What Is Congruence of Triangles Class 7: Definition & Examples

Understanding Congruence of Triangles in Class 7

Congruence means exact matching in shape and size. In Class 7 NCERT Mathematics, two triangles are said to be congruent if their corresponding sides and angles are equal. This means if you place one triangle over the other, they coincide perfectly.

Key points:

• Corresponding sides are equal in length
• Corresponding angles are equal in measure

Congruent triangles are denoted by the symbol $\cong$. For example, if triangle $ABC$ is congruent to triangle $DEF$, we write $\triangle ABC \cong \triangle DEF$.

This concept is fundamental in geometry and helps in proving many properties related to triangles and other shapes.

Criteria for Congruence of Triangles

There are four main criteria to check if two triangles are congruent. These help us avoid measuring all sides and angles.

1. SSS (Side-Side-Side) Criterion: If all three sides of one triangle are equal to all three sides of another triangle, the triangles are congruent.

2. SAS (Side-Angle-Side) Criterion: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

3. ASA (Angle-Side-Angle) Criterion: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.

4. RHS (Right angle-Hypotenuse-Side) Criterion: In right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle, the triangles are congruent.

These criteria simplify proving congruence without checking every side and angle.

How to Identify Corresponding Parts in Congruent Triangles

When two triangles are congruent, their corresponding parts match exactly. Identifying these parts correctly is important.

• Corresponding vertices: The order of letters in the triangle name shows correspondence. For example, in $\triangle ABC \cong \triangle DEF$:
• $A$ corresponds to $D$
• $B$ corresponds to $E$
• $C$ corresponds to $F$

• Corresponding sides:
• $AB$ corresponds to $DE$
• $BC$ corresponds to $EF$
• $CA$ corresponds to $FD$

• Corresponding angles:
• $\angle A$ corresponds to $\angle D$
• $\angle B$ corresponds to $\angle E$
• $\angle C$ corresponds to $\angle F$

Always write the triangle names in the correct order to match corresponding parts. This helps in solving problems involving congruence.

Worked Example: Using SSS Criterion

Let's solve a simple example to understand how to use the SSS criterion.

Example: Two triangles $\triangle PQR$ and $\triangle XYZ$ have sides:

• $PQ = 5$ cm, $QR = 7$ cm, $RP = 6$ cm
• $XY = 5$ cm, $YZ = 7$ cm, $ZX = 6$ cm

Are these triangles congruent?

Solution: Check corresponding sides:

• $PQ = XY = 5$ cm
• $QR = YZ = 7$ cm
• $RP = ZX = 6$ cm

Since all three pairs of sides are equal, by the SSS criterion,

$$\triangle PQR \cong \triangle XYZ$$

Thus, the triangles are congruent.

Importance of Congruence in Geometry and Exams

Understanding congruence of triangles is vital for Class 7 students because:

• It helps prove properties of triangles and other polygons.
• It is a foundation for learning similarity and other advanced topics.
• Many exam questions test your ability to identify and prove congruence.

For example, proving that two triangles are congruent can help find unknown side lengths or angles in geometry problems.

NCERT textbooks provide exercises to practice these concepts, which are important for CBSE exams and competitive tests.

Tips for exams:

• Memorise the four congruence criteria.
• Practice identifying corresponding parts carefully.
• Use diagrams to visualise congruence clearly.

Mastering this topic boosts confidence in geometry questions.

Summary and Formulae for Quick Revision

Here is a quick summary of what you should remember about congruence of triangles:

• Two triangles are congruent if all corresponding sides and angles are equal.
• Congruence is denoted by $\cong$.
• Four main criteria:
• SSS: Three sides equal
• SAS: Two sides and included angle equal
• ASA: Two angles and included side equal
• RHS: Right angle, hypotenuse, and one side equal

Formulae and notation:

• $\triangle ABC \cong \triangle DEF$ means:
• $AB = DE$, $BC = EF$, $CA = FD$
• $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$

Use these points to revise before exams and solve problems confidently.