In NCERT Class 7 Mathematics, the concept of Congruence of Triangles is an essential topic that helps students understand the properties of geometrical figures. Congruence means exactly equal in shape and size. Two triangles are said to be congruent if all their corresponding sides and angles are equal.
This topic is important because it forms the foundation for advanced geometry in higher classes. Understanding congruence rules helps students solve various problems related to triangles, symmetry, and construction.
What is Congruence?
The term “congruence” means that two figures have the same shape and size. If one figure is placed over another and they match exactly, they are congruent.
For example:
- Two identical coins are congruent.
- Two leaves of the same tree can be congruent.
- Two playing cards of the same deck are congruent.
When we talk about congruence in triangles, we mean that two triangles have the same corresponding sides and angles.
Congruence of Triangles
Definition
Two triangles are said to be congruent if their three sides and three angles are equal in measure.
If △ABC ≅ △DEF, then:
- AB = DE (Corresponding sides are equal)
- BC = EF (Corresponding sides are equal)
- CA = FD (Corresponding sides are equal)
- ∠A = ∠D (Corresponding angles are equal)
- ∠B = ∠E (Corresponding angles are equal)
- ∠C = ∠F (Corresponding angles are equal)
The symbol for congruence is “≅”, which means “is congruent to”.
Criteria for Triangle Congruence
There are four main rules to check if two triangles are congruent:
1. SSS (Side-Side-Side) Congruence Rule
If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.
Example:
If AB = PQ, BC = QR, and CA = RP, then △ABC ≅ △PQR by SSS rule.
2. SAS (Side-Angle-Side) Congruence Rule
If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.
Example:
If AB = PQ, ∠B = ∠Q, and BC = QR, then △ABC ≅ △PQR by SAS rule.
3. ASA (Angle-Side-Angle) Congruence Rule
If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the two triangles are congruent.
Example:
If ∠A = ∠P, AB = PQ, and ∠B = ∠Q, then △ABC ≅ △PQR by ASA rule.
4. RHS (Right Angle-Hypotenuse-Side) Congruence Rule
If one triangle is a right-angled triangle, and its hypotenuse and one side are equal to the corresponding hypotenuse and one side of another right-angled triangle, then they are congruent.
Example:
If ∠B = 90°, AC = PR, and AB = PQ, then △ABC ≅ △PQR by RHS rule.
Properties of Congruent Triangles
- Congruent triangles have equal corresponding sides and angles.
- Congruence is a rigid motion – If you flip, rotate, or slide a triangle, it remains congruent.
- Congruence helps in proving properties of geometric figures, such as proving two opposite angles are equal in a parallelogram.
- Congruent triangles help in real-life applications, such as in architecture, engineering, and design.
Examples and Applications of Congruence
Example 1: Proving Triangles Congruent
Given: △ABC and △DEF, where AB = DE, BC = EF, and CA = FD.
To prove: △ABC ≅ △DEF
Solution: Since all three sides are equal, the triangles are congruent by SSS rule.
Example 2: Identifying Congruence in Daily Life
- Bridges: Engineers use congruent triangles for strong and stable bridge structures.
- Tiles and Floor Patterns: Designs with congruent triangles create symmetry and balance.
- Construction of Houses: Walls and roof frames use congruent triangles for stability.
How to Solve NCERT Questions on Congruence of Triangles
Step 1: Read the Problem Carefully
Identify the given information about the sides and angles of the triangles.
Step 2: Identify the Congruence Rule
Check whether SSS, SAS, ASA, or RHS applies to the triangles.
Step 3: Apply the Rule and Prove Congruence
Use the relevant rule to prove that the triangles are congruent.
Step 4: Justify Each Step Clearly
While writing the solution, always provide a step-by-step justification.
Common Mistakes to Avoid in Triangle Congruence
- Confusing ASA and AAS: Remember that ASA requires an included side, while AAS does not.
- Ignoring the Corresponding Parts: Always ensure you compare the correct sides and angles.
- Forgetting the RHS Rule: This rule applies only to right-angled triangles.
- Assuming Similarity Instead of Congruence: Similar triangles have proportional sides, while congruent triangles have equal sides and angles.
Practice Questions on Congruence of Triangles
- Two triangles have sides 5 cm, 7 cm, and 10 cm. Another triangle has sides 5 cm, 7 cm, and 10 cm. Prove that they are congruent.
- Triangle ABC has angles ∠A = 50°, ∠B = 60°, and side AB = 8 cm. Triangle PQR has angles ∠P = 50°, ∠Q = 60°, and side PQ = 8 cm. Are they congruent?
- Triangle LMN and triangle XYZ have two equal sides and a common included angle. Prove they are congruent using SAS rule.
- Two right-angled triangles have hypotenuses of 12 cm and one equal side of 5 cm. Prove their congruence.
- In a parallelogram, prove that the diagonals divide the shape into two congruent triangles.
The concept of Congruence of Triangles in NCERT Class 7 Maths is crucial for understanding geometry, symmetry, and construction. By mastering the SSS, SAS, ASA, and RHS congruence rules, students can solve various mathematical problems with confidence.
Congruent triangles are widely used in architecture, design, and real-world applications, making them an essential part of mathematics and engineering. Regular practice of NCERT exercises and understanding the fundamentals of congruence will help students excel in geometry and build a strong foundation for higher studies.