Chapter 11 - Congruence of Triangles Revision Notes - Class 7 Mathematics

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📌 Key Points

Congruent triangles have identical shapes and sizes. All corresponding parts are equal.
Congruence symbol: △ABC ≅ △DEF means A↔D, B↔E, C↔F in correspondence.
SSS Criterion: All three sides equal → triangles congruent.
SAS Criterion: Two sides and INCLUDED angle equal → triangles congruent.
ASA Criterion: Two angles and INCLUDED side equal → triangles congruent.
RHS Criterion: For right triangles - hypotenuse and one leg equal → congruent.
SSA does NOT prove congruence (ambiguous case - can form two triangles).
CPCT (Corresponding Parts of Congruent Triangles): Use after proving congruence.
If congruent, corresponding sides are equal AND corresponding angles are equal.
RHS criterion ONLY applies to RIGHT triangles, not acute or obtuse.
In SAS, the angle must be BETWEEN the two sides (included angle).
In ASA, the side must be BETWEEN the two angles (included side).
Congruent triangles have equal perimeters and equal areas.
Similar triangles have same shape but not necessarily same size. Not congruent unless scale = 1.
For proving congruence: first establish which criterion applies, then conclude congruence.

📘 Important Definitions

Congruent Triangles

Two triangles are congruent if all corresponding sides are equal and all corresponding angles are equal.

SSS Criterion

Side-Side-Side criterion: If all three sides of one triangle equal corresponding sides of another, triangles are congruent.

SAS Criterion

Side-Angle-Side criterion: If two sides and the included angle equal, triangles are congruent. Angle must be between sides.

ASA Criterion

Angle-Side-Angle criterion: If two angles and the included side equal, triangles are congruent. Side must be between angles.

RHS Criterion

Right angle-Hypotenuse-Side criterion: For right triangles, if hypotenuse and one leg equal, triangles are congruent.

CPCT

Corresponding Parts of Congruent Triangles. When triangles are congruent, all their corresponding parts are equal.

Corresponding Parts

In congruent triangles △ABC ≅ △DEF, corresponding sides are AB-DE, BC-EF, CA-FD and corresponding angles are ∠A-∠D, ∠B-∠E, ∠C-∠F.

Included Angle

An angle formed by two given sides. In SAS criterion, the angle must be included (between the two sides).

🔢 Formulas & Laws

Congruence Notation

△ABC ≅ △DEF

Vertex order matters. Corresponding vertices must match.

SSS Congruence

If AB = DE, BC = EF, CA = FD → △ABC ≅ △DEF

All three sides must be equal.

SAS Congruence

If AB = DE, ∠B = ∠E, BC = EF → △ABC ≅ △DEF

Angle ∠B and ∠E must be between sides AB-BC and DE-EF respectively.

ASA Congruence

If ∠A = ∠D, AB = DE, ∠B = ∠E → △ABC ≅ △DEF

Side AB must be between angles ∠A and ∠B.

RHS Congruence

If ∠A = 90°, ∠D = 90°, BC = EF (hypotenuse), AB = DE (leg) → △ABC ≅ △DEF

Only for right triangles.

⚠️ Common Mistakes

✗ Wrong: Using SSA as a valid criterion. Thinking two sides and any angle prove congruence.

✓ Correct: SSA is NOT valid. Use SAS where angle must be INCLUDED (between the two sides).

✗ Wrong: Confusing SAS with SSA. Using non-included angle.

✓ Correct: In SAS, the angle must be between the two given sides. Example: AB, ∠B, BC (valid) NOT AB, BC, ∠A (invalid).

✗ Wrong: Writing correspondence incorrectly. △ABC ≅ △DEF but saying AB = EF.

✓ Correct: Write corresponding vertices in order. △ABC ≅ △DEF means AB = DE, BC = EF, CA = FD.

✗ Wrong: Using RHS for non-right triangles.

✓ Correct: RHS criterion ONLY applies to right triangles. For other triangles, use SSS, SAS, or ASA.

✗ Wrong: Applying CPCT without first proving congruence.

✓ Correct: Always prove congruence first using one of four criteria, then apply CPCT.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Identify correct congruence criterion from given data (very common, 2-5 marks)

5m★★★

Prove two triangles are congruent and find unknown parts using CPCT (very common, 3-5 marks)

5m★★★

Solve problems using properties of isosceles triangles via congruence (common, 3-5 marks)

5m★★

Work with right triangles and RHS criterion (common, 2-4 marks)

4m★★

MCQ: Identify congruence criterion, corresponding parts (common, 1 mark each)

1m★★★

Distinguish between congruence, similarity, and equality (occasional, 1-2 marks)

🎯 Last-Minute Recall