SSS (Side Side Side): Definition, Theorem, Similarity, Examples

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What Is SSS Theorem (Side-Side-Side Theorem) in Geometry?

The SSS theorem is called the Side-Side-Side theorem. It is a criterion used to prove triangle congruence as well as triangle similarity. However, the terms of the SSS criterion in both the cases are different. 

Congruent Triangles: Two triangles are congruent when they have the same shape and the same size. Corresponding angles and corresponding sides of two congruent triangles are also congruent. We denote the triangle congruence using the symbol ≅.

Similar Triangles: Two triangles are similar when they have the same shape but may differ in size. Corresponding angles of similar triangles are congruent. Corresponding sides of two similar triangles are proportional. We denote the triangle similarity using the symbol .

The SSS theorem helps us to prove the triangle congruence or similarity just using the lengths of the three sides. Other dimensions are not required.

SSS Definition

SSS Congruence Theorem: To prove triangle congruence, we use the SSS congruence theorem, which states that when all three sides of a triangle are equal to the corresponding sides of another triangle, and the two triangles are congruent. 

SSS congruence rule visual

SSS Similarity Theorem: To prove that two triangles are similar, we use SSS similarity criterion, which states that if the ratio of the corresponding sides of two triangles is equal, then two triangles are similar.

SSS triangle similarity - visual

SSS Congruence Theorem (SSS Congruence Rule)

SSS Congruence theorem Statement: If the three sides of a triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent. 

Example: Are the two triangles congruent?

SSS congruence rule - example

Given:

AB = PQ = 5 units

BC = QR = 4 units

AC = PR = 3 units

Thus, ▵ABC $\sim$ ▵PQR By SSS congruence theorem

SSS Similarity Rule

SSS Similarity Rule Statement: Two triangles are similar to each other, if the ratio of the corresponding sides of the two triangles is equal. 

Example: Are the given two triangles similar by SSS similarity rule?

SSS similarity rule - example

Here,

$\frac{MK}{PR} = \frac{10}{12} = \frac{5}{6}$

$\frac{KL}{RQ} = \frac{20}{24} = \frac{5}{6}$

$\frac{LM}{PQ} = \frac{15}{18} = \frac{5}{6}$

Thus, \frac{KL}{QR} = \frac{MK}{PR} = \frac{ML}{PQ} = \frac{5}{6}$

The corresponding sides are proportional.

Thus, ▵MKL $\sim$ ▵PRQ

SSS Formulas

SSS Congruence Rule: If the three sides of ΔABC are congruent to the corresponding sides of ΔXYZ, then ΔABC ≅ ΔXYZ.

If AB = XY, BC = YZ, and AC = XZ, then ΔABC ≅ ΔXYZ.

SSS Similarity Rule: If the ratio of the corresponding sides of ΔABC and  ΔXYZ is equal, then the  ΔABC ~ ΔXYZ.

If $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{ZX}$, then ΔABC $\sim$ ΔXYZ.

(Note that when we write the congruence or similarity notation, the order of the vertices is important to identify the corresponding parts.)

Facts about SSS Theorem

  • The SSS theorem can be applied to both right triangles and non-right triangles. 
  • If two triangles are congruent, they are also similar.
  • To use SSS theorem, we only need to know the lengths of all the sides of the two triangles. The angle measurements are not required.

Conclusion

In this article, we have learned about the SSS theorem, congruence rule and similarity rule with SSS rule examples. Now, let us practice solving problems on the SAS Theorem.

Solved Examples on  SSS Theorem

1. Prove that ∆ABC and ∆ADC are congruent.

Example for SSS congruence rule

Solution:

In a ΔABC and  ΔADC, 

AB = CD  (given)

BC = AD (given)

AC = AC (Common side)

Thus, by SSS congruence theorem, ΔABC ≅  ΔADC.

2. Are the given two triangles similar?

Example of SSS triangle similarity

Solution:

Let’s find the ratio of the corresponding sides.

$\frac{LM}{RS} = \frac{40}{4} = 10 : 1$

$\frac{MN}{ST} = \frac{50}{5} = 10 : 1$

$\frac{LN}{RT} = \frac{60}{6} = 10 : 1$

The ratio of corresponding sides is constant.

Thus, the two triangles are similar by SSS similarity rule.

3. Prove that ∆ABC ∆PQR.

Example of SSS congruence theorem

Solution:

AB = PQ = 9

BC = QR = 6

AC = PR = 5

Thus, ∆ABC ≅ ∆PQR by side-side-side congruence theorem.

Practice Problems on SSS Theorem

SSS (Side Side Side): Definition, Theorem, Similarity, Examples

Attend this quiz & Test your knowledge.

1

What is the full form of the SSS theorem?

Side-Angle-Side
Side-Slope-Side
Side-Side-Side
Side-Square-Side
CorrectIncorrect
Correct answer is: Side-Side-Side
SSS stands for Side-Side-Side.
2

The corresponding sides of two similar triangles are

congruent.
equal.
proportional.
similar.
CorrectIncorrect
Correct answer is: proportional.
The corresponding sides of two similar triangles are proportional.
3

Are the following pair of triangles congruent under SSS rule?

SSS (Side Side Side): Definition, Theorem, Similarity, Examples
Triangles are congruent by SSS rule.
Triangles are not congruent by SSS rule.
Insufficient information.
Triangles are similar but not congruent.
CorrectIncorrect
Correct answer is: Triangles are congruent by SSS rule.
AB = XY, BC = YZ, AC = XZ
Thus, by SSS congruence rule, the two triangles are congruent.

Frequently Asked Questions about SSS Theorem

The meaning of SSS is Side-Side-Side congruence criterion or similarity criterion.

To prove the SSS similarity rule, the lengths of all sides of a triangle should be proportional to the  lengths of corresponding sides of the other triangle.

The five rules of congruence for triangles:

  • SSS: Side-Side-Side
  • SAS: Side-Angle-Side
  • ASA: Angle-Side-Angle
  • AAS: Angle-Angle-Side
  • RHS: Right angle-Hypotenuse-Side

Yes, it is applicable to all types of triangles.

CPCTC stands for “Congruent parts of Congruent Triangles are Congruent.”