D are included between the given side pairs. - A student uses SSA as a general congruence test. Explain why this is unsafe.2 marks
The included-angle requirement of SAS has not been met.
Extra information or a valid test is needed.
A non-included angle with two sides may allow more than one triangle.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.
Official curriculum source ↗