Geometric Twins — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Geometric Twins" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Geometric Twins" — 8 important questions with detailed answers for CBSE board exam…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 7 Mathematics — Geometric Twins. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Geometric Twins — All 8 Questions Solved
Q1. What does SSS congruence criterion mean? Give an example.
SSS (Side-Side-Side) congruence criterion states that if all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.
Example:
Triangle ABC with sides AB = 5 cm, BC = 6 cm, CA = 7 cm
Triangle PQR with sides PQ = 5 cm, QR = 6 cm, RP = 7 cm
Since all three pairs of corresponding sides are equal, triangle ABC ≅ triangle PQR by SSS criterion.
Q2. Two triangles have two sides and the included angle equal. What is this congruence criterion called?
This is the SAS (Side-Angle-Side) congruence criterion.
Definition: 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:
In triangle ABC and triangle DEF:
AB = DE = 4 cm
Angle ABC = Angle DEF = 60°
BC = EF = 5 cm
Then triangle ABC ≅ triangle DEF by SAS criterion.
Note: The angle must be between the two equal sides (included angle).
Q3. Explain ASA congruence criterion and state the difference between ASA and AAS.
ASA (Angle-Side-Angle) Criterion:
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.
Example:
In triangle PQR and triangle XYZ:
Angle P = Angle X = 50°
PQ = XY = 6 cm (included side)
Angle Q = Angle Y = 70°
Then triangle PQR ≅ triangle XYZ by ASA criterion.
Difference between ASA and AAS:
ASA: Two angles and the included side (side between the angles) are equal.
AAS: Two angles and a non-included side (side not between the angles) are equal.
Both criteria make triangles congruent, but they use different sides.
Q4. What is RHS congruence criterion? Can it be applied to acute-angled triangles?
RHS (Right angle-Hypotenuse-Side) Congruence Criterion:
This criterion applies ONLY to right-angled triangles.
Definition: If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two right-angled triangles are congruent.
Example:
In right triangle ABC (right angle at B) and right triangle PQR (right angle at Q):
AC = PR = 10 cm (hypotenuses)
AB = PQ = 6 cm (one side)
Then triangle ABC ≅ triangle PQR by RHS criterion.
Can it be applied to acute-angled triangles?
No, RHS criterion CANNOT be applied to acute-angled triangles because they do not have a right angle. This criterion is specifically designed for right-angled triangles.
Q5. Triangle ABC has AB = 5 cm, BC = 7 cm, CA = 8 cm. Triangle DEF has DE = 5 cm, EF = 7 cm, FD = 8 cm. Prove the triangles are congruent and write the congruence statement.
Step 1: Compare the corresponding sides
AB = 5 cm and DE = 5 cm → AB = DE ✓
BC = 7 cm and EF = 7 cm → BC = EF ✓
CA = 8 cm and FD = 8 cm → CA = FD ✓
Step 2: Check which criterion applies
All three pairs of corresponding sides are equal.
This satisfies the SSS (Side-Side-Side) congruence criterion.
Step 3: Write the congruence statement
Triangle ABC ≅ Triangle DEF (by SSS criterion)
Note: The order of vertices in the congruence statement matters. It shows the correspondence:
A ↔ D, B ↔ E, C ↔ F
Q6. In triangles MNO and PQR, MN = PQ, NO = QR, and angle N = angle Q. Are these triangles congruent? If yes, by which criterion?
Step 1: Identify the given information
MN = PQ (one pair of sides equal)
NO = QR (another pair of sides equal)
Angle N = Angle Q (angles equal)
Step 2: Check if angle is included
Angle N is between sides MN and NO.
Angle Q is between sides PQ and QR.
Yes, the angle is between the two given sides (included angle).
Step 3: Apply congruence criterion
Two sides (MN and NO) and the included angle (angle N) are equal to the corresponding two sides (PQ and QR) and included angle (angle Q).
Conclusion:
Yes, triangle MNO ≅ triangle PQR by SAS (Side-Angle-Side) criterion.
Correspondence: M ↔ P, N ↔ Q, O ↔ R
Q7. Can two triangles be congruent if only two sides of one triangle are equal to two sides of another triangle? Explain with a sketch description.
No, two triangles cannot always be congruent if only two sides are equal.
Reason:
Congruence requires either:
(1) All three sides equal (SSS), OR
(2) Two sides and the included angle equal (SAS), OR
(3) Two angles and included side equal (ASA), OR
(4) Two angles and non-included side equal (AAS), OR
(5) For right triangles: hypotenuse and one side equal (RHS)
Example of Non-Congruence:
Consider triangle ABC with AB = 3 cm, BC = 4 cm (angle B unknown)
Consider triangle PQR with PQ = 3 cm, QR = 4 cm (angle Q unknown)
If the angles between these sides are different:
Triangle ABC with angle B = 60°
Triangle PQR with angle Q = 90°
Even though AB = PQ and BC = QR, the triangles are NOT congruent because the included angles are different.
Conclusion:
Two equal sides alone are NOT sufficient for congruence. The included angle (or the third side, or other angle information) must also be specified.
Q8. In right-angled triangle ABC with right angle at B, AB = 6 cm and AC = 10 cm. In right-angled triangle XYZ with right angle at Y, XZ = 10 cm and XY = 6 cm. Are these triangles congruent? Justify.
Step 1: Identify the sides
Triangle ABC (right angle at B):
AB = 6 cm (one leg)
AC = 10 cm (hypotenuse, as AC is opposite to the right angle)
Triangle XYZ (right angle at Y):
XY = 6 cm (one leg)
XZ = 10 cm (hypotenuse, as XZ is opposite to the right angle)
Step 2: Apply RHS criterion
For RHS congruence:
Hypotenuse of ABC = AC = 10 cm
Hypotenuse of XYZ = XZ = 10 cm
One leg of ABC = AB = 6 cm
One leg of XYZ = XY = 6 cm
Step 3: Check all conditions
Right angles: B = 90° and Y = 90° ✓
Hypotenuses equal: AC = XZ = 10 cm ✓
Corresponding legs equal: AB = XY = 6 cm ✓
Conclusion:
Yes, triangle ABC ≅ triangle XYZ by RHS (Right angle-Hypotenuse-Side) congruence criterion.
Correspondence: A ↔ X, B ↔ Y, C ↔ Z
Geometric Twins — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which of the following describes two geometric figures that are 'geometric twins'?
- A. They have the same shape but different sizes.
- B. They have the same size but different shapes.
- C. They have exactly the same shape and exactly the same size.
- D. They are mirror images of each other.
Answer: C. They have exactly the same shape and exactly the same size. — Geometric twins, in mathematics, refer to congruent figures. Congruent figures are identical in both shape and size, meaning one can be perfectly superimposed on the other.
MCQ 2. What is the symbol used to denote that two figures are congruent?
- A. ~
- B. =
- C. ≈
- D. ≅
Answer: D. ≅ — The symbol '≅' represents congruence, indicating that two figures are identical in shape and size. '~' is for similarity, '=' for equality in value, and '≈' for approximation.
MCQ 3. Two line segments are congruent if they have:
- A. The same endpoint.
- B. The same direction.
- C. The same length.
- D. Are parallel to each other.
Answer: C. The same length. — Two line segments are congruent if and only if they have the exact same length. Their position or orientation does not affect their congruence.
MCQ 4. If ∠ABC ≅ ∠PQR, which of the following statements must be true?
- A. AB = PQ
- B. BC = QR
- C. m∠ABC = m∠PQR
- D. AC = PR
Answer: C. m∠ABC = m∠PQR — When two angles are congruent, their measures are equal. The lengths of the sides forming the angles do not necessarily have to be equal for the angles themselves to be congruent.
MCQ 5. Which of the following is NOT a criterion for the congruence of triangles?
- A. SSS (Side-Side-Side)
- B. ASA (Angle-Side-Angle)
- C. AAA (Angle-Angle-Angle)
- D. RHS (Right-angle-Hypotenuse-Side)
Answer: C. AAA (Angle-Angle-Angle) — AAA (Angle-Angle-Angle) is a criterion for similarity, not congruence. Two triangles with equal angles can have different side lengths, hence different sizes.
MCQ 6. Given ΔABC ≅ ΔXYZ. Which side of ΔXYZ corresponds to side BC of ΔABC?
- A. XY
- B. YZ
- C. XZ
- D. Cannot be determined
Answer: B. YZ — When stating congruence, the order of vertices indicates corresponding parts. So, B corresponds to Y, and C corresponds to Z, meaning BC corresponds to YZ.
MCQ 7. If in ΔPQR and ΔSTU, PQ = ST, QR = TU, and RP = US, then by which congruence criterion are the triangles congruent?
- A. SAS
- B. ASA
- C. SSS
- D. RHS
Answer: C. SSS — Since all three corresponding sides of the two triangles are equal, the triangles are congruent by the SSS (Side-Side-Side) congruence criterion.
MCQ 8. In ΔABC and ΔDEF, AB = DE, ∠B = ∠E, and BC = EF. Which congruence criterion applies?
- A. ASA
- B. SSS
- C. SAS
- D. RHS
Answer: C. SAS — Here, two sides (AB, BC) and the included angle (∠B) of ΔABC are equal to the corresponding two sides (DE, EF) and the included angle (∠E) of ΔDEF. This fits the SAS (Side-Angle-Side) criterion.
MCQ 9. Two right-angled triangles are congruent if their hypotenuses and one pair of corresponding sides are equal. This is known as the:
- A. SSS criterion
- B. ASA criterion
- C. RHS criterion
- D. SAS criterion
Answer: C. RHS criterion — This specific criterion for right-angled triangles, where the Right angle, Hypotenuse, and one Side are congruent, is called the RHS (Right-angle-Hypotenuse-Side) congruence criterion.
MCQ 10. If ΔMNO ≅ ΔPQR, and M = 50°, N = 60°, then what is the measure of ∠R?
- A. 50°
- B. 60°
- C. 70°
- D. 80°
Answer: C. 70° — Since ΔMNO ≅ ΔPQR, their corresponding angles are equal. M corresponds to P, N to Q, and O to R. If ∠M = 50° and ∠N = 60°, then ∠O = 180° - (50° + 60°) = 70°. Therefore, ∠R = ∠O = 70°.
MCQ 11. A square of side 5 cm is congruent to another square of side 'x' cm. What is the value of x?
- A. 2.5 cm
- B. 5 cm
- C. 10 cm
- D. 25 cm
Answer: B. 5 cm — For two squares to be congruent, all their corresponding parts must be equal. This means their side lengths must be equal. So, x must be 5 cm.
MCQ 12. In triangles ABC and PQR, ∠B = ∠Q = 90°, AC = PR, and AB = PQ. By which congruence rule are the triangles congruent?
- A. ASA
- B. SAS
- C. SSS
- D. RHS
Answer: D. RHS — Given that both are right-angled triangles (∠B = ∠Q = 90°), the hypotenuses are equal (AC = PR), and one pair of corresponding sides are equal (AB = PQ). This satisfies the RHS (Right-angle-Hypotenuse-Side) congruence criterion.
MCQ 13. If ΔXYZ and ΔLMN are congruent such that ∠X = 40°, ∠Y = 70°, and XY = 6 cm. Which of the following information is true for ΔLMN?
- A. ∠L = 70°, ∠M = 40°, LM = 6 cm
- B. ∠L = 40°, ∠M = 70°, LM = 6 cm
- C. ∠L = 40°, ∠N = 70°, LN = 6 cm
- D. ∠M = 40°, ∠N = 70°, MN = 6 cm
Answer: B. ∠L = 40°, ∠M = 70°, LM = 6 cm — For congruent triangles, corresponding angles and sides are equal. If ΔXYZ ≅ ΔLMN, then ∠L=∠X=40°, ∠M=∠Y=70°, and LM=XY=6 cm. The order of vertices is crucial.
MCQ 14. Two circles are congruent if they have:
- A. The same center.
- B. The same radius.
- C. The same circumference.
- D. The same area.
Answer: B. The same radius. — For two circles to be congruent, they must be identical in size, which means they must have the same radius. If their radii are the same, their circumference and area will also be the same.
MCQ 15. Which congruence criterion requires two angles and the included side to be equal?
- A. ASA
- B. AAS
- C. SAS
- D. SSS
Answer: A. ASA — The ASA (Angle-Side-Angle) congruence criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
MCQ 16. Consider two triangles, ΔABC and ΔDEF. If AB = DE, BC = EF, and ∠C = ∠F, are the triangles necessarily congruent?
- A. Yes, by SAS criterion.
- B. Yes, by SSS criterion.
- C. Yes, by ASA criterion.
- D. No, this information does not guarantee congruence.
Answer: D. No, this information does not guarantee congruence. — The given information (two sides and a non-included angle) does not directly match any standard congruence criterion like SAS, ASA, SSS, or RHS. This condition is sometimes called SSA and does not always guarantee congruence.
MCQ 17. In ΔPQR, PQ = PR. If PS is the median to QR, which congruence criterion can be used to prove ΔPQS ≅ ΔPRS?
- A. ASA
- B. SAS
- C. SSS
- D. RHS
Answer: C. SSS — Since PS is a median, QS = RS. We are given PQ = PR. PS is common to both triangles. So, by SSS (PQ=PR, QS=RS, PS=PS), ΔPQS ≅ ΔPRS.
MCQ 18. If ΔLMN ≅ ΔUVW, and LM = 8 cm, ∠M = 70°, MN = 10 cm, what are the corresponding parts of ΔUVW?
- A. UV = 8 cm, ∠V = 70°, VW = 10 cm
- B. UV = 10 cm, ∠V = 70°, VW = 8 cm
- C. UW = 8 cm, ∠W = 70°, UV = 10 cm
- D. VW = 8 cm, ∠U = 70°, UW = 10 cm
Answer: A. UV = 8 cm, ∠V = 70°, VW = 10 cm — Due to congruence (ΔLMN ≅ ΔUVW), corresponding sides and angles are equal. So, LM = UV, MN = VW, LN = UW, ∠L = ∠U, ∠M = ∠V, ∠N = ∠W. Thus, UV = 8 cm, ∠V = 70°, VW = 10 cm.
MCQ 19. Can two triangles be congruent if they have the same perimeter but different side lengths?
- A. Yes, always.
- B. No, never.
- C. Sometimes, if they are isosceles.
- D. Sometimes, if they are right-angled.
Answer: B. No, never. — If triangles have different side lengths, they cannot be perfectly superimposed, thus they cannot be congruent, even if their perimeters (sum of side lengths) are the same.
MCQ 20. What is the concept of 'superposition' related to congruence?
- A. Placing one figure on top of another to compare their areas.
- B. Placing one figure on top of another to see if they exactly cover each other.
- C. Rotating a figure to match another.
- D. Enlarging a figure to match another.
Answer: B. Placing one figure on top of another to see if they exactly cover each other. — Superposition is the method of placing one geometric figure exactly over another. If they perfectly coincide, they are congruent, meaning they are 'geometric twins'.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.