Congruence of Triangles - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter introduces the concept of congruence and the four criteria for proving triangles congruent: SSS, SAS, ASA, and RHS.
1. Why this chapter matters
Congruence is the mathematical way of saying 'same shape and same size.' Understanding congruence criteria is essential for geometric proofs in higher classes. In CBSE exams, this chapter contributes 6-8 marks and is directly connected to Class 9 triangle congruence proofs.
2. What is congruence?
Two geometric figures are congruent if they have exactly the same shape and size. They can be superimposed by sliding, rotating, or flipping.
Congruence symbol
The symbol for congruence is 'approximately equal to' written as the ~ and = symbols combined.
Congruent line segments
Two line segments are congruent if they have the same length.
Congruent angles
Two angles are congruent if they have the same measure.
Congruent triangles
Two triangles are congruent if their corresponding sides are equal and corresponding angles are equal.
3. Corresponding parts
When two triangles are congruent, the vertices can be matched in a specific order. Triangle ABC is congruent to triangle PQR means:
- Vertex A corresponds to P
- Vertex B corresponds to Q
- Vertex C corresponds to R
- Side AB = PQ, BC = QR, CA = RP
- Angle A = P, Angle B = Q, Angle C = R
This is written as CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
4. Congruence criteria
SSS (Side-Side-Side) criterion
If three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent.
SAS (Side-Angle-Side) criterion
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.
Note: The angle must be INCLUDED (between the two sides).
ASA (Angle-Side-Angle) criterion
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.
RHS (Right angle-Hypotenuse-Side) criterion
For right triangles only: If the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, the triangles are congruent.
5. Criteria comparison table
| Criterion | What is needed | Type of triangle | Included element |
|---|---|---|---|
| SSS | All 3 sides | Any | Not applicable |
| SAS | 2 sides + 1 angle | Any | The angle between the two sides |
| ASA | 2 angles + 1 side | Any | The side between the two angles |
| RHS | Hypotenuse + 1 side | Right only | Right angle is fixed |
6. Conditions that do NOT prove congruence
- AAA (Angle-Angle-Angle): Triangles can be similar but not congruent (different sizes).
- SSA (Side-Side-Angle): The angle is not included and can produce two different triangles.
7. Worked examples
Example 1: Triangle ABC has AB = 5 cm, BC = 7 cm, CA = 6 cm. Triangle PQR has PQ = 5 cm, QR = 7 cm, RP = 6 cm. Are they congruent?
Yes, by SSS criterion. AB = PQ, BC = QR, CA = RP.
Example 2: In triangles ABC and DEF, AB = 4 cm, BC = 5 cm, angle B = 60. DE = 4 cm, EF = 5 cm, angle E = 60. Are they congruent?
Yes, by SAS criterion. AB = DE (side), angle B = angle E (included angle), BC = EF (side).
Example 3: In right triangles ABC and PQR, angle B = angle Q = 90. AC = 10 cm, PR = 10 cm, AB = 6 cm, PQ = 6 cm. Are they congruent?
Yes, by RHS criterion. Hypotenuse AC = PR, side AB = PQ, right angles at B and Q.
Example 4: In two triangles, all three angles match but sides are 3 cm, 4 cm, 5 cm in one and 6 cm, 8 cm, 10 cm in the other. Are they congruent?
No. AAA does not guarantee congruence. The triangles are similar but not congruent.
8. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Using SSA as a criterion | SSA does not always prove congruence. Use SAS instead |
| Placing ASA and AAS incorrectly | ASA requires the included side; AAS is different |
| Forgetting to match corresponding vertices | Write vertices in corresponding order |
| Using AAA for congruence | AAA proves similarity, NOT congruence |
| Writing incorrect correspondence | Triangle ABC congruent to PQR means A matches P, B matches Q, C matches R |
9. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Identify congruence criterion | 2 marks | 1 question |
| Prove triangles congruent | 3 marks | 1 question |
| Find missing angle/side using CPCTC | 2 marks | 1 question |
| Application in geometry | 3 marks | Occasional |
| Distinguishing congruence vs similarity | 2 marks | 1 question |
10. Self-test
- Two triangles have sides 4 cm, 6 cm, 8 cm each. Which congruence criterion applies?
- Triangle ABC has AB = 5 cm, angle A = 50, AC = 7 cm. Triangle PQR has PQ = 5 cm, angle P = 50, PR = 7 cm. Are the triangles congruent? Which criterion?
- In two right triangles, hypotenuses are 13 cm and one side is 5 cm in each. Are they congruent?
- Why does AAA not prove congruence? Give an example.
- Triangle ABC is congruent to triangle DEF. If angle A = 70, angle B = 50, find angle F.
- In the figure, AB = CD and AB is parallel to CD. Prove that triangle AOB is congruent to triangle COD.
11. Answer key
- SSS criterion.
- Yes, SAS criterion (side AB = PQ, included angle A = P, side AC = PR).
- Yes, RHS criterion.
- Two equilateral triangles of different sizes have all angles 60 but sides are different. They are similar, not congruent.
- Angle C = 180 - (70 + 50) = 60. Since triangle ABC is congruent to DEF, angle F = angle C = 60.
- AB = CD (given). Angle OAB = angle OCD (alternate interior angles as AB parallel CD). Angle OBA = angle ODC (alternate interior angles). So triangle AOB is congruent to triangle COD by ASA.
12. Quick revision
- Congruent figures have same shape and size.
- SSS: three sides equal.
- SAS: two sides and included angle equal.
- ASA: two angles and included side equal.
- RHS: hypotenuse and one side of right triangle equal.
- AAA proves similarity, not congruence.
- CPCTC: Corresponding parts of congruent triangles are congruent.
