SSS Congruence Rule
Two triangles are congruent if they are exactly the same shape and size — one can be placed on top of the other and they match perfectly. But do we always need to check all six measurements (3 sides + 3 angles) to confirm congruence?
The answer is no. The SSS (Side-Side-Side) Congruence Rule tells us that if all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent. We do not need to check the angles — they are automatically equal.
This is the simplest congruence rule to understand: if you fix all three side lengths, you can only make one triangle (up to flipping). There is no freedom left for the angles to be different.
In NCERT Class 7 Mathematics, the SSS rule is one of four congruence criteria studied in the chapter Congruence of Triangles. The others are SAS, ASA, and RHS.
What is SSS Congruence Rule?
Definition: The SSS Congruence Rule states:
If the three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.
Notation:
- If in △ABC and △PQR:
- AB = PQ, BC = QR, and CA = RP
- Then △ABC ≅ △PQR (by SSS rule)
Key terms:
- Congruent (≅): Exactly the same shape and size.
- Corresponding sides: Sides that match in position when one triangle is placed on the other.
- Corresponding angles: Angles that are at the same position in both triangles (they are automatically equal when SSS holds).
Important:
- The order of letters matters. △ABC ≅ △PQR means A↔P, B↔Q, C↔R.
- Under SSS, all corresponding angles are also equal: ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.
- SSS works for ALL triangles — acute, obtuse, and right-angled.
SSS Congruence Rule Formula
SSS Congruence Rule:
If AB = PQ, BC = QR, CA = RP, then △ABC ≅ △PQR
What follows from congruence:
- All corresponding sides are equal (this was given).
- All corresponding angles are equal: ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.
- The perimeters are equal.
- The areas are equal.
The four congruence rules (for reference):
| Rule | What is given equal |
|---|---|
| SSS | 3 sides |
| SAS | 2 sides and the included angle |
| ASA | 2 angles and the included side |
| RHS | Hypotenuse and 1 side (right triangle only) |
Derivation and Proof
Why does the SSS rule work? (Justification)
The SSS rule is accepted as a postulate (a self-evident truth) in Euclidean geometry. It is not proved from other theorems but can be understood intuitively.
Activity to verify:
- Take three sticks of lengths 5 cm, 7 cm, and 9 cm.
- Try to form a triangle with these sticks.
- No matter how you arrange them, you will get only one triangle (or its mirror image).
- Now take another set of three sticks with the same lengths: 5 cm, 7 cm, and 9 cm.
- The triangle formed is exactly the same as the first one.
- This means: if all three sides are fixed, the triangle is completely determined.
Key insight:
- Once three side lengths are fixed (and they satisfy the triangle inequality), there is only ONE possible triangle shape.
- The angles are automatically fixed. There is no room for the angles to vary.
- This is why SSS guarantees congruence.
Contrast with SSA:
- Two sides and a non-included angle (SSA) do NOT guarantee congruence — more than one triangle may be possible. This is called the ambiguous case.
Types and Properties
Types of problems on SSS Congruence:
1. Proving triangles congruent:
- Given measurements of both triangles, show that all three pairs of sides are equal.
- Conclude congruence by SSS.
2. Finding corresponding parts:
- Once congruence is established, use CPCT (Corresponding Parts of Congruent Triangles) to find equal angles or sides.
3. SSS in geometric figures:
- In parallelograms, rhombuses, and other shapes, diagonals create triangles that can be proved congruent by SSS.
4. Isosceles and equilateral triangles:
- If an equilateral triangle has side length a, any other equilateral triangle with side a is congruent to it by SSS.
5. Construction-based problems:
- Construct a triangle with given three sides (using compass and ruler) and verify it matches another triangle with the same sides.
6. Identifying which congruence rule applies:
- Given certain information, decide whether SSS, SAS, ASA, or RHS is the appropriate rule.
Solved Examples
Example 1: Example 1: Proving congruence by SSS
Problem: In △ABC and △DEF: AB = DE = 5 cm, BC = EF = 7 cm, and CA = FD = 6 cm. Are the triangles congruent?
Solution:
Given:
- AB = DE = 5 cm (Side 1)
- BC = EF = 7 cm (Side 2)
- CA = FD = 6 cm (Side 3)
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: Finding equal angles using CPCT
Problem: △PQR ≅ △XYZ by SSS (PQ = XY, QR = YZ, RP = ZX). If ∠P = 70°, find ∠X.
Solution:
Given:
- △PQR ≅ △XYZ (by SSS)
- ∠P = 70°
By CPCT (Corresponding Parts of Congruent Triangles):
- P ↔ X, Q ↔ Y, R ↔ Z
- ∠P = ∠X
- ∠X = 70°
Answer: ∠X = 70°.
Example 3: Example 3: Diagonal of a rectangle
Problem: ABCD is a rectangle. Show that △ABC ≅ △CDA.
Solution:
In a rectangle ABCD:
- AB = CD (opposite sides of a rectangle are equal)
- BC = DA (opposite sides of a rectangle are equal)
- AC = CA (common side)
All three pairs of sides are equal.
By SSS Congruence Rule:
- △ABC ≅ △CDA
Therefore (by CPCT): ∠BAC = ∠DCA and ∠BCA = ∠DAC.
Answer: △ABC ≅ △CDA by SSS rule.
Example 4: Example 4: Isosceles triangle and median
Problem: In △ABC, AB = AC = 8 cm and BC = 6 cm. M is the midpoint of BC. Show that △ABM ≅ △ACM.
Solution:
Given:
- AB = AC = 8 cm
- BM = MC = 3 cm (M is the midpoint of BC)
- AM = AM (common side)
In △ABM and △ACM:
- AB = AC = 8 cm ✓
- BM = MC = 3 cm ✓
- AM = AM (common) ✓
By SSS Congruence Rule:
- △ABM ≅ △ACM
Therefore (by CPCT): ∠AMB = ∠AMC. Since these are supplementary (linear pair), each equals 90°. So AM ⊥ BC.
Answer: △ABM ≅ △ACM by SSS. The median AM is also the perpendicular bisector of BC.
Example 5: Example 5: Checking if SSS applies
Problem: In △ABC and △DEF: AB = 4 cm, BC = 6 cm, ∠B = 60°, DE = 4 cm, EF = 6 cm, ∠E = 60°. Can we use SSS rule?
Solution:
Given information:
- AB = DE = 4 cm (one pair of sides)
- BC = EF = 6 cm (second pair of sides)
- ∠B = ∠E = 60° (included angle)
Analysis: We know only 2 pairs of sides and 1 pair of angles. We do NOT know the third pair of sides (CA and FD).
SSS requires all 3 pairs of sides. Here, we have 2 sides and 1 included angle — this is the SAS rule, not SSS.
Answer: No, we cannot use SSS. The correct rule here is SAS.
Example 6: Example 6: Equilateral triangles
Problem: Two equilateral triangles have the same side length of 10 cm. Are they congruent?
Solution:
Triangle 1:
- All sides = 10 cm → sides are 10, 10, 10
Triangle 2:
- All sides = 10 cm → sides are 10, 10, 10
Comparing:
- Side 1 = Side 1 = 10 cm ✓
- Side 2 = Side 2 = 10 cm ✓
- Side 3 = Side 3 = 10 cm ✓
By SSS: △1 ≅ △2
Answer: Yes. Two equilateral triangles with the same side length are always congruent by SSS.
Example 7: Example 7: Rhombus diagonals
Problem: ABCD is a rhombus with diagonals intersecting at O. Show that △AOB ≅ △COD.
Solution:
In a rhombus:
- All sides are equal: AB = BC = CD = DA
- Diagonals bisect each other: AO = OC and BO = OD
In △AOB and △COD:
- AO = OC (diagonals bisect each other) ✓
- OB = OD (diagonals bisect each other) ✓
- AB = CD (sides of a rhombus) ✓
By SSS Congruence Rule:
- △AOB ≅ △COD
Answer: △AOB ≅ △COD by SSS rule.
Example 8: Example 8: Correct correspondence
Problem: In △ABC and △QPR: AB = QP = 5 cm, BC = PR = 8 cm, CA = RQ = 6 cm. Write the congruence statement with correct correspondence.
Solution:
Matching sides:
- AB = QP → A ↔ Q, B ↔ P
- BC = PR → B ↔ P, C ↔ R
- CA = RQ → C ↔ R, A ↔ Q
Correspondence: A ↔ Q, B ↔ P, C ↔ R
Congruence statement:
- △ABC ≅ △QPR
This means:
- ∠A = ∠Q, ∠B = ∠P, ∠C = ∠R
Answer: △ABC ≅ △QPR (by SSS).
Example 9: Example 9: Not congruent
Problem: △ABC has sides 3 cm, 4 cm, 5 cm. △DEF has sides 3 cm, 4 cm, 6 cm. Are they congruent by SSS?
Solution:
Comparing sides:
- 3 = 3 ✓
- 4 = 4 ✓
- 5 ≠ 6 ✗
The third pair of sides is NOT equal.
Answer: No, the triangles are NOT congruent. All three pairs of sides must be equal for SSS to apply.
Example 10: Example 10: Two triangles sharing a common side
Problem: Points A and B lie on opposite sides of line segment CD. If CA = CB = 7 cm and DA = DB = 5 cm, show that △ACD ≅ △BCD.
Solution:
Given:
- CA = CB = 7 cm
- DA = DB = 5 cm
- CD = CD (common side)
In △ACD and △BCD:
- CA = CB = 7 cm ✓
- DA = DB = 5 cm ✓
- CD = CD (common) ✓
By SSS Congruence Rule:
- △ACD ≅ △BCD
Therefore (by CPCT): ∠ACD = ∠BCD. This means CD bisects ∠ACB.
Answer: △ACD ≅ △BCD by SSS rule.
Real-World Applications
Real-world uses of SSS congruence:
- Construction and building: When building identical roof trusses, ensuring all three sides are the same length guarantees the triangles are identical — no angle measurement needed.
- Bridge engineering: Triangular supports in bridges must be identical. Measuring three sides ensures congruence and structural stability.
- Surveying: Surveyors use triangulation. If two triangles in a survey have all three sides equal, the measurements are consistent.
- Carpentry: Making identical triangular shelves or brackets — cut three pieces of wood to the same lengths, and the resulting triangles are congruent.
- Manufacturing: Identical triangular parts in machinery. Checking three edge measurements is enough to confirm the parts are identical.
- Origami and paper folding: When paper is folded symmetrically, the resulting triangles have the same three sides, making them congruent by SSS.
Key Points to Remember
- SSS Congruence Rule: If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
- SSS stands for Side-Side-Side.
- Once congruent, all corresponding angles are also equal (CPCT).
- The order of vertices in the congruence statement indicates the correspondence.
- SSS is one of four congruence rules: SSS, SAS, ASA, RHS.
- If three side lengths are fixed, only one triangle is possible (unique triangle).
- SSA is NOT a valid congruence rule — knowing two sides and a non-included angle does not guarantee congruence.
- A common side shared by two triangles counts as one equal pair of sides for SSS.
- SSS works for ALL types of triangles — acute, obtuse, right-angled, isosceles, equilateral, and scalene.
- After proving congruence by SSS, use CPCT to deduce other equal parts.
Practice Problems
- In △ABC and △XYZ: AB = XY = 6 cm, BC = YZ = 9 cm, CA = ZX = 7 cm. Are the triangles congruent? State the rule.
- ABCD is a quadrilateral where AB = CD and AD = BC. Diagonal AC is drawn. Show that △ABC ≅ △CDA using SSS.
- In △PQR, PQ = PR = 10 cm and QR = 12 cm. S is the midpoint of QR. Prove that △PQS ≅ △PRS.
- △ABC has sides 5 cm, 12 cm, and 13 cm. △DEF has sides 12 cm, 5 cm, and 13 cm. Are they congruent? Write the correct correspondence.
- In the figure, AB = CD and AC = BD. Prove that △ABC ≅ △DCB.
- Two equilateral triangles have perimeters 24 cm each. Are they congruent? Explain using SSS.
- Can two triangles with sides 3, 4, 5 and 5, 4, 3 be proved congruent by SSS? Explain.
- Why is SSA (two sides and a non-included angle) not a valid congruence rule?
Frequently Asked Questions
Q1. What is the SSS congruence rule?
The SSS (Side-Side-Side) congruence rule states that if all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent. This means they have the same shape and size.
Q2. Do we need to check angles in SSS?
No. If all three sides are equal, the angles are automatically equal. Once the three sides are fixed, the triangle is completely determined — there is only one possible shape. So checking angles is not needed.
Q3. What does CPCT mean?
CPCT stands for Corresponding Parts of Congruent Triangles. Once two triangles are proved congruent (by any rule including SSS), all their corresponding parts (sides and angles) are equal. CPCT is used to deduce additional equal parts.
Q4. Why does the order of letters matter in the congruence statement?
The order tells us which vertices correspond. △ABC ≅ △PQR means A↔P, B↔Q, C↔R. This determines which sides and angles are equal. Writing △ABC ≅ △QRP would imply a different (possibly incorrect) correspondence.
Q5. Is SSS the only congruence rule?
No. There are four congruence rules: SSS (3 sides), SAS (2 sides and included angle), ASA (2 angles and included side), and RHS (hypotenuse and one side for right triangles). Each requires different information to prove congruence.
Q6. 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. Since two different triangles are possible, SSA does not guarantee congruence.
Q7. Can SSS be used for right triangles?
Yes. SSS works for all triangles including right triangles. However, for right triangles, you can also use the RHS rule (which needs only the hypotenuse and one other side). Both are valid.
Q8. If two triangles have equal perimeters, are they congruent?
Not necessarily. Equal perimeters mean the sum of sides is the same, but the individual sides could be different. For example, triangles with sides 3-4-5 and 4-4-4 both have perimeter 12 but are not congruent. For SSS, each individual side must match.
