Have you ever observed that two ATM cards issued by the same bank are identical?

Remember that whenever identical objects are to be produced, the concept of congruence is taken into consideration.

Congruence of triangles means that:

- All corresponding angle pairs are equal.
- All corresponding sides are proportional.

However, in order to be sure that the two triangles are similar, we do not necessarily need to have information about all sides and all angles.

In this mini-lesson, we will learn about the SSS similarity theorem in the concept of the SSS rule of congruence, using similar illustrative examples.

Check out the interactive simulation to explore more congruent shapes and do not forget to try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What Is the SSS Rule of Congruence?**

The word **"congruent"** means equal in every aspect or figure in terms of shape and size.

Congruence** **is the term used to describe the relation of two figures that are congruent.

Now let's discuss the congruence of triangles.

Look at \(\Delta ABC\) and \(\Delta PQR\) below.

These two triangles are of the same size and shape.

Thus, we can say that they are congruent.

They can be considered as congruent triangle examples.

We can represent this in a mathematical form using the congruent triangles symbol (≅).

\(\Delta ABC \cong \Delta PQR\) |

This means \(A\) falls on \(P\), \(B\) falls on \(Q\) and \(C\) falls on \(R\).

Also, \(AB\) falls on \(PQ\), \(BC\) falls on \(QR\) and \(AC\) falls on \(PR\).

This indicates that the corresponding parts of congruent triangles are equal.

**SSS Criterion**

SSS Criterion stands for side side side congruence postulate.

Under this criterion, if all the three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent.

**How to Prove SSS Rule of Congruence? **

Let's perform an activity to show SSS proof.

Draw two right-angled triangles with the hypotenuse of 6 inches and one side of 4 inches each.

Cut these triangles and try to place one triangle over the other such that equal sides are placed over one another.

Do you observe that these two triangles superimpose on each other completely?

This means these two triangles are congruent.

This completes the SSS proof.

Do you want to explore more congruency rules?

Here is a simulation for you to explore these properties of congruent triangles.

**SSS Similarity Theorem**

SSS similarity theorem says that "if we have two triangles such that the sides of one triangle are proportional to the sides of the other triangle, then these two triangles are similar to each other."

If two triangles are similar, then their corresponding angles are also equal.

**Important Notes**

**Two figures are congruent if they have the same shape and size.****Two triangles are congruent if their shape and size are exactly the same.****We represent the congruent triangle's mathematical form using the congruent triangles symbol (≅).****If all the three sides of a triangle are equal to the three sides of another triangle then the triangles are congruent by SSS Criterion for Congruence.**

**Solved Examples**

Example 1 |

The two points P and Q are on the opposite sides of the line segment AB.

The points P and Q are equidistant from points A and B.

Can you prove that \(\Delta PAQ\) is congruent to the \(\Delta PBQ\)?**Solution**

As the two points P and Q are equidistant from the endpoints of the line segment AB.

Therefore,

\[\begin{aligned}AP&=BP\\AQ&=BQ\end{aligned}\]

Now the side PQ is common in both the triangles \(\Delta PAQ\) and \(\Delta PBQ\).

Therefore according to the SSS postulate, the two triangles are congruent.

Hence,

\[\Delta PAQ\cong\Delta PBQ\]

\(\therefore\) \(\Delta PAQ\cong\Delta PBQ\). |

Example 2 |

Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A.

Can you prove that \(\Delta ADB\) is congruent to the \(\Delta ADC\)?

**Solution**

The triangle, ABC is an isosceles triangle therefore, \(AB=AC\).

Now the side AD is common in both the triangles \(\Delta ADB\) and \(\Delta ADC\).

As the line segment AD is the angle bisector of the angle A then it divides the line segment BC into two equal parts BD and CD.

Therefore,

\[\begin{aligned}BD&=CD\\AB&=AC\end{aligned}\]

Now according to the SSS postulate, the two triangles are congruent.

Hence,

\[\Delta ADB\cong \Delta ADC\]

\(\therefore\) \(\Delta ADB\cong \Delta ADC\). |

- Can you determine the difference between SSS congruency rule and the SSS similarity rule?
- Can you prove the SSS congruency rule using any other congruency rule?

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of the SSS rule of congruence. The math journey around the SSS rule starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp but will also stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs) **

## 1. What is the SSS criterion?

SSS Criterion stands for side side side congruence postulate.

Under the SSS theorem, if all the three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent.

## 2. How do you prove SSS similarity?

SSS similarly can be proved by showing that the side lengths of one triangle are proportional to the side length of the other triangle.

## 3. What is the difference between SAS and SSS?

Both SAS and SSS rules are the triangle congruence rules. SAS full form is "side-angle-side" and SSS full form is "side-side-side."

- In the SAS postulate, two sides and the angle between them in a triangle are equal to the corresponding two sides and the angle between them in another triangle.
- In the SSS postulate, all three sides of one triangle are equal to the three corresponding sides of another triangle.

## 4. What are the three triangle similarity theorems?

The three triangle similarity theorems are:

- Angle-Angle (AA)
- Side-Angle-Side (SAS)
- Side-Side-Side (SSS)