Congruent Triangles

Congruent Triangles

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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?

Explaining the concept of congruence using real life example of photographs

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

Explaining the concept of congruence using real life example of ATM Cards

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.

Explaining the concept of congruence using real life example of ice-cubes

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.

Explaining the congruent definition using circles with same radii

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.

Explaining the congruence of triangles

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.

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

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.

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

Explaining the congruence of triangles using SSS Criterion of 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

Explaining the congruence of triangles using SAS Criterion of 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

Explaining the congruence of triangles using ASA Criterion of 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

Explaining the congruence of triangles using AAS Criterion of 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

Explaining the congruence of triangles using RHS Criterion of 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.

 

 
important notes to remember
Important Notes
  1. Two figures are congruent if they have the same shape and size.
  2. Two angles are congruent if their measures are exactly the same.
  3. 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.

Congruent Triangles: 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.

Example of congruent triangles of class 8 - Triangle PQR is an isosceles triangle. L and M are the midpoints of the equal sides, and N is 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.

Examples of congruent triangles of class 8 - Two right angled triangles are shown where 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.

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

Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What are congruent triangles?

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

2. What are the properties of congruent triangles?

The properties of congruent triangles are:

  1. SSS Criterion for Congruence
  2. SAS Criterion for Congruence
  3. ASA Criterion for Congruence
  4. AAS Criterion for Congruence
  5. RHS Criterion for Congruence

3. Give examples of congruent triangles.

The following pairs of triangles are congruent.

Examples of Congruent Triangles of class 8.

 
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