Congruence of Triangles
1. What Is Congruence?
Two figures are CONGRUENT if they have EXACTLY the SAME shape and size.
Symbol: ≅ means 'is congruent to'.
Congruent Figures
- Two line segments of the SAME length are congruent.
- Two circles of the SAME radius are congruent.
- Two triangles of the SAME shape and size are congruent.
Correspondence
When we say ΔABC ≅ ΔDEF, it means:
- A corresponds to D, B to E, C to F (vertices).
- AB = DE, BC = EF, AC = DF (sides).
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (angles).
Important
The ORDER of vertices MATTERS. ΔABC ≅ ΔDEF means AB corresponds to DE, NOT to DF.
2. Conditions for Congruence of Triangles
SSS Criterion (Side-Side-Side)
If ALL THREE sides of one triangle are EQUAL to the corresponding three sides of another triangle, the triangles are CONGRUENT.
Example: ΔABC with sides 5, 6, 7 cm and ΔDEF with sides 5, 6, 7 cm → ΔABC ≅ ΔDEF (SSS).
SAS Criterion (Side-Angle-Side)
If TWO sides and the INCLUDED angle of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are CONGRUENT.
Important: The angle MUST be between the two given sides (INCLUDED angle).
Example: AB = 5 cm, BC = 6 cm, ∠B = 60° in one triangle. DE = 5 cm, EF = 6 cm, ∠E = 60° in another. ΔABC ≅ ΔDEF (SAS).
ASA Criterion (Angle-Side-Angle)
If TWO angles and the INCLUDED side of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are CONGRUENT.
Example: ∠A = 50°, ∠B = 60°, AB = 5 cm in ΔABC. ∠D = 50°, ∠E = 60°, DE = 5 cm in ΔDEF. ΔABC ≅ ΔDEF (ASA).
RHS Criterion (Right angle-Hypotenuse-Side)
In RIGHT triangles only: If the HYPOTENUSE and one SIDE of one right triangle are equal to the corresponding hypotenuse and side of another, the triangles are CONGRUENT.
3. Comparison of Congruence Criteria
| Criterion | What You Need | Works For |
|---|---|---|
| SSS | All 3 sides | Any triangle |
| SAS | 2 sides + INCLUDED angle | Any triangle |
| ASA | 2 angles + INCLUDED side | Any triangle |
| AAS | 2 angles + NON-included side | Any triangle (ASA equivalent) |
| RHS | Hypotenuse + 1 side | Right triangles ONLY |
NOT a Criterion
- AAA (Angle-Angle-Angle): This gives SIMILAR triangles (same shape, possibly different size), NOT congruent.
- SSA (Side-Side-Angle): Does NOT guarantee congruence (ambiguous case).
4. Worked Examples (ICSE Focus)
Example 1 (ICSE 2024, 3 marks)
'In ΔABC and ΔDEF, AB = DE = 5 cm, AC = DF = 4 cm, and ∠A = ∠D = 60°. Are the triangles congruent? Which criterion?'
Solution: AB = DE (given), AC = DF (given), ∠A = ∠D (given, included angle). YES, by SAS criterion: ΔABC ≅ ΔDEF.
Example 2 (ICSE 2023, 4 marks)
'In the figure, AB = AC and AD is the angle bisector of ∠A. Prove that BD = DC.'
Solution: In ΔABD and ΔACD:
- AB = AC (given)
- ∠BAD = ∠CAD (AD bisects ∠A)
- AD = AD (common side) Therefore ΔABD ≅ ΔACD (SAS criterion). Hence BD = DC (corresponding parts of congruent triangles).
Example 3: Application in Geometry
'P is the midpoint of AB and CD. Prove that AC = BD.'
Solution: In ΔAPC and ΔBPD:
- AP = BP (P is midpoint of AB)
- CP = DP (P is midpoint of CD)
- ∠APC = ∠BPD (vertically opposite angles) Therefore ΔAPC ≅ ΔBPD (SAS). Hence AC = BD (corresponding parts).
5. CPCT (Corresponding Parts of Congruent Triangles)
Once two triangles are proved congruent, the REMAINING corresponding parts (sides and angles) are also EQUAL.
This is the KEY to solving many geometry problems.
Steps to Prove
- Identify which two triangles to prove congruent.
- List the equal parts (sides and angles) with reasons.
- State the congruence criterion (SSS, SAS, ASA, RHS).
- Conclude using CPCT.
6. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Identifying congruence criterion | 2 marks | High |
| Proving triangles congruent | 3-4 marks | Very High |
| CPCT applications | 3-4 marks | Very High |
| RHS criterion in right triangles | 2-3 marks | Medium |
Common Mistakes
- SAS confusion: the angle must be INCLUDED (between the two sides).
- AAA is NOT a congruence criterion — only similarity.
- Wrong correspondence: matching wrong vertices leads to wrong conclusions.
- Not giving REASONS for equal parts (given, common, vertically opposite, etc.).
Self-Test (5 Questions)
Q1. Which criterion proves ΔABC ≅ ΔDEF if AB = DE, BC = EF, AC = DF? (1 mark)
- A) SAS
- B) SSS
- C) ASA
- D) RHS
Q2. State TRUE/FALSE: 'AAA is a valid congruence criterion.' (1 mark)
Q3. 'Can SAS be used if the given angle is NOT between the two sides?' (1 mark)
Q4. 'In ΔABC, AB = AC. AD ⊥ BC. Prove that BD = DC.' (3 marks)
Q5. 'In the figure, O is the midpoint of both AB and CD. Prove that AC = DB.' (4 marks)
Answers
A1. B) SSS. (All three sides are equal.) A2. FALSE. AAA gives similarity, not congruence. A3. NO. The angle must be the INCLUDED angle between the two given sides. A4. In ΔABD and ΔACD: AB = AC (given), AD = AD (common), ∠ADB = ∠ADC = 90° (AD ⊥ BC). By RHS: ΔABD ≅ ΔACD. Hence BD = DC (CPCT). A5. In ΔAOC and ΔBOD: AO = OB (O is midpoint), CO = OD (O is midpoint), ∠AOC = ∠BOD (vertically opposite). By SAS: ΔAOC ≅ ΔBOD. Hence AC = DB (CPCT).
