NCERT Solutions for Class 7 Mathematics Ganita Prakash II Chapter 1
NCERT solutions for Class 7 Mathematics Chapter Geometric Twins – complete answers & explanations
In Class 7 Mathematics, Chapter Geometric Twins introduces students to the concept of congruence and how shapes can be exactly the same in size and shape. This chapter helps students understand how measurements like sides and angles are used to recreate figures and verify whether two figures are identical. It also explains important ideas like congruent triangles and the conditions required to prove congruence. Learning this chapter builds a strong foundation in geometry and improves logical thinking skills, which are essential for solving mathematical problems. This blog provides clear and reliable NCERT solutions that help students understand each concept step by step. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance.
What this NCERT chapter covers?
1. Understanding how to recreate figures
using measurements like lengths and angles.
2. Learning the concept of congruence and how figures can be identical in shape and size.
3. Identifying congruent figures by comparing sides and angles.
4. Understanding why arm lengths alone are not sufficient without the included angle.
5. Exploring congruence of triangles using different conditions like SSS, SAS, ASA, AAS, and RHS.
6. Learning how to construct triangles using given measurements.
7. Understanding when triangles are not congruent, especially in SSA cases.
8. Studying properties of isosceles triangles and equal angles.
9. Learning that all angles in an equilateral triangle are equal and measure 60°.
10. Applying congruence concepts to solve real-life and diagram-based problems.
11. Identifying corresponding vertices, sides, and angles in congruent triangles.
12. Using geometric reasoning to justify congruence and equality of parts.
How to use these NCERT solutions?
1. Start by carefully reading each
question in the worksheet and try solving it on your own.
2. Use these NCERT solutions to check your answers after attempting the questions.
3. Compare your steps with the given answers to understand the correct approach.
4. Focus on understanding why a particular rule like SSS or SAS is applied.
5. Practice drawing figures neatly to match the given conditions.
6. Parents and teachers can guide students by explaining each step in simple terms.
7. Use these answers to revise concepts before tests and exams.
8. Follow the same order of questions as in the worksheet to avoid confusion.
9. Re-attempt questions where mistakes are made to improve accuracy.
Important tips & tricks for students
1. Always check whether both sides and angles are given before concluding congruence.
2. Do not assume figures are congruent just by appearance; verify using measurements.
3. Remember all congruence rules clearly: SSS, SAS, ASA, AAS, and RHS.
4. Be careful while matching corresponding vertices in triangles.
5. Draw diagrams neatly and label all points correctly.
6. Avoid skipping steps while explaining answers in geometry.
7. Practice identifying congruent figures in different orientations.
8. Revise angle sum property of triangles regularly.
9. Understand the difference between included and non-included angles.
10. Pay attention to construction steps when drawing triangles.
NCERT solutions – complete answer key
1.1 Geometric Twins
To recreate the symbol, we can measure the arm lengths AB and BC, and the angle ∠ABC. These measurements are enough to construct an exact replica of the symbol.
AB and BC (arm lengths)
∠ABC (angle ABC∠ABC (the angle between the arms)
These measurements are sufficient to recreate the exact figure.
No, the arm lengths AB and BC alone are not sufficient to exactly recreate the figure.
Reason : If only arm lengths are known, different shapes can still be formed because the angle between them can vary.
Yes, the measure of the angle ∠ABC, along with the arm lengths AB and BC, will fix the shape and size of the figure.
Yes, with these three measurements, we can create an exact replica of the symbol. The three measurements define the shape and size uniquely, ensuring the figure is congruent.
No, only same arm lengths are not enough; the symbols are congruent only when both the arm lengths and the included angle are the same.
Figure It Out
1. The figures are congruent if their corresponding sides and angles are equal. You can compare their sides and angles to check for congruence.
2. The congruent pairs are:
The two cloud shapes (top right)
The two leaf shapes (bottom right)
(The other shapes are not congruent because their sizes or shapes differ.)
3. (a) Circle: Measure the radius (or diameter).
Congruent circles: Two circles are congruent if their radii (or diameters) are equal.
(b) Rectangle: Measure the length and breadth.
Congruent rectangles: Two rectangles are congruent if their length and breadth are equal.
4 Check: Compare the lengths of the arms and the angle between them.
Answer: The pair with same arm lengths and same angle are congruent; others are not.
1.2 Congruence of Triangles
They can measure the three sides and angles of the triangular frame and then draw the same measurements on cardboard to make an identical (congruent) cutout.
Yes, I agree with Meera.
A triangle is uniquely determined by its three side lengths (SSS), so the angles are not required to construct a congruent triangle.
Even with side lengths 4 cm, 6 cm, and 8 cm, a unique triangle can be constructed, so it will still be congruent by SSS rule.
Yes, the information is sufficient.
Using the three sides (4 cm, 6 cm, 8 cm), we can construct a triangle (SSS rule).
However, the two triangles ΔABE and ΔABF are mirror images of each other, so they are still congruent.
Yes, ΔABE and ΔABF are congruent.
They have the same three side lengths, so by the SSS (Side-Side-Side) rule, they are congruent.
Conventions to Express Congruence
Overlap the vertices as follows:
A → X, B → Y, C → Z.
No, only the correct matching of corresponding vertices will make them overlap exactly.
(a) A ↔ X, B ↔ Y, C ↔ Z
(b) AB ↔ XY, BC ↔ YZ, AC ↔ XZ
(c) ∠A ↔ ∠X, ∠B ↔ ∠Y, ∠C ↔ ∠Z
Congruent triangles: ΔABD ≅ ΔCDB
Reason: AB = CD, AD = CB (rectangle property) and BD is common, so by SSS rule, the triangles are congruent.
This superimposition is incorrect because the corresponding sides are not matched properly, so it does not prove congruence.
Correct correspondence of vertices: A ↔ C, B ↔ D, D ↔ B
Congruence statement: ΔABD ≅ ΔCDB
Figure it Out
1. Since corresponding vertices are H ↔ B, E ↔ I, N ↔ G, the other correct ways are:
ΔENH ≅ ΔIGB
ΔNHE ≅ ΔGBI
ΔHNE ≅ ΔBGI
ΔEHN ≅ ΔIBG
ΔNEH ≅ ΔGIB
2 Both triangles have sides 3.5 cm, 5 cm, and 6 cm, so they are congruent by SSS rule.
Congruence: ΔRED ≅ ΔJAM
3 Congruent triangles: ΔABC ≅ ΔADC
Reason: AB = AD, CB = CD, and AC is common → SSS rule
Angle bisection: Yes, AC divides ∠BAD and ∠BCD into two equal parts because corresponding angles of congruent triangles are equal.
4 Yes, ΔDFE ≅ ΔGED.
Reason: DF = DG, FE = GE, and DE is common → SSS rule, so the triangles are congruent.
Measuring the Angles
No, we cannot create an exact copy using only angles (30°, 70°, 80°).
Triangles with the same angles can have different sizes, so they are not necessarily congruent.
Measuring Two Sides and the Included Angle
Yes, ΔABC ≅ ΔXYZ.
Since two sides and the included angle are equal, by SAS (Side-Angle-Side) rule, the triangles are congruent.
Construction: Draw AB = 6 cm, construct ∠A = 30°, then mark AC = 5 cm on that ray and join BC.
Answer: Yes, all such triangles are congruent because they have the same two sides and included angle → SAS rule.
Measuring Two Sides and a Non-included Angle
No, they are not necessarily congruent.
This is the SSA case (two sides and a non-included angle), which does not guarantee congruence.
Yes, non-congruent triangles can exist.
In the SSA case, two different triangles can be constructed with the same given measurements, so they are not necessarily congruent.
Two different (non-congruent) triangles can be formed, so SSA does not guarantee congruence.
Two Angles and the Included Side
Yes, ΔABC ≅ ΔXYZ.
Since two angles and the included side are equal, by ASA rule, the triangles are congruent.
No, non-congruent triangles cannot exist with these measurements.
All such triangles are congruent because two angles and the included side are equal → ASA rule.
Since AO = OD and BO = OC, and ∠AOB = ∠DOC (vertically opposite angles),
ΔAOB ≅ ΔDOC (SAS rule) ⇒ AB = CD.
Yes, there are other equal parts:
Since ΔAOB ≅ ΔDOC (by SAS), we get AB = CD, ∠BAO = ∠CDO, and ∠ABO = ∠DCO.
Figure it Out
1 Both triangles have two sides (7 cm, 5 cm) and the included angle (47°) equal.
So, they are congruent by SAS rule.
Congruence: ΔABC ≅ ΔXZY
2 Since AB ∥ CD, the alternate angles are equal:
∠ABO = ∠DCO and ∠BAO = ∠CDO
Also, AB = CD and BO = AO (common intersection gives vertical angles equality context)
So, triangles are congruent
3 Given : ∠ABC = ∠DBC and ∠ACB = ∠DCB, and BC is common,
⇒ ΔABC ≅ ΔDBC by ASA rule
Hence, ∠BAC = ∠BDC.
4 Given: ∠ABD = ∠DCA and ∠ACB = ∠DBC
Equal parts:
∠ABD = ∠DCA
∠ACB = ∠DBC
BC = CB (common side)
Thus: ΔABD ≅ ΔDCA by ASA rule.
Measuring Two Angles and a Non-Included Side
∠B = 70° and ∠Y = 70°.
Yes, this helps.
Now ∠B = ∠Y, ∠C = ∠Z, and BC = YZ, so triangles are congruent by ASA rule ⇒ ΔABC ≅ ΔXYZ.
Measuring Two Sides in a Right Triangle
Yes, ΔABC ≅ ΔXYZ.
They are right triangles with equal hypotenuse (AC = XZ) and one corresponding side (BC = YZ), so by RHS (Right angle–Hypotenuse–Side) rule, they are congruent.
No, non-congruent triangles cannot exist.
With a right angle, equal hypotenuse, and one side, the triangle is uniquely determined → RHS rule, so all such triangles are congruent.
No, non-congruent triangles cannot exist; all such triangles are congruent by RHS (Right angle–Hypotenuse–Side) rule.
1.3 Angles of Isosceles and Equilateral Triangles
Since AB = AC, ΔABC is isosceles, so base angles are equal.
∠B = ∠C = 50°.
Yes. Since ∠B = ∠C and total angles = 180°,
∠B = ∠C = (180° − 80°) ÷ 2 = 50°.
Angles opposite equal sides are equal.
So, if AB = AC ⇒ ∠B = ∠C
and if AB = BC ⇒ ∠A = ∠C.
Each angle = 60°.
Pyramid & Louvre: triangular faces are congruent
Dome: repeating congruent triangular frames
Rangoli: identical congruent triangle patterns
Bridge: congruent triangles in the truss structure
Figure it Out
1 Corresponding vertices : A ↔ F, I ↔ L, R ↔ Y
Corresponding sides : AI ↔ FL, IR ↔ LY, AR ↔ FY
Corresponding angles : ∠A ↔ ∠F, ∠I ↔ ∠L, ∠R ↔ ∠Y
2 (a) AB = DE, BC = EF, CA = DF → SSS ⇒ ΔABC ≅ ΔDEF
(b) AB = EF, ∠A = ∠E, AC = ED → SAS ⇒ ΔABC ≅ ΔFED
(c) AB = DF, ∠B = ∠D = 90°, AC = FE → RHS ⇒ ΔABC ≅ ΔDFE
(d) ∠A = ∠D, ∠B = ∠E, AC = DF → AAS/ASA ⇒ ΔABC ≅ ΔDEF
(e) AB = DF, ∠B = ∠F, AC = DE → SSA (not sufficient) ⇒ Not congruent
3 Given OB = OC and OA = OD, and ∠AOB = ∠DOC (vertically opposite),
⇒ ΔAOB ≅ ΔDOC (SAS)
So, ∠BAO = ∠CDO (alternate angles) ⇒ AB ∥ CD.
4 Since ABCD is a square,
AB = BC = CD = DA and AC is common
⇒ ΔABC ≅ ΔADC (by SSS)
Yes, ΔABC is also congruent to ΔCDA (same triangle with different order of vertices).
Example: An equilateral triangle — its triangles can be written congruent in 6 different ways due to equal sides and angles.
5 Since A is the centre, AB = AC (radii), so ΔABC is isosceles.
Given : ∠A = 120°,
∠B = ∠C = (180° − 120°) ÷ 2 = 30°.
Answer: ∠B = 30°, ∠C = 30°.
6 Missing angles are: 34°, 44°, 46°, 56°, 30°, 68°, 98°, 90°.
Explanation: The missing angles are found using triangle angle sum (180°), linear pair (180°), vertically opposite angles (equal), and equal sides ⇒ equal angles (isosceles triangles).
Why NCERT solutions help students?
NCERT solutions help students build strong understanding of concepts and improve their problem-solving skills. They make it easier to prepare for exams by providing clear and correct approaches to answering questions. With proper practice, students gain confidence and develop accuracy in solving geometry problems.
Help your child master geometry concepts with expert-guided Mathematics learning support.