In this mini-lesson, we will explore the world of congruent triangles. Starting with the definition, we will understand the properties of congruent triangles and learn interesting facts about them.

Before we move ahead, let’s peak in your refrigerator. You may have noticed ice trays in it. The molds inside the tray create the same ice cubes, similar in size as well as shape. That means the ice cubes so produced are congruent.

So is the case with a pizza, we get it with an equal number of slices. Each slice is congruent to all others.

Now that you have some idea about congruence, let’s move ahead and learn more about congruent triangles.

**Table of Contents**

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**Understanding Congruent Triangles with Cuemath**

The most common, primary shapes we learn about are triangles.

Now, let's learn about the meaning of congruent triangles with Cuemath.

Watch this interesting video to understand more about this concept.

**Introduction to ****Congruence**

Have you ever observed that two copies of a single photograph of the same size are identical?

Similarly, ATM cards issued by the same bank are identical.

Such figures are called **congruent figures**.

You may have noticed an ice tray in your refrigerator.

The moulds inside the tray that is used for making ice are congruent.

Have you struggled to replace a new refill in an older pen?

This could have happened because the new refill is not the same size as the one you want to replace.

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

**What is ****Congruent?**

**Congruent: Definition**

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.

Let us do a small activity.

Draw two circles of the same radius and place one on another.

Do they cover each other completely?

Yes, they do.

Hence, we can say that they are congruent circles.

Use the following simulation to explore more congruent shapes.

**Congruent Triangles**

Now let's discuss congruence of two 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.

We can represent this in a mathematical form using the congruent symbol.

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**.

Congruent Parts of \(\mathbf{\Delta ABC}\) and \(\mathbf{\Delta PQR}\) | |
---|---|

Corresponding Vertices |
\(A\) and \(P\) \(B\) and \(Q\) \(C\) and \(R\) |

Corresponding Sides |
\(\overline{AB}\) and \(\overline{PQ}\) \(\overline{BC}\) and \(\overline{QR}\) \(\overline{AC}\) and \(\overline{PR}\) |

Corresponding Angles |
\(\angle A\) and \(\angle P\) \(\angle B\) and \(\angle Q\) \(\angle C\) and \(\angle R\) |

Remember that it is incorrect to write \(\Delta BAC \cong \Delta PQR\) because \(A\) corresponds to \(P\), \(B\) corresponds to \(Q\) and \(C\) corresponds to \(R\).

**Properties of Congruent Triangles**

**Property 1**

**SSS Criterion for Congruence**

SSS Criterion stands for **Side-Side-Side** Criterion.

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

**Property 2**

**SAS Criterion for Congruence**

SAS Criterion stands for **Side-Angle-Side** Criterion.

Under this criterion, if the two sides and the angle between the sides of one triangle are equal to the two corresponding sides and the angle between the sides of another triangle, the two triangles are congruent.

**Property 3**

**ASA Criterion for Congruence**

ASA Criterion stands for **Angle-Side-Angle **Criterion.

Under this criterion, if the two angles and the side included between them of one triangle are equal to the two corresponding angles and the side included between them of another triangle, the two triangles are congruent.

**Property 4**

**AAS Criterion for Congruence**

AAS Criterion stands for **Angle-Angle-Side **Criterion.

Under this criterion, if the two angles and the non-included side of one triangle are equal to the two corresponding angles and the non-included side of another triangle, the triangles are congruent.

**Property 5**

**RHS Criterion for Congruence**

RHS Criterion stands for **Right Angle-Hypotenuse-Side **Criterion.

Under this criterion, if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.

Explore these properties of congruent using the simulation below.

- Two figures are congruent if they have the same shape and size.
- Two angles are congruent if their measures are exactly the same.
- Two triangles with equal corresponding angles may not be congruent to each other because one triangle might be an enlarged copy of the other. Hence, there is no AAA Criterion for Congruence.
**We represent the congruent triangles mathematical form by using the congruent triangles symbol (≅).**

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**Solved Examples**

Example 1 |

Jolly was doing geometrical construction assignments in her notebook.

She drew an isosceles triangle \(PQR\) on a page.

She marked \(L\) and \(M\) as the mid points of the equal sides of the triangle.

She marked \(N\) as the mid point of the third side.

She states that \(LN=MN\).

Is she right?

**Solution:**

We will prove that

\(\Delta LPN \cong \Delta MRN\)

We know that \(\Delta PQR\) is an isosceles triangle and \(PQ=QR\).

Angles opposite to equal sides are equal.

Thus,

\(\angle QPR=\angle QRP\)

Since \(L\) and \(M\) are the mid points of \(PQ\) and \(QR\) respectively,

\(\begin{align}PL=LQ=QM=MR=\frac{QR}{2}\end{align}\)

\(N\) is the mid point of \(PR\), hence,

\(PN=NR\)

In \(\Delta LPN\) and \(\Delta MRN\),

- \(LP=MR\)
- \(\angle LPN=\angle MRN\)
- \(PN=NR\)

Thus, by SAS Criterion of Congruence,

\(\Delta LPN \cong \Delta MRN\)

Since congruent parts of congruent triangles are equal, \(LN=MN\).

Yes, she is right that \(LN=MN\). |

Example 2 |

Jack wanted to make a paper plane.

He cuts two right-angled triangles out of paper.

He cuts them in such a way that one side and an acute angle of one of the triangles is equal to the corresponding side and angle of the other triangle.

Are both triangles congruent to each other?

**Solution:**

In \(\Delta ABC\) and \(\Delta DEF\),

- \(\angle ABC=\angle DEF = 90^{\circ}\)
- \(AB=DE\)
- \(\angle CAB=\angle FDE\)

By AAS Criterion of Congruence,

\(\Delta ABC \cong \Delta DEF\)

\(\therefore\) Both triangles are congruent to each other. |

Go through the following tips that may help you while proving congruence of triangles.

Example 3 |

James wanted to know which congruency rule explains why these triangles are congruent. Let's help him.

**Solution:**

\begin{aligned}

&\text { Here, } E F = M N =3 \mathrm{in}\\

& F G = N O =4.5 \mathrm{in}\\

&\angle E F G=\angle M N O =110^{\circ}\\

&\triangle \mathrm{EFG}\cong\triangle \mathrm{MNO}(\text { By SAS rule })

\end{aligned}

\(\therefore\) These triangles are congruent by SAS rule |

Example 4 |

Olivia drew a figure with two congruent triangles sharing a common side. State the rule of congruence followed by congruent triangles ABC and DCB.

**Solution:**

\begin{aligned}

&\ln \Delta \mathrm{ABC} \text { and } \Delta \mathrm{DCB}\\

&\mathrm{AB}=\mathrm{DC}\\

&\mathrm{AC}=\mathrm{DB}\\

&\mathrm{BC}=\mathrm{BC}\\

&\Delta \mathrm{ABC}\cong\Delta \mathrm{DCB}

\end{aligned} (by SSS)

\(\therefore\) \(\Delta ABC \cong \Delta DCB\) by SSS rule |

Example 5 |

While solving a problem for which figure is given below, Noah came to the conclusion that ΔABC & ΔXYZ are congruent by ASA rule, with BC = YZ = 4 units. Help him finding the value of a and b if ΔABC ≅ ΔXYZ.

**Solution:**

\begin{aligned}

&\Delta \mathrm{ABC}\cong\Delta \mathrm{XYZ} \text { (By ASA rule) }\\

&\angle \mathrm{B}=\angle \mathrm{Y}=65^{\circ} \text { (given) }\\

&\mathrm{BC}^{-}=\mathrm{YZ}^{-}=4 \mathrm{units} \text { (given) }\\

&\angle \mathrm{a}=35^{\circ} \text { (for ASA rule) }\\

&\text { Now in } \Delta \text { XYZ }\\

&\!\angle \mathrm{X}\!+\!\!\angle \mathrm{Y}\!+\!\!\angle \mathrm{Z}\!\!=\!\!180^{\circ}\!\! \text { (Angle sum property) }\\

&\angle b +65^{\circ}+\angle a=180^{\circ}\\

&\angle b+65^{\circ}+35^{\circ}=180^{\circ}\\

&\angle b+100^{\circ}=180^{\circ}\\

&\angle b=180^{\circ}-100^{\circ}=80^{\circ}\\

&\text { Hence, } a=35^{\circ} \text { and } b=80^{\circ}

\end{aligned}

\(\therefore\) \(a=35^{\circ} \text{ and } b=80^{\circ}\) |

- To prove if two triangles are congruent, mark the information given in the statement in your diagram.
- Remember all the criterions for congruence.
- If you need to prove any specific part of triangles are equal, try proving the triangles which contain those specific parts are congruent.
- If two triangles are overlapping, draw them separately to get a better look at the given information.

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**Practice Questions**

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**Maths Olympiad Sample Papers**

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**Frequently Asked Questions (FAQs)**

### What makes triangles congruent?

Two triangles are congruent if they are exact copies of each other and when superimposed, they cover each other completely.

### How do you know if a triangle is congruent?

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

### What is the common side of the two triangles in the example?

In the following example:

In ΔABCΔABC and ΔDEFΔDEF

ΔABC≅ΔDEFΔABC≅ΔDEF by SSS

BC is the common side.

### What are congruent triangles?

Two triangles are congruent if they are exact copies of each other and when superimposed, they cover each other exactly.

### What are the properties of congruent triangles?

The properties of congruent triangles are:

- SSS Criterion for Congruence
- SAS Criterion for Congruence
- ASA Criterion for Congruence
- AAS Criterion for Congruence
- RHS Criterion for Congruence

### Give examples of congruent triangles.

The following pairs of triangles are congruent.