Congruence of Triangles
Congruence means having the same shape and the same size. Two figures are congruent if one can be placed exactly over the other so that they match perfectly.
Two triangles are congruent if all three sides and all three angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle.
In Class 7 Mathematics (NCERT), congruence of triangles is studied as a separate chapter. You will learn the conditions under which two triangles are congruent, without needing to check all six measurements (3 sides + 3 angles).
There are four congruence rules: SSS, SAS, ASA, and RHS. Each rule specifies a minimum set of measurements that guarantee congruence.
What is Congruence of Triangles?
Definition: Two triangles are said to be congruent if they have the same shape and size — that is, all corresponding sides are equal and all corresponding angles are equal.
Notation:
- If triangle ABC is congruent to triangle PQR, we write: △ABC ≅ △PQR
- The order of vertices matters: A corresponds to P, B to Q, C to R.
- This means: AB = PQ, BC = QR, AC = PR, ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.
CPCT — Corresponding Parts of Congruent Triangles:
- CPCT states that if two triangles are congruent, then all their corresponding parts (sides and angles) are equal.
- This is used AFTER proving congruence, to deduce that specific sides or angles are equal.
- Example: If △ABC ≅ △DEF, then by CPCT, AB = DE, ∠A = ∠D, etc.
Corresponding parts:
- The parts that occupy the same position in two congruent triangles.
- Matching is determined by the order of vertices in the congruence statement.
Congruence of Triangles Formula
Four congruence rules (criteria):
SSS, SAS, ASA (or AAS), RHS
1. SSS (Side-Side-Side):
- If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
2. SAS (Side-Angle-Side):
- If two sides and the included angle (the angle between them) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
3. ASA (Angle-Side-Angle):
- If two angles and the included side (the side between them) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
4. RHS (Right Angle-Hypotenuse-Side):
- If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, they are congruent.
- This rule applies only to right-angled triangles.
Derivation and Proof
Why do these rules work?
Why SSS guarantees congruence:
- If you know all three sides, you can construct only ONE triangle (up to reflection).
- Try it: take 3 sticks of fixed lengths. You can form only one triangle shape with them.
- So if two triangles have all three sides equal, they must be identical in shape and size.
Why SAS guarantees congruence:
- If you fix two sides and the angle between them, the third side and remaining angles are automatically determined.
- You cannot form two different triangles with the same SAS combination.
Why ASA guarantees congruence:
- If you fix two angles and the side between them, the remaining side lengths and angle are determined.
- The two angles fix the shape, and the included side fixes the size.
Why AAA (Angle-Angle-Angle) does NOT guarantee congruence:
- Two triangles with the same three angles can have different sizes — they are similar, not necessarily congruent.
- Example: A triangle with angles 60°-60°-60° could have sides 3-3-3 or 6-6-6. Same angles, different sizes.
Why SSA (Side-Side-Angle) does NOT guarantee congruence:
- Two sides and a non-included angle can sometimes form two different triangles (the ambiguous case).
- This is why SAS requires the included angle — the angle must be between the two given sides.
Types and Properties
Types of congruence problems:
1. Identifying the congruence rule:
- Given information about two triangles, determine which rule (SSS, SAS, ASA, RHS) applies.
2. Proving triangles congruent:
- List the equal parts, state the rule, and write the congruence statement.
- The order of vertices in the statement must match the correspondence.
3. Using CPCT to find unknown values:
- First prove triangles congruent, then use CPCT to find unknown sides or angles.
4. Checking if information is sufficient:
- Determine whether given information about two triangles is enough to prove congruence.
- Remember: AAA and SSA are NOT sufficient.
5. Real-life congruence:
- Identifying congruent objects in daily life — tiles, stamps, coins, etc.
Solved Examples
Example 1: Example 1: SSS congruence
Problem: In △ABC and △DEF: AB = DE = 5 cm, BC = EF = 7 cm, AC = DF = 6 cm. Are the triangles congruent? State the rule.
Solution:
Given:
- AB = DE = 5 cm
- BC = EF = 7 cm
- AC = DF = 6 cm
All three pairs of corresponding sides are equal.
By SSS congruence rule: △ABC ≅ △DEF.
Answer: Yes, the triangles are congruent by SSS rule.
Example 2: Example 2: SAS congruence
Problem: In △PQR and △XYZ: PQ = XY = 4 cm, ∠Q = ∠Y = 60°, QR = YZ = 6 cm. Are they congruent?
Solution:
Given:
- PQ = XY = 4 cm (side)
- ∠Q = ∠Y = 60° (included angle)
- QR = YZ = 6 cm (side)
Two sides and the included angle are equal.
By SAS congruence rule: △PQR ≅ △XYZ.
Answer: Yes, congruent by SAS rule.
Example 3: Example 3: ASA congruence
Problem: In △ABC and △DEF: ∠A = ∠D = 50°, AB = DE = 5 cm, ∠B = ∠E = 70°. Are they congruent?
Solution:
Given:
- ∠A = ∠D = 50° (angle)
- AB = DE = 5 cm (included side — between ∠A and ∠B)
- ∠B = ∠E = 70° (angle)
Two angles and the included side are equal.
By ASA congruence rule: △ABC ≅ △DEF.
Answer: Yes, congruent by ASA rule.
Example 4: Example 4: RHS congruence
Problem: In right triangles △ABC and △PQR: ∠B = ∠Q = 90°, AC = PR = 13 cm (hypotenuse), AB = PQ = 5 cm. Are they congruent?
Solution:
Given:
- ∠B = ∠Q = 90° (right angle)
- AC = PR = 13 cm (hypotenuse)
- AB = PQ = 5 cm (one side)
Both are right triangles with equal hypotenuse and one equal side.
By RHS congruence rule: △ABC ≅ △PQR.
Answer: Yes, congruent by RHS rule.
Example 5: Example 5: Using CPCT to find unknown side
Problem: △ABC ≅ △PQR. If AB = 6 cm, BC = 8 cm, and PQ = 6 cm, find QR.
Solution:
Given:
- △ABC ≅ △PQR
- AB = PQ = 6 cm (corresponding sides — confirmed)
By CPCT:
- BC corresponds to QR (second sides in the congruence statement).
- BC = QR
- QR = 8 cm
Answer: QR = 8 cm.
Example 6: Example 6: Using CPCT to find unknown angle
Problem: △LMN ≅ △XYZ. If ∠L = 55° and ∠M = 75°, find ∠Z.
Solution:
Step 1: Find ∠N:
- ∠L + ∠M + ∠N = 180°
- 55° + 75° + ∠N = 180°
- ∠N = 50°
Step 2: By CPCT:
- ∠N corresponds to ∠Z
- ∠Z = ∠N = 50°
Answer: ∠Z = 50°.
Example 7: Example 7: AAA does not prove congruence
Problem: △ABC has angles 60°, 70°, 50°. △PQR also has angles 60°, 70°, 50°. Are they necessarily congruent?
Solution:
- Both triangles have the same three angles.
- However, AAA is not a congruence rule.
- △ABC could have sides 3, 4, 5 and △PQR could have sides 6, 8, 10. Same angles, different sizes.
Answer: No, they are not necessarily congruent. They are similar (same shape) but may have different sizes.
Example 8: Example 8: Identifying the congruence rule
Problem: In △ABC and △DEF: BC = EF, ∠B = ∠E, ∠C = ∠F. Which congruence rule applies?
Solution:
Given:
- ∠B = ∠E (angle)
- BC = EF (included side — between ∠B and ∠C)
- ∠C = ∠F (angle)
We have two angles and the included side equal.
Answer: ASA congruence rule.
Example 9: Example 9: Isosceles triangle proof
Problem: In △ABC, AB = AC. M is the midpoint of BC. Prove that ∠ABM = ∠ACM.
Solution:
In △ABM and △ACM:
- AB = AC (given)
- BM = CM (M is the midpoint of BC)
- AM = AM (common side)
By SSS congruence: △ABM ≅ △ACM.
By CPCT: ∠ABM = ∠ACM. Hence proved.
Example 10: Example 10: Finding which rule to use
Problem: In △ABC and △DEF: AB = DE, AC = DF, ∠A = ∠D. Which rule proves congruence?
Solution:
Given:
- AB = DE (side)
- ∠A = ∠D (included angle — between AB and AC, and between DE and DF)
- AC = DF (side)
Two sides and the included angle are equal.
Answer: SAS congruence rule. △ABC ≅ △DEF.
Real-World Applications
Real-world applications of congruence:
- Manufacturing: Machine parts must be congruent to be interchangeable. Every nut of the same size is congruent to others.
- Construction: Identical bricks, tiles, and beams are congruent shapes. Congruent triangular trusses ensure structural symmetry.
- Art and design: Patterns, tessellations, and symmetrical designs use congruent shapes.
- Stamps and printing: Every impression from a stamp or seal is congruent to others.
- Surveying: Congruent triangles help calculate distances that are difficult to measure directly.
- Mirror images: Your reflection is congruent to you (same shape and size, but flipped).
- Geometry proofs: Proving triangles congruent is one of the most powerful techniques in Euclidean geometry — used to prove equal sides, equal angles, and parallel lines.
Key Points to Remember
- Two triangles are congruent if they have the same shape and size.
- Congruent triangles have all corresponding sides and angles equal.
- The symbol for congruence is ≅.
- There are 4 congruence rules: SSS, SAS, ASA, RHS.
- SSS: All 3 sides equal.
- SAS: 2 sides and the included angle equal.
- ASA: 2 angles and the included side equal.
- RHS: Right angle + hypotenuse + one side equal (only for right triangles).
- AAA is NOT a congruence rule — same angles can give different sizes (similar, not congruent).
- CPCT (Corresponding Parts of Congruent Triangles) is used after proving congruence to deduce equal parts.
- The order of vertices in △ABC ≅ △PQR determines which parts correspond.
Practice Problems
- In △ABC and △DEF: AB = DE = 4 cm, BC = EF = 6 cm, CA = FD = 5 cm. State the congruence rule.
- In △PQR and △XYZ: ∠P = ∠X = 40°, PQ = XY = 5 cm, ∠Q = ∠Y = 80°. Are they congruent? Which rule?
- △ABC ≅ △DEF. If AC = 7 cm, ∠B = 65°, find DF and ∠E.
- In right triangles △ABC and △PQR: ∠B = ∠Q = 90°, BC = QR = 8 cm, AC = PR = 10 cm. Prove congruence.
- Can we conclude that two triangles with ∠A = ∠D, ∠B = ∠E, ∠C = ∠F are congruent? Justify.
- In △ABC, AB = AC. D is the midpoint of BC. Prove that AD ⊥ BC.
- State whether the following information is sufficient to prove congruence: two sides and a non-included angle.
- △XYZ ≅ △LMN. List all pairs of equal sides and equal angles.
Frequently Asked Questions
Q1. What does congruence mean?
Congruence means having the same shape and the same size. Two congruent figures can be placed on top of each other and they will match exactly.
Q2. What are the four rules for triangle congruence?
The four rules are: SSS (3 sides equal), SAS (2 sides + included angle), ASA (2 angles + included side), and RHS (right angle + hypotenuse + one side). Each is sufficient to prove congruence.
Q3. What is CPCT?
CPCT stands for Corresponding Parts of Congruent Triangles. Once two triangles are proved congruent, CPCT allows us to conclude that their corresponding sides and angles are equal.
Q4. Why is AAA not a congruence rule?
AAA (three angles equal) does not guarantee congruence because triangles with the same angles can have different sizes. They are similar (same shape) but not necessarily congruent (same size).
Q5. What does 'included angle' mean in SAS?
The included angle is the angle formed between the two given sides. In SAS, the angle must be the one that is 'sandwiched' between the two sides being compared.
Q6. When is RHS used?
RHS is used only for right-angled triangles. If two right triangles have equal hypotenuse and one equal side, they are congruent. The right angle provides the necessary third condition.
Q7. Does the order of letters matter in a congruence statement?
Yes. In △ABC ≅ △PQR, A corresponds to P, B to Q, C to R. This determines which sides and angles are equal. Writing △ABC ≅ △QPR would mean a different correspondence.
Q8. Are congruent triangles also similar?
Yes. All congruent triangles are similar (same shape). But not all similar triangles are congruent — similar triangles may have different sizes.
Q9. Give an example of congruent shapes in daily life.
Two coins of the same denomination are congruent. Tiles on a floor, pages of a book, stamps, and factory-made nuts and bolts are all examples of congruent shapes.
Q10. Why is SSA not a valid congruence rule?
SSA (two sides and a non-included angle) can sometimes produce two different triangles with the same measurements. This is called the 'ambiguous case,' so SSA does not guarantee congruence.
