Congruent Triangles

Congruent Triangles
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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.

Watch this interesting video to understand about the concept of Congruent Triangles.

Lesson Plan

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.

Now that you are familiar with the word "congruent," you will be able to generalize "Congruent triangles definition."

Let's have a look at congruent triangles definition in the following section.

Real-life congruent triangle examples include their applications in construction, architecture and other such purposes.

Also, congruent triangles examples in the solved examples section would help you to have better understanding of congruent triangles geometry.


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. 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.

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 \(R\).

The next section will give an overview of the properties of congruent triangles, this being the most important aspect of congruent triangles geometry.


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

HL Criterion for Congruence

Explaining the congruence of triangles using RHS Criterion of Congruence

HL Criterion stands for Hypotenuse-Leg 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 congruence 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. We represent the congruent triangles mathematical form by using the congruent triangles symbol (≅).
  4. 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.

Solved Examples

Example 1

 

 

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 2

 

 

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 3

 

 

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}

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

Example 4

 

 

Jolly was doing geometrical construction assignments in her notebook.

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

She marked \(L\), \(M\) as the mid points of the equal sides of the triangle and \(N\) 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 5

 

 

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 by AAS rule.
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 parts of triangles are equal, try proving that 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.

Interactive Questions

Here are few activities for you to practice.

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

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about congruent triangles with the simulations and practice questions. Now you will be able to easily solve problems on congruent triangles definition, congruent triangles symbol, congruent triangles Class 8, congruent triangles geometry, congruent triangles examples and two triangles are congruent.

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Frequently Asked Questions(FAQs)

1. What makes triangles congruent?

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

2. 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.

3. What is the common side of the two triangles in the example?

In the following example:

In \(\Delta ABC\) and \(\Delta DEF\) 

\(\Delta ABC \cong \Delta DEF\) by SSS

BC is the common side.

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