Class 7 Maths Ch 7: A Tale of Three Intersecting Lines – learn to construct and analyse triangles, use triangle inequality, angle sum and exterior angle properties, and draw altitudes, with clear notes, solved examples and quiz for CBSE Exam
Complete Chapter 7 guide: basics of triangles (sides, vertices, angles, naming), constructing equilateral and general triangles using ruler and compass, when a triangle is possible or impossible (triangle inequality with tent–tree–pole story), finding the third angle using a parallel line and proving angle sum property ∠A+∠B+∠C=180°, exterior angles equal to the sum of two interior opposite angles, and altitudes/heights of a triangle with paper‑folding and set‑square constructions, plus solved examples and practice for CBSE Class 7 Maths
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A Tale of Three Intersecting Lines
Class 7 Mathematics Chapter 7 | Complete Guide | Triangles, Constructions, Inequality, Angles, Altitudes 2025
Chapter at a Glance – A Tale of Three Intersecting Lines
This chapter introduces triangles, their constructions using compass and ruler, triangle inequality for existence, angle sum property, exterior angles, altitudes, and classification based on sides and angles.
Main Topics Covered
- Definition of triangles: vertices, sides, angles
- Equilateral triangles and construction
- SSS construction
- Triangle inequality
- SAS and ASA constructions
- Angle sum property (180°)
- Exterior angles
- Altitudes using set square
- Types: equilateral, isosceles, scalene; acute, right, obtuse
Key Takeaways for Exams
Triangle Inequality
Each side < sum of other two
Angle Sum
∠A + ∠B + ∠C = 180°
Exterior Angle
Equal to sum of opposite interior angles
Altitude
Perpendicular from vertex to opposite side
Right Triangle
One 90° angle
Equilateral
All sides equal, all angles 60°
Key Concepts & Rules – A Tale of Three Intersecting Lines
Important definitions, construction steps, inequalities, and properties for quick reference.
Key Rules
- Triangle Inequality: For sides a,b,c: a + b > c; a + c > b; b + c > a
- Angle Sum Property: Sum of interior angles = 180°
- Exterior Angle: Equal to sum of two opposite interior angles
- SSS Construction: Draw base, arcs from ends with other sides
- SAS Construction: Draw base, angle at one end, mark second side
- ASA Construction: Draw base, angles at both ends, intersection
- Altitude: Perpendicular to opposite side; in right triangle, legs are altitudes
- Types by Sides: Equilateral (all equal), Isosceles (two equal), Scalene (all different)
- Types by Angles: Acute (all <90°), Right (one 90°), Obtuse (one >90°)
Construction Tools
| Tool | Use |
|---|---|
| Compass | Draw arcs for equal lengths |
| Ruler | Measure and draw sides |
| Set Square | Draw perpendiculars for altitudes |
| Protractor | Measure angles for SAS/ASA |
Golden Rules for Exams
"Check triangle inequality before construction: Longest side < sum of others."
"Third angle = 180° - sum of two given angles."
Concept Cards – Quick Explanations
Triangle Basics
3 vertices, 3 sides, 3 angles.
Equilateral
All sides equal; Construct with compass arcs.
SSS Construction
Base + two arcs intersect.
Triangle Inequality
Side < sum of others; Ensures arcs intersect.
SAS Construction
Two sides + included angle.
ASA Construction
Two angles + included side; Sum <180°.
Angle Sum
180°; Use parallel line proof.
Exterior Angle
= Sum of two interior opposites.
Altitude
Perp. to base; Paper fold or set square.
Types by Sides
Equilateral, Isosceles, Scalene.
Types by Angles
Acute, Right, Obtuse.
Examples + Solutions
Example 1: Equilateral Construction
Problem: Construct equilateral triangle with side 4 cm.
Solution:
- Draw base AB = 4 cm.
- Arc radius 4 cm from A.
- Arc radius 4 cm from B; intersect at C.
- Join AC, BC.
Answer: Equilateral ΔABC.
Example 2: SSS Construction
Problem: Construct triangle with sides 4 cm, 5 cm, 6 cm.
Solution:
- Base AB = 4 cm.
- Arc 5 cm from A.
- Arc 6 cm from B; intersect at C.
- Join AC, BC.
Answer: Scalene ΔABC.
Example 3: Triangle Inequality Check
Problem: Can triangle have sides 3 cm, 4 cm, 8 cm?
Solution:
- 3 + 4 = 7 < 8; Fails inequality.
- Arcs won't intersect.
Answer: No.
Example 4: SAS Construction
Problem: Construct ΔABC with AB=5 cm, AC=4 cm, ∠A=45°.
Solution:
- Draw AB=5 cm.
- At A, draw 45° ray.
- Mark C on ray at 4 cm from A.
- Join BC.
Answer: Acute triangle.
Example 5: ASA Construction
Problem: Construct ΔABC with AB=5 cm, ∠A=45°, ∠B=80°.
Solution:
- Draw AB=5 cm.
- Draw 45° at A, 80° at B.
- Intersect at C.
- Third ∠=55° (180°-125°).
Answer: Acute triangle.
Example 6: Angle Sum Property
Problem: If ∠A=50°, ∠B=60°, find ∠C and exterior at C.
Solution:
- ∠C=180°-110°=70°
- Exterior at C=180°-70°=110°=50°+60°
Answer: ∠C=70°, exterior=110°.
Example 7: Altitude Construction
Problem: Construct altitude from A to BC in arbitrary triangle.
Solution:
- Align ruler to BC.
- Slide set square until vertical edge through A.
- Draw line.
Answer: Perpendicular AD.
Figure it Out Solutions (All Solved)
Page 5: Isosceles/Equilateral with Circles
1. Form isosceles triangles using points on circle/center.
Any two points with center form isosceles (radii equal).
2. Isosceles/equilateral with same size circles.
A,B centers: Join centers and points for isosceles.
A,B,C centers: Equilateral if centers form equilateral.
Page 6: Triangle Existence
1. No triangles for 3,4,8 and 2,3,6 by construction.
3+4=7<8; 2+3=5<6; Fails inequality.
2. Existence check:
- (a) 10+10=20<25? No
- (b) 5+10=15<20? No
- (c) 12+20=32<40? No
3. Always at least two where direct < sum; Longest may fail.
Sort lengths; Check longest < sum of others.
Page 11: Triangle Existence
1. Which lengths form triangle?
- (a) 2+2=4<5? No
- (b) 3+4=7>6, etc. Yes
- (c) 2+4=6<8? No
- (d) 5+5=10>8, etc. Yes
- (e) 10+20=30>25, etc. Yes
- (f) 10+20=30<35? No
- (g) 24+26=50>28, etc. Yes
Page 14: More on Existence
1. Check existence:
- (a) 1+100=101>100, etc. Yes
- (b) 3+6=9=9? Equal, degenerate
- (c) 1+1=2<5? No
- (d) 5+10=15>12, etc. Yes
2. Equilateral always exists for any equal sides >0.
50+50>50, etc. Yes.
3. Third side ranges:
- (a) >99, <101
- (b) >0, <10
- (c) >4, <10
Page 16: SAS Constructions
1. Construct SAS:
(a) 3 cm, 75°, 7 cm; Always exists.
(b) 6 cm, 25°, 3 cm; Exists.
(c) 3 cm, 120°, 8 cm; Exists.
Page 17: ASA Constructions
1. Construct ASA:
(a) 75°,5 cm,75°; Exists (sum 150°<180°).
(b) 25°,3 cm,60°; Exists.
(c) 120°,6 cm,30°; Exists.
Page 18: Angle Pairs for ASA
1. For given angle, possible/not:
(a) 30°: Possible <150°; Not ≥150°.
(b) 70°: Possible <110°; Not ≥110°.
(c) 54°: Possible <126°; Not ≥126°.
(d) 144°: Possible <36°; Not ≥36°.
2. Pairs possible:
- (a) 35°+150°=185°>180° No
- (b) 70°+30°=100°<180° Yes
- (c) 90°+85°=175°<180° Yes
- (d) 50°+150°=200°>180° No
Page 20: Third Angle
1. Third angles:
- (a) 72°
- (b) 15°
- (c) 60°
- (d) 60°
2. All 70°? Third=40°, no. Equal: 60° each.
Equilateral has 60° angles.
3. ∠B=∠C=65° (180°-50°)/2.
Isosceles base angles equal.
Page 22: Exterior Angle
Find exterior for different ∠A,∠B.
Exterior = ∠A + ∠B.
Page 25: Altitudes and Types
1. Altitude from A to BC in 5,6,5 cm triangle.
Use set square; Isosceles.
2. Altitude in TRY with RY=4, TR=7, ∠R=140°.
Obtuse; Extend base.
3. Right triangle with ∠B=90°, AC=5 cm.
Infinite; ∠A can vary 0°-90°.
4. Equilateral right/obtuse? No. Isosceles right/obtuse? Yes.
Equilateral all acute; Isosceles can be right (45-45-90) or obtuse.
Extra Practice Questions (Exam-Ready) – Chapter 7 A Tale of Three Intersecting Lines
20+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. Sum of triangle angles?
2. Equilateral angles?
3. Triangle inequality for a,b,c?
4. Altitude definition?
5. Right triangle has how many altitudes as legs?
2-Mark Questions (Short Answer)
6. Can sides 5,5,10 form triangle?
7. Third angle if 40°, 60°?
8. Exterior angle if interiors 50°,70°?
9. Classify triangle with sides 3,4,5.
10. SAS needs what?
3-Mark Questions (Reasoning / Explanation)
11. Why sum of two angles <180° for ASA?
12. Prove angle sum 180° using parallel.
13. Why equilateral not obtuse?
14. Altitude in obtuse triangle?
15. Check 4,5,6 triangle; Type?
4–5 Mark Questions (Application / Word Problems)
16. Third side range for 7,9.
17. Construct SAS 5 cm,60°,4 cm.
18. Why 3,3,7 no triangle?
19. Angles 30°,70°; Third? Exterior at third?
20. Classify: Sides 5,5,5; Angles 90°,45°,45°.
Common Mistakes & How to Avoid
Mistake 1: Ignoring Inequality
Trying to construct without check.
Avoid: Always verify a+b>c etc.
Mistake 2: ASA Sum ≥180°
Not checking angle sum.
Avoid: Two angles <180°.
Mistake 3: Wrong Third Angle
Miscalculating 180° - sum.
Avoid: Double-check arithmetic.
Mistake 4: Altitude in Obtuse
Not extending base.
Avoid: Check if foot outside.
Mistake 5: Confusing Types
Isosceles as equilateral.
Avoid: Count equal sides/angles.
Mistake 6: Exterior Wrong
Not summing opposites.
Avoid: Remember = remote interiors sum.
Quick Revision One-Pager & Mind Map
| Concept | Key Points |
|---|---|
| Triangle Inequality | a + b > c etc.; For existence |
| Constructions | SSS: Arcs intersect; SAS: Angle + sides; ASA: Angles + side |
| Angle Sum | 180°; Parallel proof |
| Exterior Angle | = Sum opposite interiors |
| Altitudes | 3 per triangle; Set square |
| Types | Sides: Equi/Iso/Scal; Angles: Acute/Right/Obtuse |
Mind Map
Central: Triangles
- Basics:
- Vertices/Sides/Angles
- Equilateral construction
- Constructions:
- SSS/SAS/ASA
- Inequality check
- Properties:
- Angle sum 180°
- Exterior = interiors
- Altitudes perp.
- Types:
- Sides: Equi/Iso/Scal
- Angles: Acute/Right/Obtuse
Interactive Chapter Review Quiz
Inequality • Constructions • Angle Sum • Exterior Angles • Altitudes • Types
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