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Congruence of Triangles – CBSE NCERT Study Resources

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7th

7th - Mathematics

Congruence of Triangles

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Overview of the Chapter: Congruence of Triangles

This chapter introduces the concept of congruence in triangles, a fundamental topic in geometry. Students will learn how to identify congruent triangles based on specific criteria and understand the properties that make two triangles congruent. The chapter covers practical applications and problem-solving techniques to reinforce the concepts.

Key Concepts

Congruent Figures: Two figures are congruent if they have the same shape and size, i.e., their corresponding sides and angles are equal.

Congruence of Triangles: Two triangles are congruent if their corresponding sides and angles are equal. This can be determined using specific criteria such as SSS, SAS, ASA, and RHS.

Criteria for Congruence of Triangles

  • SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.
  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  • RHS (Right Angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle are equal to the corresponding parts of another right-angled triangle, the triangles are congruent.

Applications of Congruence

Understanding congruence helps in solving geometric problems, constructing shapes, and proving theorems. It is also used in real-life scenarios such as architecture and engineering.

Solved Examples

Example 1: Prove that two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding parts of the other triangle (ASA criterion).

Solution: By the ASA criterion, since the angles and the included side match, the triangles must be congruent.

Practice Questions

  1. State whether the following pairs of triangles are congruent. If yes, state the criterion used.
  2. In triangle ABC and DEF, AB = DE, BC = EF, and ∠B = ∠E. Are the triangles congruent? If yes, by which criterion?

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
In real life, where do we see congruent triangles?
Answer:

Bridge trusses, roof designs

Question 2:
If two triangles have equal areas, are they always congruent?
Answer:

No, shapes may differ

Question 3:
What is the full form of SSS in triangle congruence?
Answer:
Side-Side-Side
Question 4:
If ΔABC ≅ ΔPQR by ASA, which angles are equal?
Answer:

∠A = ∠P, ∠B = ∠Q

Question 5:
Which congruence rule applies if two sides and included angle are equal?
Answer:
SAS
Question 6:
In ΔXYZ ≅ ΔLMN by RHS, what must be right angles?
Answer:

∠Y and ∠M

Question 7:
Are all equilateral triangles congruent?
Answer:

No, only if sides are equal

Question 8:
If ΔDEF ≅ ΔJKL, what is the length of JK if DE = 5 cm?
Answer:
5 cm
Question 9:
Which congruence criterion requires a right angle, hypotenuse, and one side?
Answer:
RHS
Question 10:
Can two triangles be congruent if only three angles are equal?
Answer:

No, sides must also match

Question 11:
If ΔABC ≅ ΔDEF by SSS, which side equals BC?
Answer:

EF

Question 12:
What is the minimum number of equal parts needed for triangle congruence?
Answer:
3
Question 13:
If ΔABC ≅ ΔPQR by SAS, which sides and angles are equal?
Answer:

AB=PQ, ∠B=∠Q, BC=QR

Question 14:
Two triangles are congruent if their corresponding angles and sides are equal. (True/False)
Answer:

True

Question 15:
Which congruence rule applies if two right triangles have equal hypotenuse and one leg?
Answer:

RHS Congruence

Question 16:
If ΔDEF ≅ ΔMNO by ASA, what must be equal?
Answer:

∠D=∠M, DE=MN, ∠E=∠N

Question 17:
Can two triangles with sides 3cm, 4cm, 5cm and 4cm, 5cm, 3cm be congruent?
Answer:

Yes (SSS)

Question 18:
What is the minimum number of equal parts needed to prove triangle congruence?
Answer:

3

Question 19:
In real life, why do engineers use congruent triangles in bridges?
Answer:

For equal weight distribution

Question 20:
If two angles and a side of one triangle equal another, which congruence rule applies?
Answer:

AAS Congruence

Question 21:
Are all equilateral triangles congruent? (Yes/No)
Answer:

No (sides may differ)

Question 22:
What is the symbol for congruence?
Answer:

≅

Question 23:
If ΔXYZ has XY=5cm, ∠Y=60°, YZ=7cm and ΔUVW has UV=5cm, ∠V=60°, VW=7cm, are they congruent?
Answer:

Yes (SAS)

Question 24:
State the condition under which two triangles are congruent by the SAS criterion.
Answer:

Two triangles are congruent by the SAS (Side-Angle-Side) criterion if two sides and the included angle of one triangle are equal to the corresponding sides and included angle of the other triangle.

Question 25:
If ΔABC ≅ ΔPQR by SSS congruence, what can you say about their corresponding angles?
Answer:

If ΔABC ≅ ΔPQR by SSS (Side-Side-Side) congruence, then all their corresponding angles are equal.
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.

Question 26:
What is the minimum number of pairs of equal sides/angles required to prove two triangles congruent using ASA?
Answer:

For ASA (Angle-Side-Angle) congruence, we need two angles and the included side of one triangle equal to the corresponding parts of the other triangle.

Question 27:
Can two triangles be congruent if all three angles of one triangle are equal to all three angles of another? Justify.
Answer:

No, two triangles with all angles equal are similar but not necessarily congruent. Congruence requires equal sides as well (e.g., by AAA, only similarity is guaranteed).

Question 28:
In ΔABC and ΔDEF, AB = DE, BC = EF, and ∠B = ∠E. Which congruence criterion applies here?
Answer:

The given condition satisfies the SAS (Side-Angle-Side) congruence criterion because two sides and the included angle are equal.

Question 29:
If two right-angled triangles have their hypotenuses and one pair of legs equal, which congruence criterion is satisfied?
Answer:

This satisfies the RHS (Right angle-Hypotenuse-Side) congruence criterion, specific to right-angled triangles.

Question 30:
Write the full form of RHS in the context of triangle congruence.
Answer:

RHS stands for Right angle-Hypotenuse-Side, a congruence criterion for right-angled triangles where the hypotenuse and one leg are equal.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
State the SSS Congruence Rule for triangles.
Answer:

The SSS Congruence Rule states that if the three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Question 2:
What is the SAS Congruence Rule? Explain with an example.
Answer:

The SAS Congruence Rule states that 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 triangles are congruent.
Example: If ΔABC has AB = 5 cm, ∠B = 60°, BC = 4 cm and ΔPQR has PQ = 5 cm, ∠Q = 60°, QR = 4 cm, then ΔABC ≅ ΔPQR by SAS rule.

Question 3:
How does the ASA Congruence Rule differ from the AAS Congruence Rule?
Answer:

The ASA Congruence Rule requires two angles and the included side to be equal, while the AAS Congruence Rule requires two angles and a non-included side to be equal.
Example:
ASA: ∠A = ∠P, AB = PQ, ∠B = ∠Q
AAS: ∠A = ∠P, ∠B = ∠Q, BC = QR

Question 4:
If two triangles are congruent under the RHS Congruence Rule, what must be true about their angles?
Answer:

In the RHS Congruence Rule, one angle must be a right angle (90°), and the hypotenuse and one side of the right-angled triangle must be equal to the corresponding parts of the other triangle.

Question 5:
Can two triangles be congruent if all their corresponding angles are equal? Justify.
Answer:

No, two triangles with all corresponding angles equal are similar, but not necessarily congruent. For congruency, at least one pair of corresponding sides must also be equal (as per AAA similarity).

Question 6:
In ΔABC and ΔDEF, AB = DE, BC = EF, and ∠B = ∠E. Which congruence rule applies here?
Answer:

The SAS Congruence Rule applies here because two sides (AB = DE, BC = EF) and the included angle (∠B = ∠E) are equal.

Question 7:
If ΔPQR ≅ ΔXYZ by ASA rule, what can you say about their corresponding parts?
Answer:

If ΔPQR ≅ ΔXYZ by ASA rule, then:
∠P = ∠X
PQ = XY
∠Q = ∠Y
This means all other corresponding sides and angles are also equal due to congruency.

Question 8:
Why is there no SSA Congruence Rule? Explain with an example.
Answer:

SSA is not a valid congruence rule because two triangles can have two sides and a non-included angle equal but still not be congruent.
Example: Two triangles can have sides 5 cm, 7 cm, and a non-included angle of 30°, but the third side may differ, making them non-congruent.

Question 9:
In a right-angled triangle ΔABC (∠B = 90°), AB = 6 cm and AC = 10 cm. If ΔPQR is congruent to ΔABC by RHS rule, what must be the measures of PQ and PR?
Answer:

For ΔPQR ≅ ΔABC by RHS rule:
PQ = AB = 6 cm (one side)
PR = AC = 10 cm (hypotenuse)
∠Q must also be 90°.

Question 10:
If two triangles have their corresponding sides equal, are their perimeters also equal? Why?
Answer:

Yes, if two triangles are congruent (all sides equal), their perimeters must also be equal because the perimeter is the sum of all sides, and corresponding sides are equal.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
State the SSS Congruence Rule for triangles and explain why it guarantees congruence.
Answer:

The SSS Congruence Rule states that if all three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent.


This guarantees congruence because the lengths of the sides uniquely determine the shape and size of a triangle. If all sides match, the angles must also match due to the rigidity of triangles.

Question 2:
If two triangles ABC and PQR are congruent under the ASA condition, what can you conclude about their corresponding parts?
Answer:

If ABC ≅ PQR by ASA Congruence, then:

  • ∠A = ∠P (included angle)
  • Side AB = PQ (one adjacent side)
  • ∠B = ∠Q (other adjacent angle)

All other corresponding sides and angles will also be equal due to the congruence.

Question 3:
Draw two congruent triangles and label their corresponding equal sides and angles. State the congruence criterion used.
Answer:

Consider ΔABC ≅ ΔDEF by SAS Congruence:

  • AB = DE (equal sides)
  • ∠B = ∠E (equal included angles)
  • BC = EF (equal sides)

Corresponding angles and remaining sides will automatically be equal.

Question 4:
Explain why AAA is not a valid congruence criterion for triangles with an example.
Answer:

AAA is not a congruence criterion because triangles can have the same angles but different side lengths (similar but not congruent).


Example: An equilateral triangle with sides 2cm and another with sides 4cm have equal angles (60° each) but are not congruent.

Question 5:
In ΔABC and ΔPQR, if AB = PQ, BC = QR, and ∠C = ∠R, can we say the triangles are congruent? Justify your answer.
Answer:

No, the triangles are not necessarily congruent because the given condition is SSA, which is not a valid congruence rule.


For congruence, the equal angle must be included between the two sides (SAS rule). Here, ∠C and ∠R are not included angles.

Question 6:
State the SSS Congruence Rule and explain why it guarantees two triangles are congruent.
Answer:

The SSS Congruence Rule states that if all three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent.


This works because the lengths of the sides completely determine the shape and size of a triangle. If all sides match, the angles must also match due to rigidity, ensuring identical triangles.

Question 7:
If ΔABC ≅ ΔPQR under the SAS Congruence Rule, what must be the given conditions?
Answer:

For ΔABC ≅ ΔPQR under SAS Congruence, the following conditions must hold:


  • Two sides of ΔABC must be equal to two corresponding sides of ΔPQR.
  • The included angle (the angle between the two sides) in ΔABC must be equal to the corresponding angle in ΔPQR.
Question 8:
Explain why AAA (Angle-Angle-Angle) is not a valid congruence rule for triangles.
Answer:

AAA only ensures that the triangles have the same shape (they are similar) but not necessarily the same size. Two triangles can have all angles equal but different side lengths, like a small and a large equilateral triangle. Thus, AAA does not guarantee congruence.

Question 9:
If two right-angled triangles have their hypotenuses and one corresponding leg equal, which congruence rule applies? Justify.
Answer:

This is the RHS Congruence Rule (Right Angle-Hypotenuse-Side).

Since both are right-angled, one pair of legs and the hypotenuse are equal. The right angle ensures the triangles' shapes are fixed, making them congruent.

Question 10:
Draw two congruent triangles ΔXYZ and ΔLMN where XY = LM, YZ = MN, and ∠Y = ∠M. Which congruence rule is satisfied here?
Answer:

The given conditions satisfy the SAS Congruence Rule.

Here:
XY = LM (one pair of equal sides),
YZ = MN (second pair of equal sides),
∠Y = ∠M (included angles are equal).
Thus, ΔXYZ ≅ ΔLMN by SAS.

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Explain the SSS congruence rule with an example from our textbook. How can this rule be applied in real-life situations?
Answer:
Introduction

We studied that two triangles are congruent if all three sides of one triangle are equal to the corresponding sides of another. This is the SSS rule.


Argument 1

Our textbook shows an example where ΔABC and ΔPQR have AB = PQ, BC = QR, and AC = PR. By SSS rule, ΔABC ≅ ΔPQR.


Argument 2

In real life, this rule helps in constructing identical structures like bridges or roofs, ensuring stability.


Conclusion

SSS rule is a simple yet powerful tool to verify congruence in triangles.

Question 2:
Using the SAS congruence rule, prove that ΔABC ≅ ΔDEF if AB = DE, ∠B = ∠E, and BC = EF. Include a diagram.
Answer:
Introduction

We know that two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding parts of another.


Argument 1

Given AB = DE, ∠B = ∠E, and BC = EF, by SAS rule, ΔABC ≅ ΔDEF. [Diagram: Two triangles with marked equal sides and angles]


Argument 2

This rule is used in construction to ensure identical triangular frames for doors or windows.


Conclusion

SAS rule helps confirm congruence when specific sides and angles are known.

Question 3:
Describe the ASA congruence rule with a textbook example. How is it different from AAS?
Answer:
Introduction

The ASA rule states that two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding parts of another.


Argument 1

Our textbook shows ΔLMN ≅ ΔXYZ where ∠L = ∠X, LM = XY, and ∠M = ∠Y. This satisfies ASA.


Argument 2

Unlike ASA, AAS requires two angles and a non-included side. ASA is stricter as it demands the side to be included.


Conclusion

ASA ensures congruence when angles and the side between them are equal.

Question 4:
Prove that ΔABC ≅ ΔDEF using the SAS Congruence Rule given AB = DE, ∠B = ∠E, and BC = EF. Include a diagram.
Answer:
Introduction

We know that two triangles are congruent if two sides and the included angle are equal, called the SAS Congruence Rule.


Argument 1

Given AB = DE, ∠B = ∠E, and BC = EF, we can align ΔABC and ΔDEF such that they overlap perfectly.


Argument 2

[Diagram: Two triangles with sides AB = DE, BC = EF, and ∠B = ∠E.] This proves ΔABC ≅ ΔDEF by SAS rule.


Conclusion

SAS rule is useful for proving congruence when angle and sides are known.

Question 5:
Describe the ASA Congruence Rule with a textbook example. How is it different from the AAS rule?
Answer:
Introduction

The ASA Congruence Rule states that two triangles are congruent if two angles and the included side are equal.


Argument 1

Our textbook shows ΔLMN and ΔXYZ where ∠L = ∠X, LM = XY, and ∠M = ∠Y. By ASA rule, ΔLMN ≅ ΔXYZ.


Argument 2

AAS rule requires two angles and any side, while ASA requires the side to be included between the angles.


Conclusion

ASA is stricter but equally important for congruence proofs.

Question 6:
Describe the SAS Congruence Rule with a diagram. How is it different from the ASA rule?
Answer:
Introduction

The SAS Congruence Rule states that two triangles are congruent if two sides and the included angle are equal.


Argument 1

[Diagram: Two triangles with marked equal sides and included angle.] Our textbook example shows ΔDEF ≅ ΔLMN because DE = LM, EF = MN, and ∠E = ∠M.


Argument 2

Unlike SAS, the ASA rule requires two angles and the included side to be equal.


Conclusion

SAS and ASA are distinct but both ensure triangle congruence under specific conditions.

Question 7:
Prove that ΔABC ≅ ΔDEF using the RHS Congruence Rule. What real-life objects follow this rule?
Answer:
Introduction

The RHS Congruence Rule applies to right-angled triangles when their hypotenuse and one side are equal.


Argument 1

Given ΔABC and ΔDEF are right-angled at B and E, with AC = DF (hypotenuse) and AB = DE (one side). By RHS, ΔABC ≅ ΔDEF.


Argument 2

Real-life examples include identical door frames or ladder placements, where right angles and side lengths matter.


Conclusion

RHS rule is crucial for verifying congruence in right-angled structures.

Question 8:
Explain the SSS congruence rule with an example from our textbook. How is it useful in real-life situations?
Answer:
Introduction

We studied that two triangles are congruent if their corresponding sides are equal. This is the SSS rule.


Argument 1

Our textbook shows ΔABC and ΔDEF where AB = DE, BC = EF, and AC = DF. By SSS rule, ΔABC ≅ ΔDEF.


Argument 2

In real life, engineers use SSS to ensure stability in structures like bridges by checking triangular supports.


Conclusion

SSS helps verify exact shape and size equality, crucial in design and construction.

Question 9:
Prove that ΔPQR ≅ ΔSTU using the SAS congruence rule, given PQ = ST, ∠Q = ∠T, and QR = TU. Include a diagram.
Answer:
Introduction

We know two triangles are congruent if two sides and the included angle are equal (SAS rule).


Argument 1

Given PQ = ST, ∠Q = ∠T, and QR = TU, ΔPQR and ΔSTU satisfy SAS.


Argument 2

[Diagram: Two triangles with sides PQ=ST, QR=TU and included angle ∠Q=∠T.]


Conclusion

Since all conditions match, ΔPQR ≅ ΔSTU by SAS rule.

Question 10:
Describe the ASA congruence rule with a real-life example. How is it different from AAS?
Answer:
Introduction

The ASA rule states two triangles are congruent if two angles and the included side are equal.


Argument 1

For example, in book pages, folded corners create congruent triangles by ASA (equal angles and common side).


Argument 2

AAS requires two angles and any side, while ASA strictly needs the included side.


Conclusion

ASA ensures precise alignment, useful in manufacturing identical parts.

Question 11:
Prove that the two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle (ASA Congruence Rule). Explain with a diagram and step-by-step reasoning.
Answer:

To prove the ASA Congruence Rule, let us consider two triangles, △ABC and △DEF, such that:


  • ∠B = ∠E (Given)
  • ∠C = ∠F (Given)
  • BC = EF (Given, included side)

Now, let us superimpose △ABC on △DEF such that side BC coincides with EF and point B falls on E, and point C falls on F.


Since ∠B = ∠E, the side BA will coincide with ED.


Similarly, since ∠C = ∠F, the side CA will coincide with FD.


Thus, point A will coincide with point D, making all corresponding sides and angles equal.


Therefore, △ABC ≅ △DEF by ASA Congruence Rule.


Diagram: Draw two triangles with the given conditions, labeling angles and sides accordingly.

Question 12:
In a quadrilateral ABCD, diagonal AC divides it into two congruent triangles, △ABC and △ADC. Prove that the quadrilateral is a parallelogram using the properties of congruent triangles.
Answer:

Given: △ABC ≅ △ADC by SSS Congruence Rule (since AC is common, AB = AD, and BC = DC as per the problem).


From the congruence, we can derive the following:


  • ∠BAC = ∠DAC (Corresponding angles of congruent triangles)
  • ∠BCA = ∠DCA (Corresponding angles of congruent triangles)

Now, since alternate interior angles (∠BAC and ∠DCA) are equal, AB ∥ DC.


Similarly, since ∠BCA = ∠DAC, AD ∥ BC.


Since both pairs of opposite sides are parallel, ABCD is a parallelogram.


Note: This proof uses the property that if both pairs of opposite sides of a quadrilateral are parallel, it is a parallelogram.

Question 13:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the Congruence of Triangles concept, prove that both triangles are congruent. State the criterion used and justify your answer with a diagram.
Answer:

To prove that ΔABC ≅ ΔPQR, we use the SAS (Side-Angle-Side) congruence criterion.


Given:
AB = PQ (Side)
BC = QR (Side)
∠B = ∠Q (Included Angle)


Proof:
Since two sides and the included angle of ΔABC are equal to the corresponding sides and angle of ΔPQR, by the SAS rule, both triangles are congruent.


Diagram:
(Draw two triangles with sides AB = PQ, BC = QR, and ∠B = ∠Q marked as equal.)


Conclusion:
Hence, ΔABC ≅ ΔPQR by SAS congruence.

Question 14:
If ΔDEF ≅ ΔLMN by ASA congruence rule, what additional information is needed if ∠D = ∠L and ∠E = ∠M? Explain with steps and a diagram.
Answer:

For ΔDEF ≅ ΔLMN by the ASA (Angle-Side-Angle) rule, we need:


Given:
∠D = ∠L (Angle)
∠E = ∠M (Angle)


Additional Information Required:
The included side between the two given angles must be equal, i.e., DE = LM.


Proof:
1. Two angles and the included side of ΔDEF must match those of ΔLMN.
2. Since ∠D = ∠L and ∠E = ∠M, the side between them (DE) must equal LM.


Diagram:
(Draw two triangles with angles ∠D = ∠L and ∠E = ∠M, and mark DE = LM.)


Conclusion:
With DE = LM, ΔDEF ≅ ΔLMN by ASA congruence.

Question 15:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the SAS congruence rule, prove that the two triangles are congruent. Also, state the remaining matching parts.
Answer:

Given: In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q.

To prove: ΔABC ≅ ΔPQR by SAS congruence rule.


Proof:


1. According to the SAS (Side-Angle-Side) rule, if two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.


2. Here, AB = PQ (given), BC = QR (given), and ∠B = ∠Q (given).


3. Since the angle is included between the two sides, the SAS condition is satisfied.


4. Therefore, ΔABC ≅ ΔPQR.


Remaining matching parts:

  • AC = PR (by CPCT)
  • ∠A = ∠P (by CPCT)
  • ∠C = ∠R (by CPCT)
Question 16:
Two triangles ΔXYZ and ΔLMN are such that XY = LM, YZ = MN, and XZ = LN. Prove that the triangles are congruent using the SSS congruence rule. Also, explain why congruence is important in real-life applications.
Answer:

Given: In ΔXYZ and ΔLMN, XY = LM, YZ = MN, and XZ = LN.

To prove: ΔXYZ ≅ ΔLMN by SSS congruence rule.


Proof:


1. According to the SSS (Side-Side-Side) rule, if all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.


2. Here, XY = LM, YZ = MN, and XZ = LN (given).


3. Since all three sides are equal, the SSS condition is satisfied.


4. Therefore, ΔXYZ ≅ ΔLMN.


Real-life importance of congruence:

  • Congruent triangles ensure identical shapes and sizes, which is crucial in construction (e.g., roof trusses, bridges).
  • Used in manufacturing to create identical parts (e.g., car components, furniture).
  • Helps in accurate measurements in maps, engineering designs, and art.
Question 17:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the SAS congruence rule, prove that the two triangles are congruent. Also, state the other pairs of corresponding parts that are equal.
Answer:

Given: ΔABC and ΔPQR where AB = PQ, BC = QR, and ∠B = ∠Q.

To prove: ΔABC ≅ ΔPQR by SAS congruence rule.


Proof:


1. According to the given, AB = PQ (Side).
2. ∠B = ∠Q (Included Angle).
3. BC = QR (Side).


Since two sides and the included angle of ΔABC are equal to the corresponding parts of ΔPQR, by the SAS congruence rule, ΔABC ≅ ΔPQR.


Other corresponding equal parts:


1. AC = PR (By CPCT).
2. ∠A = ∠P (By CPCT).
3. ∠C = ∠R (By CPCT).


Thus, all corresponding sides and angles are equal, confirming the congruence.

Question 18:
Two triangles, ΔXYZ and ΔLMN, are such that ∠X = ∠L, ∠Y = ∠M, and XY = LM. Prove that the two triangles are congruent using the ASA congruence rule. Also, name the other equal parts.
Answer:

Given: ΔXYZ and ΔLMN where ∠X = ∠L, ∠Y = ∠M, and XY = LM.

To prove: ΔXYZ ≅ ΔLMN by ASA congruence rule.


Proof:


1. ∠X = ∠L (Given as one angle).
2. XY = LM (Given as the included side).
3. ∠Y = ∠M (Given as the other angle).


Since two angles and the included side of ΔXYZ are equal to the corresponding parts of ΔLMN, by the ASA congruence rule, ΔXYZ ≅ ΔLMN.


Other corresponding equal parts:


1. XZ = LN (By CPCT).
2. YZ = MN (By CPCT).
3. ∠Z = ∠N (By angle sum property and CPCT).


Thus, the triangles are congruent, and all corresponding parts are equal.

Question 19:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the SAS congruence rule, prove that the two triangles are congruent. Also, state the other pairs of equal parts if ΔABC ≅ ΔPQR.
Answer:

Given: ΔABC and ΔPQR where AB = PQ, BC = QR, and ∠B = ∠Q.

To prove: ΔABC ≅ ΔPQR using the SAS congruence rule.


Proof:


1. According to the SAS rule, two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and angle of the other triangle.


2. Here, AB = PQ (given), BC = QR (given), and ∠B = ∠Q (given).


3. Since the included angle ∠B is equal to ∠Q, and the sides adjacent to these angles are equal, ΔABC ≅ ΔPQR by SAS congruence.


Other equal parts:


If ΔABC ≅ ΔPQR, then:

  • AC = PR (by CPCT)
  • ∠A = ∠P (by CPCT)
  • ∠C = ∠R (by CPCT)
Question 20:
Two triangles, ΔXYZ and ΔLMN, are such that ∠X = ∠L, ∠Y = ∠M, and XY = LM. Prove that the two triangles are congruent using the ASA congruence rule. Also, explain why the third pair of sides must be equal.
Answer:

Given: ΔXYZ and ΔLMN where ∠X = ∠L, ∠Y = ∠M, and XY = LM.

To prove: ΔXYZ ≅ ΔLMN using the ASA congruence rule.


Proof:


1. According to the ASA rule, two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding angles and side of the other triangle.


2. Here, ∠X = ∠L (given), ∠Y = ∠M (given), and XY = LM (given, which is the included side between the two angles).


3. Thus, ΔXYZ ≅ ΔLMN by ASA congruence.


Third pair of sides:


Since the triangles are congruent, the third pair of sides XZ = LN and YZ = MN must be equal due to CPCT (Corresponding Parts of Congruent Triangles). This is because all corresponding sides and angles of congruent triangles are equal.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that ΔABC ≅ ΔPQR. Also, name the criterion used.
Answer:
Problem Interpretation

We studied that two triangles are congruent if their corresponding sides and angles are equal.


Mathematical Modeling

Given: AB = PQ, BC = QR, ∠B = ∠Q.


Solution

By the SAS (Side-Angle-Side) congruence rule, since two sides and the included angle are equal, ΔABC ≅ ΔPQR.

Question 2:
A ladder leans against a wall, forming ΔABC with the ground. Another ladder of the same length forms ΔPQR. If AB = PQ and ∠A = ∠P, prove ΔABC ≅ ΔPQR using the congruence rule.
Answer:
Problem Interpretation

Our textbook shows real-life examples of congruent triangles, like ladders against walls.


Mathematical Modeling

Given: AB = PQ (ladder length), ∠A = ∠P (angle with ground).


Solution

By the ASA (Angle-Side-Angle) rule, since two angles and the included side are equal, ΔABC ≅ ΔPQR.

Question 3:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that ΔABC ≅ ΔPQR. Explain why this rule applies.
Answer:
Problem Interpretation

We need to prove ΔABC ≅ ΔPQR using given sides and angles.

Mathematical Modeling

Given: AB = PQ (Side), BC = QR (Side), ∠B = ∠Q (Angle).

Solution

By the SAS (Side-Angle-Side) congruence rule, if two sides and the included angle of one triangle are equal to another, they are congruent. Here, AB = PQ, BC = QR, and ∠B = ∠Q, so ΔABC ≅ ΔPQR.

Question 4:
A ladder leans against a wall forming ΔABC. Another ladder of the same length leans similarly forming ΔPQR. If AB = PQ and ∠BAC = ∠QPR, which congruence rule proves ΔABC ≅ ΔPQR? Justify.
Answer:
Problem Interpretation

We compare two triangles formed by ladders against a wall.

Mathematical Modeling

Given: AB = PQ (Side), ∠BAC = ∠QPR (Angle), and both ladders are equal (Hypotenuse).

Solution

Using the RHS (Right Angle-Hypotenuse-Side) rule, since both are right-angled triangles with equal hypotenuse and one side, ΔABC ≅ ΔPQR. Our textbook shows this applies when right angle, hypotenuse, and one side match.

Question 5:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that the triangles are congruent. Also, if AB = 5 cm and ∠A = 50°, find the length of PR and ∠R.
Answer:
Problem Interpretation

We studied that two triangles are congruent if their corresponding sides and angles are equal. Here, ΔABC and ΔPQR satisfy the SAS rule.


Mathematical Modeling
  • Given: AB = PQ, BC = QR, ∠B = ∠Q
  • By SAS rule, ΔABC ≅ ΔPQR

Solution

Since ΔABC ≅ ΔPQR, PR = AC = 5 cm (as AB = PQ). Also, ∠R = ∠C = 180° - (50° + ∠B).

Question 6:
A ladder leans against a wall, forming ΔABC with the ground. If another ladder forms ΔPQR with the same wall and ground, and AB = PQ, BC = QR, prove their congruency. What can you say about the heights AC and PR?
Answer:
Problem Interpretation

Our textbook shows real-life examples of congruent triangles. Here, both ladders form right-angled triangles with the wall and ground.


Mathematical Modeling
  • Given: AB = PQ, BC = QR, ∠B = ∠Q = 90°
  • By RHS rule, ΔABC ≅ ΔPQR

Solution

Since ΔABC ≅ ΔPQR, their corresponding sides are equal. Thus, the heights AC and PR must be equal.

Question 7:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that ΔABC ≅ ΔPQR. Also, explain why this rule is useful in real-life constructions.
Answer:
Problem Interpretation

We studied that two triangles are congruent if their corresponding sides and angles are equal. Here, AB = PQ, BC = QR, and ∠B = ∠Q.


Mathematical Modeling

By the SAS (Side-Angle-Side) congruence rule, ΔABC ≅ ΔPQR.


Solution
  • Given: AB = PQ, BC = QR, ∠B = ∠Q.
  • By SAS rule, ΔABC ≅ ΔPQR.

This rule helps in constructing identical structures like bridges or roofs.

Question 8:
A ladder leans against a wall, forming ΔABC with the ground. Another ladder of the same length forms ΔPQR. If AB = PQ and ∠B = ∠Q, prove ΔABC ≅ ΔPQR using the RHS congruence rule.
Answer:
Problem Interpretation

We have two right-angled triangles ΔABC and ΔPQR with hypotenuses AB = PQ and one side BC = QR.


Mathematical Modeling

Since both are right-angled and satisfy RHS (Right-Hypotenuse-Side) conditions, they are congruent.


Solution
  • Given: AB = PQ (hypotenuse), BC = QR (side), ∠C = ∠R = 90°.
  • By RHS rule, ΔABC ≅ ΔPQR.

This ensures stability in ladder placements.

Question 9:
In ΔABC and ΔPQR, AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that the triangles are congruent. Also, explain why this rule applies.
Answer:
Problem Interpretation

We studied that two triangles are congruent if their corresponding sides and angles are equal.

Mathematical Modeling

Given: AB = PQ, BC = QR, and ∠B = ∠Q.

Solution

By the SAS (Side-Angle-Side) congruence rule, ΔABC ≅ ΔPQR. This rule applies because two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle.

Question 10:
A ladder leans against a wall, forming ΔABC with the ground. Another ladder of the same length leans the same way, forming ΔPQR. If AB = PQ and ∠A = ∠P, prove ΔABC ≅ ΔPQR using the congruence rule.
Answer:
Problem Interpretation

Our textbook shows that real-life objects like ladders can form congruent triangles.

Mathematical Modeling

Given: AB = PQ (ladder length), ∠A = ∠P (angle with ground).

Solution

By the ASA (Angle-Side-Angle) rule, ΔABC ≅ ΔPQR. Since two angles and the included side are equal, the triangles are congruent.

Question 11:
In a park, two triangular flower beds, ΔABC and ΔPQR, are designed such that AB = PQ, BC = QR, and ∠B = ∠Q. Using the congruence rule, prove that the two flower beds are congruent. Also, mention the real-life application of congruent triangles in design.
Answer:

Given: ΔABC and ΔPQR with AB = PQ, BC = QR, and ∠B = ∠Q.

To prove: ΔABC ≅ ΔPQR.


Proof:


1. By the given conditions, AB = PQ (Side).
2. ∠B = ∠Q (Angle).
3. BC = QR (Side).


Since two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle, the triangles are congruent by the SAS (Side-Angle-Side) congruence rule.


Real-life application: Congruent triangles are used in design to ensure symmetry and balance, such as in tiling patterns, architectural structures, and artwork, where identical shapes are needed for aesthetic or functional purposes.

Question 12:
Two students, Riya and Aman, drew triangles ΔXYZ and ΔLMN respectively. Riya measured XY = 5 cm, YZ = 7 cm, and ∠Y = 60°, while Aman measured LM = 5 cm, MN = 7 cm, and ∠M = 60°. Are the two triangles congruent? Justify your answer using the appropriate congruence rule and explain why this rule is important in geometry.
Answer:

Given: ΔXYZ with XY = 5 cm, YZ = 7 cm, and ∠Y = 60°.
ΔLMN with LM = 5 cm, MN = 7 cm, and ∠M = 60°.


Analysis:


1. XY = LM (Both are 5 cm).
2. YZ = MN (Both are 7 cm).
3. ∠Y = ∠M (Both are 60°).


Since two sides and the included angle of ΔXYZ are equal to the corresponding parts of ΔLMN, the triangles are congruent by the SAS (Side-Angle-Side) congruence rule.


Importance of SAS rule: This rule helps in proving congruence when specific measurements of sides and angles are known, ensuring accuracy in geometric constructions and real-world applications like engineering and design, where precise measurements are crucial.

Question 13:
In a park, two triangular flower beds, ΔABC and ΔPQR, are designed such that AB = PQ, BC = QR, and ∠B = ∠Q. A gardener claims they are congruent. Is he correct? Justify using the appropriate congruence rule.
Answer:

The gardener is correct because the two triangles ΔABC and ΔPQR satisfy the SAS (Side-Angle-Side) congruence rule.


Justification:


Given:

  • AB = PQ (One pair of equal sides)
  • BC = QR (Second pair of equal sides)
  • ∠B = ∠Q (Included angles are equal)

Since two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle, the triangles are congruent by SAS rule.

Question 14:
Two triangles, ΔXYZ and ΔLMN, have XY = LM, YZ = MN, and XZ = LN. A student says they are congruent by SSS rule. Is the student right? Explain with reasoning.
Answer:

Yes, the student is correct because the given triangles satisfy the SSS (Side-Side-Side) congruence rule.


Explanation:


Given:

  • XY = LM (One pair of equal sides)
  • YZ = MN (Second pair of equal sides)
  • XZ = LN (Third pair of equal sides)

Since all three sides of ΔXYZ are equal to the corresponding sides of ΔLMN, the triangles are congruent by SSS rule. This means all corresponding angles will also be equal.

Question 15:
In a park, two triangular flower beds, ΔABC and ΔPQR, are designed such that AB = PQ, BC = QR, and ∠B = ∠Q. A student claims that both triangles are congruent. Is the student correct? Justify your answer using the appropriate congruence rule.
Answer:

The student is correct. The given triangles ΔABC and ΔPQR satisfy the SAS (Side-Angle-Side) congruence rule.


Justification:


1. AB = PQ (Given, one pair of equal sides).
2. ∠B = ∠Q (Given, included angle between the sides is equal).
3. BC = QR (Given, second pair of equal sides).


Since two sides and the included angle of one triangle are equal to the corresponding parts of the other triangle, the triangles are congruent by the SAS rule.

Question 16:
Two triangles, ΔXYZ and ΔLMN, have XY = LM, YZ = MN, and XZ = LN. A gardener wants to confirm if both triangles are congruent to ensure symmetry in the garden layout. Which congruence rule applies here? Explain with reasoning.
Answer:

The gardener can confirm that ΔXYZ and ΔLMN are congruent using the SSS (Side-Side-Side) congruence rule.


Explanation:


1. XY = LM (Given, first pair of equal sides).
2. YZ = MN (Given, second pair of equal sides).
3. XZ = LN (Given, third pair of equal sides).


Since all three sides of one triangle are equal to the corresponding sides of the other triangle, the triangles are congruent by the SSS rule. This ensures perfect symmetry in the garden layout.

Question 17:
Riya and Priya are drawing two triangles, ΔABC and ΔPQR, respectively. Riya measures AB = 5 cm, BC = 7 cm, and ∠B = 60°. Priya measures PQ = 5 cm, QR = 7 cm, and ∠Q = 60°. They claim their triangles are congruent. Is their claim correct? Justify using the appropriate congruence rule.
Answer:

Yes, Riya and Priya's claim is correct. The triangles ΔABC and ΔPQR are congruent by the SAS (Side-Angle-Side) congruence rule.


Justification:


1. Given: AB = PQ = 5 cm (Side)
2. Given: BC = QR = 7 cm (Side)
3. Given: ∠B = ∠Q = 60° (Angle)

Since two sides and the included angle of ΔABC are equal to the corresponding sides and included angle of ΔPQR, the triangles are congruent by SAS rule.

Question 18:
In ΔXYZ and ΔLMN, XY = LM, YZ = MN, and ∠Y = ∠M. Are the triangles congruent? If yes, state the congruence rule applied. If not, explain why.
Answer:

Yes, ΔXYZ and ΔLMN are congruent by the SAS (Side-Angle-Side) congruence rule.


Explanation:


1. Given: XY = LM (Side)
2. Given: YZ = MN (Side)
3. Given: ∠Y = ∠M (Angle)

Since two sides and the included angle of ΔXYZ are equal to the corresponding sides and included angle of ΔLMN, the triangles satisfy the SAS rule. Hence, they are congruent.

Question 19:
Riya and Priya are drawing two triangles ABC and PQR respectively. Riya's triangle ABC has sides AB = 5 cm, BC = 7 cm, and AC = 6 cm. Priya's triangle PQR has sides PQ = 5 cm, QR = 7 cm, and PR = 6 cm. Without measuring angles, can they conclude that the triangles are congruent? Justify your answer using the appropriate congruence rule.
Answer:

Yes, Riya and Priya can conclude that triangles ABC and PQR are congruent without measuring the angles.


Here's why:


Both triangles have the same side lengths:

  • AB = PQ = 5 cm
  • BC = QR = 7 cm
  • AC = PR = 6 cm

This satisfies the SSS (Side-Side-Side) congruence rule, which states that if all three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent.


Since all corresponding sides are equal, ΔABC ≅ ΔPQR by SSS congruence.

Question 20:
In a park, two slides for children are built with triangular supports. The first slide's support triangle has angles of 60° and 70° with an included side of 4 meters. The second slide's support triangle has angles of 60° and 70° with the same included side length. Are these support triangles congruent? Explain using the appropriate congruence criterion.
Answer:

Yes, the two support triangles are congruent.


Here's the explanation:


Both triangles have:

  • Two angles equal (60° and 70° each)
  • The included side between these angles equal (4 meters)

This satisfies the ASA (Angle-Side-Angle) congruence criterion, which states that 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 triangles are congruent.


To verify further:


Third angle in both triangles = 180° - (60° + 70°) = 50°
All corresponding angles are equal and one corresponding side is equal, satisfying ASA rule.


Thus, the support triangles are congruent by ASA criterion.

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