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Congruence of Triangles Class 7 Notes Maths

7. Congruence of Triangles

Congruence of Plane Figures

Congruence of Triangles Class 7 Notes Maths

If two objects are of exactly the same shape and size, they are said to be congruent.

If two objects are of exactly the same shape and size, they are said to be congruent.

The relation between two congruent objects is called congruence. The method of superposition examines the congruence of plane figures, line segments and angles. Two plane figures are congruent if each, when superimposed on the other, covers it exactly.

If two line segments have the same or equal length, they are congruent. Also, if two line segments are congruent, then they have the same length.

Eg: Two line segments, say, PQ & RS are congruent if they have equal lengths. We write this as

Congruence of Triangles Class 7 Notes Maths

Congruence of Triangles Class 7 Notes Maths

If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are the same.

Eg:

Two angles, say, ∠PQR and ∠XYZ, are congruent if their measures are equal. We write this as or as m∠PQR = m∠XYZ. However, commonly, we write

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Congruence of Triangles Class 7 Notes Maths

A plane figure is any shape that can be drawn in two dimensions.

Congruence of Plane Figures

Congruence of Triangles Class 7 Notes Maths

A plane figure is any shape that can be drawn in two dimensions.

A plane figure is any shape that can be drawn in two dimensions.

Examples: Rectangle, square, triangle, rhombus, etc.

If two objects are of exactly the same shape and size, they are said to be congruent, and the relation of the two objects being congruent is called congruence. Or we can state it as: "Two plane figures are congruent if each, when superimposed on the other, covers it exactly." Congruence is denoted by .

However, how do you check if two figures drawn on a paper are congruent or not? One method is to make a traced copy of one of the figures on a tracing paper and place it over the other.

The other method is to cut out one of these figures and place it over the other. However, take care not to twist, bend or stretch the traced or cut image.

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Congruence of Triangles Class 7 Notes Maths

If two angles have the same measure, then they are congruent. Also, if two angles are congruent, then their measures are the same. If two angles are congruent, then the lengths of their arms do not matter.

Congruence of Triangles Class 7 Notes Maths

Criteria for Congruence of Triangles

Congruence of Triangles Class 7 Notes Maths

Two triangles are congruent if all the sides and all the angles

Congruence of triangles:

Consider triangles ABC and XYZ. Cut triangle ABC and place it over XYZ. The two triangles cover each other exactly, and they are of the same shape and size. Also notice that A falls on X, B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ. So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as  Congruence of Triangles Class 7 Notes Maths

So, in general, we can say that two triangles are congruent if all the sides and all the angles of one triangle are equal to the corresponding sides and angles of the other triangle.

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In two congruent triangles ABC and XYZ, the corresponding vertices are A and X, B and Y, and C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z. So, we write ABC corresponds to XYZ.

We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom.

SSS congruence criterion: Two triangles are congruent if three sides of one triangle are equal to the three corresponding sides of the other triangle. 

SAS congruence criterion: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

RHS congruence criterion: Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

ASA congruence criterion: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Congruence of Triangles Class 7 Notes Maths
Congruence of Triangles Class 7 Notes Maths
Congruence of Triangles Class 7 Notes Maths
Congruence of Triangles Class 7 Notes Maths

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We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom.

SSS congruence criterion: Two triangles are congruent if three sides of one triangle are equal to the three corresponding sides of the other triangle.

SAS congruence criterion: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

RHS congruence criterion: Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.

ASA congruence criterion: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Congruence of Triangles Class 7 Notes Maths
Congruence of Triangles Class 7 Notes Maths
Congruence of Triangles Class 7 Notes Maths

The document Congruence of Triangles Class 7 Notes Maths is a part of the Class 7 Course Mathematics (Maths) Class 7.
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FAQs on Congruence of Triangles Class 7 Notes Maths

1. What is congruence of triangles?
Ans. Congruence of triangles is a term used in geometry to refer to the fact that two triangles are identical in shape and size. This means that all the corresponding sides and angles of the two triangles are equal.
2. What are the criteria for congruence of triangles?
Ans. There are several criteria for congruence of triangles, including SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), and RHS (right angle-hypotenuse-side). These criteria state that if the corresponding sides and angles of two triangles are equal, then the two triangles are congruent.
3. How is congruence of triangles used in real life?
Ans. Congruence of triangles is used in a variety of real-world applications, including architecture, engineering, and construction. For example, when building a bridge or a building, it is important to ensure that the different parts of the structure are congruent in order to maintain stability and safety. Congruence of triangles is also used in the field of computer graphics to create 3D models of objects.
4. What is the importance of congruence of triangles in mathematics?
Ans. Congruence of triangles is an important concept in mathematics because it allows us to prove mathematical theorems and solve geometric problems. By knowing that two triangles are congruent, we can use this information to make conclusions about their properties, such as their angles and side lengths. Congruence of triangles is also a fundamental concept in geometry, providing a basis for many other geometrical concepts and proofs.
5. How can I use congruence of triangles to solve problems in geometry?
Ans. To solve problems involving congruence of triangles, you need to identify the criteria for congruence that apply to the given triangles. Once you have determined that the triangles are congruent, you can use this information to make conclusions about their corresponding sides and angles. For example, if you know that two triangles are congruent by the SAS criterion, then you can conclude that the two corresponding angles and the side between them are equal. You can then use this information to solve for the other sides and angles of the triangles.
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