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Chapter 7 – Congruence of Triangles

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Take out two pages from a notebook and place one above one another. Can you see that they are wholly superimposed? Such congruent objects are exact copies of one another, and this relation of two objects being congruent is called congruence. The example mentioned above illustrates the congruence of plane figures. Similarly, you can find congruence among line segments, congruence of angles, and congruence of triangles. While studying triangles, there are different criteria for the congruence of triangles.

Congruence can be displayed among line segments as well. How can we prove congruence among line segments? Take two lines, AB = 5 cm, and CD = 5 cm. It can be written as AB ≅ CD. If both the line segments have equal lengths, they are congruent. If two line segments are congruent, then they will, in turn, have the same length.

Congruence can be displayed among line segments as well. How can we prove congruence among line segments? Take two lines, AB = 5 cm, and CD = 5 cm. It can be written as AB ≅ CD. If both the line segments have equal lengths, they are congruent. If two line segments are congruent, then they will, in turn, have the same length.

Like we studied congruence for plane figures and line segments, we can check the congruence of angles.

Source: Class 7 Maths NCERT

In the triangles shown above, you can see that ∠ABC and ∠PQR are equal, and hence, congruent. This can be represented as ∠ABC ≅ ∠PQR.

This shows that if two angles are equal, they must be congruent. Also, if two angles are congruent, their angles must be equal.

Similarly, you can apply the above methods to find the congruence of triangles.

Source: Class 7 Maths NCERT

The triangles shown above are similar in structure with equal sides and equal angles. Hence, they are congruent.

Hence, ΔABC ≅ ΔPQR

Draw two triangles randomly in your notebook. How can you check their congruence by not cutting them out and using superimposition? There must be some criteria for the congruence of triangles. Let us take a look.

Source: Class 7 Maths NCERT

Suppose you have two triangles named ABC and PQR, as shown in the figure above.

AB = PR = 3.5 cm

BC = PQ = 7.1 cm

AC = QR = 5 cm

Hence, ΔABC ≅ ΔRPQ

According to the SSS congruence theorem, two triangles are congruent if three sides of one triangle are equal in measurement to the three corresponding sides of another triangle.

What will you do if you now have two triangles named ABC and DEF, in which an angle is mentioned along with two sides?

Source: Class 7 Maths NCERT

AB = EF = 7 cm

BC = DE = 5 cm

∠B = ∠E = 50°

Since two sides and one angle are equal, these triangles are congruent by the SAS congruence theorem.

ΔABC ≅ ΔFED

According to the SAS congruence theorem,

Two triangles are congruent if two sides and an interior angle of a triangle are equal to two corresponding sides and the interior angle of another triangle.

You have been given two triangles with two angles and one side.

Source: Class 7 Maths NCERT

Here,

∠ABC = ∠DEF = 60°

∠BAC = ∠DFE = 40°

AB = EF = 3.5 cm

Since two angles and one side are equal, these triangles are congruent by the SAS congruence theorem.

ΔABC ≅ ΔDEF

According to the ASA congruence theorem, two triangles are congruent if two angles and a side of a triangle are equal to two corresponding angles and the side of another triangle.

According to AAS Rule, If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.

Source – https://www.teachoo.com/8813/2824/AAS-Congruence-rule/category/ASA-Congruency-Criteria/

In the figure, 

∠ ABC = ∠ PQR 

∠ ACB = ∠PRQ

AC = PR (given)

Hence, ∆ABC ≅ ∆PQR by AAS Rule

Now, how will you prove congruence among right-angled triangles?

Source: Class 7 Maths NCERT

In the above-mentioned right-angled triangles RPQ and ABC,

∠RQP = ∠ABC = 90°

RP = AB = 3 cm

RQ = AC = 8 cm

ΔRPQ ≅ ΔABC

According to the RHS congruence theorem, two triangles are congruent if the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle.

Congruence is observed in various objects around us if they are completely superimposed. Congruence of a line segment, angle, and triangle can be found by comparing the sides, and angles for each of them, respectively. Different rules are used to check the congruence of triangles, including the SSS rule, SAS rule, ASA rule, and RHS congruence rule.

  1. What are the four tests of congruence in a triangle?
    The four tests of congruence in a triangle are as follows:

      • Side Side Side (SSS) rule
      • Side Angle Side (SAS) rule
      • Angle Side Angle (ASA) rule
      • Angle Angle Side (AAS) rule
  2. What are the five triangle congruence theorems?
    The four tests of congruence in a triangle are as follows:

      • Side Side Side (SSS) congruence theorem
      • Side Angle Side (SAS) congruence theorem
      • Angle Side Angle (ASA) congruence theorem
      • Angle Angle Side (AAS) congruence theorem
      • RHS Congruence theorem
  3. How do you write congruent triangles?
    If two triangles named ABC and DEF are congruent to each other by any of the congruence theorems, it is written as ≅.∆ABC ≅ ∆DEF
  4. Is AAA a congruence theorem?
    No, there is no AAA congruence theorem, because when two triangles have equal corresponding angles, one of the two triangles can be an enlarged copy of the other.
  5. What is the ASA congruence rule?
    According to the ASA congruence rule, two triangles are congruent if two angles and the side included between them are equal to the corresponding angles. The side is formed between the other triangles.

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