← Unit 4 Congruence and Similarity Tests

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MA5-GEO-C-01

Congruence and Similarity Tests

⏱ 25 min📚 Year 9📈
Think First

Two triangles have sides 5, 7, 9 and 10, 14, 18. Are they congruent? Similar? Neither?

💡 Revision: Ensure you understand corresponding sides and angles in triangles.

Learning Intentions

Know

  • SSS, SAS, ASA, RHS congruence
  • SSS, SAS, AA similarity
  • Corresponding parts

Understand

  • Why AAA is similarity but not congruence
  • How to choose the appropriate test

Can Do

  • Prove triangles congruent using SSS, SAS, ASA, RHS
  • Prove triangles similar using SSS, SAS, AA
  • Write formal geometric proofs
SSSSASASARHSAA
Learn Phase
1

Congruence Tests

Four ways to prove triangles congruent

Two triangles are congruent (identical) if:

  • SSS: All three sides equal
  • SAS: Two sides and the included angle equal
  • ASA: Two angles and the included side equal
  • RHS: Right angle, hypotenuse, and one side equal

Note: AAA (three angles) proves similarity, not congruence — the triangles could be different sizes.

2

Similarity Tests

Three ways to prove triangles similar

Two triangles are similar if:

  • SSS: All sides in proportion
  • SAS: Two sides in proportion and included angles equal
  • AA (or AAA): Two (or three) angles equal

If triangles are similar, corresponding sides are in the same ratio.

3

Writing Proofs

Structured geometric reasoning

A formal proof follows a logical structure:

  1. Given: State what is known
  2. To prove: State the goal
  3. Proof: Step-by-step reasoning with reasons
  4. Conclusion: State what has been proved

Example: Given AB = CD and angle ABC = angle DCB, prove triangle ABC congruent to triangle DCB.

Proof: AB = CD (given), BC = CB (common), angle ABC = angle DCB (given). Therefore triangles are congruent (SAS).

✓

Check Understanding

Try it yourself

Triangles ABC and DEF have AB=DE=5, BC=EF=7, angle B = angle E = 60°. Are they congruent? Which test?

Worked Example

Congruence and Similarity

1

Triangles have sides 3,4,5 and 6,8,10. Are they similar? Which test?

Ratios: $6/3 = 2$, $8/4 = 2$, $10/5 = 2$

All sides in proportion. Similar by SSS.

2

In triangle ABC, D and E are midpoints of AB and AC. Prove triangle ADE ~ triangle ABC.

AD/AB = 1/2, AE/AC = 1/2. Angle A is common.

Therefore similar by SAS.

3

Two right triangles have hypotenuses 13 and 26, and one side 5 and 10. Are they congruent? Similar?

Ratios: $26/13 = 2$, $10/5 = 2$. RHS holds for similarity.

Similar (not congruent) by RHS similarity.

Common Misconceptions

✕

Use AAA to prove congruence. No — AAA only proves similarity. Triangles with the same angles can be different sizes.

✕

Confuse SAS congruence with SAS similarity. SAS congruence requires equal sides; SAS similarity requires sides in proportion.

✕

Assume all equilateral triangles are congruent. They are all similar, but only congruent if sides are equal.

Your Turn

Practice — Congruence Tests

Work through each question in your book or digitally. Answers are in the Questions phase.

1Prove two triangles with sides 5,12,13 and 10,24,26 are similar.
2In a parallelogram ABCD, prove triangle ABC congruent to triangle CDA.
3Two triangles have angles 40°, 60°, 80°. Are they congruent? Similar? What can you conclude?
Real-World Anchor

Engineering and Surveying

Surveyors use congruence and similarity to calculate inaccessible distances. By creating similar triangles with known base lengths, they can determine the height of mountains or the width of rivers without direct measurement. The Great Trigonometric Survey of India used these principles.

📓 Copy Into Your Books

▼

Congruence

  • SSS - three sides
  • SAS - two sides, included angle
  • ASA - two angles, included side
  • RHS - right, hyp, side

Similarity

  • SSS - sides in proportion
  • SAS - two sides proportional, included angle
  • AA - two angles equal

Proof structure

  • Given
  • To prove
  • Proof with reasons
  • Conclusion
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
AAA proves:
Sides 4,5,6 and 8,10,12. Test:
Two angles and included side equal:
RHS applies to:
SAS similarity requires:
All equilateral triangles are:
If triangle ABC ≅ triangle DEF, then AB =
Two sides in proportion + included angle equal:
1Prove that two isosceles triangles with equal vertex angles are similar.
2In triangle ABC, D is on AB and E is on AC such that DE || BC. Prove ADE ~ ABC.
3Explain why SSA is not a valid congruence test.

Comprehensive Answers

1Isosceles triangles, equal vertex angles.
Base angles equal (angles sum to 180°). All angles equal, so similar by AA.
2DE || BC.
Angle ADE = angle ABC (corresponding), angle AED = angle ACB (corresponding), angle A common. Similar by AA.
3Why not SSA.
Two different triangles can have two sides and a non-included angle equal (ambiguous case).
MC 1AAA proves.
Similarity. Answer: B
MC 2Sides 4,5,6 and 8,10,12.
SSS similarity. Answer: B
MC 3Two angles + included side.
ASA. Answer: C
MC 4RHS applies to.
Right triangles only. Answer: B
MC 5SAS similarity.
Sides in proportion, included angle equal. Answer: B
MC 6Equilateral triangles.
Similar. Answer: B
MC 7ABC ≅ DEF, AB =.
DE (corresponding sides). Answer: A
MC 8Two sides proportion + included angle.
SAS similarity. Answer: B
SA 1Isosceles similar.
AA - base angles equal.
SA 2DE || BC.
AA - corresponding angles equal.
SA 3Why not SSA.
Ambiguous case - two different triangles possible.
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