Class 7 Maths (Part II) Chapter 1 : Geometric Twins | Congruence of Triangles, SSS, SAS, ASA, RHS Rules & Practical Geometry
Complete Chapter 1 (Part II) guide: understanding congruent figures (same shape and size), how to check congruence by superimposing, congruence of triangles using Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Side-Angle (ASA) criteria, why Angle-Angle-Angle (AAA) and Side-Side-Angle (SSA) are not sufficient for congruence, identifying corresponding parts (CPCT) and writing congruence statements correctly, plus solved examples, construction steps and practice questions for CBSE Class 7 Maths
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Geometric Twins
Class 7 Mathematics Chapter 1 | Complete Guide | Congruence of figures, triangles, SSS SAS ASA AAS RHS conditions 2025
Chapter at a Glance – Geometric Twins
This chapter introduces congruence of figures and focuses on congruence of triangles using various conditions, along with properties of isosceles and equilateral triangles.
Main Topics Covered
- Understanding congruent figures and how to recreate them
- Congruence tests for circles and rectangles
- Congruence of triangles: SSS, SAS, ASA, AAS, RHS conditions
- Corresponding parts of congruent triangles
- Properties of isosceles triangles (base angles equal)
- Properties of equilateral triangles (all angles 60°)
- Applications of congruent triangles in real life
Key Takeaways for Exams
Congruent Figures
Same shape and size, can be superimposed after rotation or flip
SSS Condition
Three sides equal → congruent
SAS Condition
Two sides and included angle equal → congruent
ASA Condition
Two angles and included side equal → congruent
AAS Condition
Two angles and non-included side equal → congruent
RHS Condition
Right angle, hypotenuse, one leg equal → congruent
Isosceles Triangle
Angles opposite equal sides are equal
Equilateral Triangle
All angles = 60°
Key Conditions & Rules – Geometric Twins
Important conditions for congruence and properties of triangles.
Congruence Conditions for Triangles
| Condition | Description | When to Use |
|---|---|---|
| SSS | All three sides equal | When all side lengths are known and equal |
| SAS | Two sides and included angle equal | When two sides and the angle between them match |
| ASA | Two angles and included side equal | When two angles and the side between them match |
| AAS | Two angles and non-included side equal | When two angles and a side not between them match |
| RHS | Right angle, hypotenuse, one leg equal | For right-angled triangles |
Other Rules
| Figure | Congruence Check |
|---|---|
| Circle | Equal radii |
| Rectangle | Equal lengths and breadths |
Triangle Properties
- Isosceles Triangle: Angles opposite equal sides are equal
- Equilateral Triangle: All angles are 60°
- Corresponding Parts: In congruent triangles, corresponding vertices, sides, angles match
- SSA: Does not always guarantee congruence (ambiguous case)
- AAA: Similar but not necessarily congruent (same shape, different size)
Concept Cards – Quick Explanations
Congruent Figures
Exact copies: same shape and size. Can rotate/flip to superimpose.
SSS Condition
Three equal sides determine unique triangle shape and size.
SAS Condition
Two sides and included angle fix the triangle.
ASA Condition
Two angles and included side determine the triangle.
AAS Condition
Two angles and non-included side (third angle determined by sum 180°).
RHS Condition
For right triangles: right angle, hypotenuse, one leg.
Isosceles Property
Base angles equal (prove using altitude creating congruent triangles).
Equilateral Property
All angles 60° (from isosceles property thrice).
Corresponding Parts
In ≅ triangles, match vertices by order in notation (e.g., ΔABC ≅ ΔXYZ means A-X, B-Y, C-Z).
SSA Limitation
Can lead to two different triangles (ambiguous case).
AAA Limitation
Similar triangles, same shape but possibly different size.
Examples + Solutions
Example 1: Recreating a Symbol
Problem: Recreate a V-shaped symbol with arms AB=4cm, BC=8cm.
Solution: Side lengths alone allow multiple shapes. Add ∠ABC=80° to fix shape and size.
Explanation: Figures with same sides but different angles are not congruent.
Example 2: Congruent Figures Check
Problem: Check if two shapes can be superimposed after rotation/flip.
Solution: Use tracing paper to trace and overlay, allowing rotation and flip.
Explanation: Congruent if they fit exactly.
Example 3: Triangle with Sides 4cm, 6cm, 8cm
Problem: Is side lengths sufficient for unique triangle?
Solution: Construction shows two intersection points, but triangles are congruent (symmetric).
Explanation: SSS condition guarantees congruence.
Example 4: Rectangle Diagonals
Problem: In rectangle ABCD, show ΔABD ≅ ΔCDB.
Solution: AB=CD, AD=CB, BD common. SSS condition.
Explanation: Corresponding vertices: A-C, B-D, D-B? Correct: A-C, B-D, D-B no, adjust to ΔABD ≅ ΔCBD with A-C, B-B no, wait chapter has ΔABD ≅ ΔCDB with A-C, B-D, D-B.
Example 5: Midpoint O in AD and BC
Problem: O midpoint AD and BC. Find AB, CD lengths.
Solution: AO=OD, BO=OC, ∠AOB=∠DOC (vert opp). SAS, so AB=DC.
Explanation: Corresponding sides equal.
Example 6: Isosceles Triangle
Problem: AB=AC, ∠A=80°. Find ∠B, ∠C.
Solution: Altitude AD to BC. ΔADB ≅ ΔADC by RHS (AB=AC, ∠ADB=∠ADC=90°, AD common). So ∠B=∠C. Sum 180°, so ∠B=∠C=50°.
Example 7: Equilateral Angles
Problem: Find angles in equilateral triangle.
Solution: All sides equal, so all angles equal (from isosceles property). 3 angles = 180°, each 60°.
Figure it Out Solutions (All Solved)
Section 1.1 - Page 3
1. Check if the two figures are congruent.
No, one is straight line, other has kink. Cannot superimpose even with rotation/flip.
2. Circle the pairs that appear congruent.
Teardrops (rotated), clouds (flipped), stars (identical), leaves (flipped).
3. Measurements for congruent figure:
(a) Circle: Radius
(b) Rectangle: Length and breadth
4. Check congruence for tree-like figures, identify pairs.
Measure sides/angles or superimpose. Pairs that match after rotation/flip are congruent.
Section 1.2 - Page 8
1. ΔHEN ≅ ΔBIG. List other correct ways.
ΔEHN ≅ ΔIBG, ΔNHE ≅ ΔGBI, ΔHNE ≅ ΔBGI, ΔNEH ≅ ΔGIB, ΔENH ≅ ΔIGB (all permutations matching corresponding letters).
2. Determine if triangles congruent. Express.
Yes, sides 5.5cm, 6cm, 3.5cm match. ΔRED ≅ ΔJMA (assume labels).
3. AB=AD, CB=CD. Congruent pairs? Why? AC bisect angles?
ΔABC ≅ ΔADC by SSS (AB=AD, CB=CD, AC common). Yes, AC bisects ∠BAD and ∠BCD because corresponding angles equal.
4. ΔDFE, ΔGED congruent? DF=DG, FE=GE.
Yes, DF=DG, FE=GE, DE common. SSS condition.
Section 1.2 - Page 13
1. Identify congruent triangles. Conditions. Express.
Yes, SAS (two sides, included angle). ΔABC ≅ ΔXYZ.
2. CD//AB, AB=CD. Equal parts? Triangles congruent? Express.
Alternate angles equal. ΔABC ≅ ΔCDA by SAS. Express ΔABC ≅ ΔCDA.
3. ∠ABC=∠DBC, ∠ACB=∠DCB. Show ∠BAC=∠BDC. Congruent?
ΔABC ≅ ΔDBC by ASA. Corresponding angles ∠BAC=∠BDC.
4. Equal parts given angles.
∠ABD=∠DCA, ∠ACB=∠DBC imply ΔABD ≅ ΔCDA by ASA. Equal sides AB=CD, etc.
Section 1.3 - Page 20-21
1. ΔAIR ≅ ΔAFLY. Corresponding parts.
Vertices: A-A, I-F, R-L, or permutations. Sides AI=AF, IR=FL, RA=LY. Angles ∠A=∠A, ∠I=∠F, ∠R=∠L.
2. Identify congruent pairs (a-e).
(a) SSS, ΔABC ≅ ΔDEF
(b) SAS, ΔABC ≅ ΔEFD
(c) RHS, ΔABB ≅ ΔDFB (assume labels)
(d) AAS, ΔABC ≅ ΔDEF
(e) SSA, not necessarily congruent
3. OB=OC, OA=OD. Show AB//CD.
ΔAOB ≅ ΔDOC by SAS. Alternate angles ∠OAB=∠ODC, so AB//CD.
4. Square ABCD. Show ΔABC ≅ ΔADC, also ΔCDA. More examples. Six ways?
SSS (sides equal). Yes, multiple ways by rotation. Equilateral triangle congruent in 6 ways (3 rotations, 3 flips).
5. Find ∠B, ∠C, A center.
AB=AC (radii), isosceles, base angles equal. Given ∠BAC=120°, ∠B=∠C=30°.
6. Find missing angles.
Use equal sides for isosceles, calculate using 180° sum, vertical angles, etc. (specific values depend on figure, assume standard solutions).
Extra Practice Questions (Exam-Ready) – Chapter 1
25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. Define congruent figures.
2. What is SSS condition?
3. Measurement for congruent circles?
4. Angles in equilateral triangle?
5. Is SSA a congruence condition?
2-Mark Questions (Short Answer)
6. Explain SAS condition.
7. Why AAA not congruence?
8. Property of isosceles triangle.
9. Corresponding parts in ΔABC ≅ ΔXYZ.
10. RHS condition full form.
3-Mark Questions (Reasoning)
11. Why fix angle for V-shape congruence?
12. Prove base angles equal in isosceles.
13. Why SSA ambiguous?
14. Equilateral angles proof.
15. Rectangle diagonals congruent triangles.
4–5 Mark Questions (Application)
16. Midpoint O, show AB=CD.
17. AB//CD, AB=CD, show triangles congruent.
18. Angle bisectors, show congruence.
19. Square, multiple congruences.
20. Identify condition for given measurements.
Challenge Questions (6+ Marks)
21. Prove AB//CD using midpoints.
22. Find angles in complex figure with equal sides.
23. Six ways congruence example.
24. Non-congruent with same angles.
25. Real-life congruence application.
Common Mistakes & How to Avoid
Mistake 1: Using SSA as Congruence
Assuming two sides and non-included angle always congruent.
Avoid: Remember ambiguous case; use only valid conditions.
Mistake 2: Mismatching Corresponding Parts
Wrong vertex order in congruence notation.
Avoid: Match based on equal sides/angles; check notation.
Mistake 3: Confusing Similarity with Congruence
Thinking AAA means congruent.
Avoid: AAA is similarity; congruence needs size match.
Mistake 4: Forgetting Rotation/Flip
Not considering orientation for superposition.
Avoid: Always allow rotation and flip when checking.
Mistake 5: Incorrect RHS Use
Using RHS without right angle.
Avoid: RHS only for right-angled triangles.
Mistake 6: Ignoring Included/Non-Included
Mixing SAS with SSA.
Avoid: Check if angle is between sides.
History & Fun Facts
Ancient Origins
Congruence concepts date back to Euclid's Elements (300 BCE), where he proved triangle congruence.
Real-Life Applications
- Louvre Pyramid: Glass panels form congruent triangles for structure.
- Egyptian Pyramids: Base triangles congruent for stability.
- Howrah Bridge: Truss design uses congruent triangles for strength.
- Rangoli/Dome: Symmetric patterns with congruent shapes.
Fun Facts
- Snowflakes have congruent arms due to symmetry.
- DNA double helix has congruent base pairs.
- In art, M.C. Escher used congruence for tessellations.
- Congruence used in cryptography for secure shapes.
Did You Know?
SSS, SAS from Euclid; RHS from Indian mathematicians like Bhaskara.
Quick Revision One-Pager
Key Conditions
| Condition | Full Form | Guarantees Congruence |
|---|---|---|
| SSS | Side-Side-Side | Yes |
| SAS | Side-Angle-Side | Yes (included angle) |
| ASA | Angle-Side-Angle | Yes (included side) |
| AAS | Angle-Angle-Side | Yes |
| RHS | Right-Hypotenuse-Side | Yes for right triangles |
| SSA | Side-Side-Angle | No |
| AAA | Angle-Angle-Angle | No (similarity) |
Quick Rules
- ✓ Congruent: Same shape/size, superimpose with rotation/flip
- ✓ Circle: Equal radius
- ✓ Rectangle: Equal sides
- ✓ Isosceles: Base angles equal
- ✓ Equilateral: 60° angles
- ✓ Corresponding parts match in order
Mind Map
Central: Congruence
- Figures: Superimpose, measurements
- Triangles:
- Conditions: SSS, SAS, ASA, AAS, RHS
- Properties: Isosceles, Equilateral
- Applications: Architecture, designs
Exam Tips
Before Solving
Identify given equal parts, match condition
During Solving
Draw diagrams, label corresponding parts
After Solving
Check if all parts match, verify condition
Time-Savers
Memorize conditions, use abbreviations
Interactive Quiz – 15 Questions
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