In Class 7 Mathematics, congruence of triangles is an important topic. Two triangles are said to be congruent if they have the same shape and size, even if their orientation is different. This topic provides a list of important questions and answers related to the congruence of triangles, helping students strengthen their understanding.
Basic Concept of Congruence
Two triangles are congruent if their corresponding sides and angles are equal. The symbol for congruence is ≅.
Congruence Criteria
There are four main criteria to prove two triangles are congruent:
- SSS (Side-Side-Side) Criterion – If three sides of one triangle are equal to the three sides of another, the triangles are congruent.
- 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, the triangles are congruent.
- 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, the triangles are congruent.
- RHS (Right-Angle-Hypotenuse-Side) Criterion – If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another, the triangles are congruent.
Important Questions and Answers
Question 1: What is the meaning of congruent triangles?
Answer:
Two triangles are congruent when all their corresponding sides and angles are equal.
Question 2: What are the four main congruence rules?
Answer:
The four main congruence rules are:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- RHS (Right-Angle-Hypotenuse-Side)
Question 3: Are two triangles with equal perimeters always congruent?
Answer:
No, two triangles with the same perimeter may have different shapes and angles, so they may not be congruent.
Question 4: What is the difference between similarity and congruence?
Answer:
- Congruent triangles have the same shape and size.
- Similar triangles have the same shape but may have different sizes.
Question 5: Which congruence criterion is applicable for right-angled triangles?
Answer:
The RHS (Right-Angle-Hypotenuse-Side) criterion is used to prove the congruence of right-angled triangles.
Question 6: Two triangles have two equal angles and one equal side. Are they congruent?
Answer:
Yes, they are congruent by the ASA (Angle-Side-Angle) Criterion.
Question 7: If two triangles are congruent, what can we say about their areas?
Answer:
If two triangles are congruent, their areas are always equal.
Question 8: If two triangles have two equal sides, are they always congruent?
Answer:
No, for congruence, either the included angle must also be equal (SAS Criterion) or the third side must be equal (SSS Criterion).
Question 9: Can two triangles be congruent if only their angles are equal?
Answer:
No, only equal angles do not prove congruence. The triangles may have different sizes.
Question 10: If two triangles have equal bases and equal heights, are they congruent?
Answer:
No, equal bases and heights only ensure equal areas, but the triangles may not be congruent in shape.
Practice Problems
Question 11:
Two triangles ABC and PQR have:
- AB = PQ
- BC = QR
- ∠B = ∠Q
Are they congruent? If yes, by which criterion?
Solution:
Yes, they are congruent by the SAS (Side-Angle-Side) Criterion because two sides and the included angle are equal.
Question 12:
Triangle XYZ has:
- XY = 6 cm, YZ = 8 cm, XZ = 10 cm
Triangle PQR has: - PQ = 6 cm, QR = 8 cm, PR = 10 cm
Are these two triangles congruent?
Solution:
Yes, they are congruent by the SSS (Side-Side-Side) Criterion as all three sides are equal.
Question 13:
A right-angled triangle has:
- Hypotenuse = 13 cm
- One side = 5 cm
Another right-angled triangle has:
- Hypotenuse = 13 cm
- One side = 5 cm
Are the triangles congruent?
Solution:
Yes, they are congruent by the RHS (Right-Angle-Hypotenuse-Side) Criterion.
Understanding congruence of triangles is crucial in mathematics. By practicing these questions, students can grasp the fundamental concepts and apply the congruence rules effectively. Keep solving problems to strengthen your knowledge!