The idea, step by step
Sufficient information
Congruent triangles match exactly in size and shape. Useful tests are SSS, SAS, ASA, AAS and RHS for right triangles. In SAS the equal angle must lie between the two known sides.
AAA establishes similarity, not congruence. SSA without extra restrictions is not a general congruence test because different triangles may satisfy the same information.
Use the result, not the target
List the given facts first. Choose a test from facts already established, then state the triangle correspondence. Only after congruence is proved may you conclude that other corresponding sides or angles are equal.
A proof is circular if it assumes the very side equality it is trying to establish. Similarity instead permits proportional corresponding sides; it does not turn unequal-sized figures into congruent ones.
A common trap
Using AAA as a congruence test.
Instead: Equal angles fix shape but not size. A triangle and a twice-as-large copy share all angles.
See the thinking
A shared diagonal
In quadrilateral ABCD, AB=AD and BC=CD. Prove ∠BAC=∠CAD.
- In triangles ABC and ADC, AB=AD and BC=DC are given.
- AC=AC is shared, so the triangles are congruent by SSS.
- The corresponding angles BAC and CAD are equal.
Lengths from similarity
Right triangles have corresponding legs 3, 4 and 6, 8. Explain why they are similar but not congruent.
- Both right angles are included between the listed legs.
- The leg ratios 3/6 and 4/8 both equal 1/2, so SAS similarity applies.
- The scale factor is 2 rather than 1, so they are not congruent.
Check your understanding
Two right triangles have equal hypotenuses and one equal corresponding leg. Which congruence test applies?
Show answer
RHS
Right angle, hypotenuse, and a corresponding side are given.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Two triangles have corresponding sides 5, 7, 8 and 5, 7, 8. Name the congruence test.2 marks
- Two triangles share side AC. AB=AD and BC=DC. Prove triangles ABC and ADC congruent.3 marks
- Explain why sides 5 and 7 with a non-included 30° angle do not automatically prove congruence.3 marks
- Triangles ABC and DEF have A↔D, B↔E, C↔F and are congruent. ∠B=52° and AC=9 cm. State ∠E and DF.2 marks
- Two triangles have proportional sides 2:3, 4:6, 6:9. Can they be non-degenerate triangles? Explain before applying a similarity test.3 marks
- Two triangles have corresponding angles 70° and 45°, and the included corresponding side is 6 cm in each. Are they congruent?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Two triangles have corresponding sides 5, 7, 8 and 5, 7, 8. Name the congruence test.2 marks
All three pairs of corresponding sides match.
SSS - Two triangles share side AC. AB=AD and BC=DC. Prove triangles ABC and ADC congruent.3 marks
AB=AD and BC=DC are given.
AC=AC is shared.
SSS congruence. - Explain why sides 5 and 7 with a non-included 30° angle do not automatically prove congruence.3 marks
The angle is not included between the known sides.
An ambiguous second triangle may exist.
This is SSA, not a general congruence test. - Triangles ABC and DEF have A↔D, B↔E, C↔F and are congruent. ∠B=52° and AC=9 cm. State ∠E and DF.2 marks
Use corresponding angles and sides after congruence.
52° and 9 cm - Two triangles have proportional sides 2:3, 4:6, 6:9. Can they be non-degenerate triangles? Explain before applying a similarity test.3 marks
2+4=6 violates the strict triangle inequality.
The listed points would be collinear.
No. - Two triangles have corresponding angles 70° and 45°, and the included corresponding side is 6 cm in each. Are they congruent?2 marks
Two angles and their included side match.
Yes, by ASA.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗