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 Updated: 3 months ago
Categories: Class 7 Mathematics, NCERT Maths Notes 2025, Geometry – Triangles, Constructions & Angle Properties, CBSE Exam Preparation, Practice Questions & Quizzes
Tags: class 7 maths chapter 7 notes, a tale of three intersecting lines class 7, definition of triangle sides vertices angles, equilateral isosceles scalene triangles examples, construct triangle with given sides ruler and compass, triangle inequality rule sum of two sides greater than third, angle sum property of triangle 180 degrees, find third angle using parallel line, exterior angle equals sum of interior opposite angles, altitudes and heights of triangle, perpendicular from vertex to base, paper folding activity for angle sum, ganita prakash grade 7 triangles, cbse class 7 mathematics triangles chapter, ncert a tale of three intersecting lines solutions, extra questions mcq worksheet, class 7 maths blog summary quiz
Class 7 Mathematics Chapter 7: A Tale of Three Intersecting Lines – Complete Notes, Solutions, Questions & Answers 2025
Chapter at a Glance
Key Concepts & Rules
Concept Cards
Examples + Solutions
Figure it Out Solutions
Extra Practice Questions
Common Mistakes
Quick Revision + Mind Map
Interactive Quiz (15 Q)
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°
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.
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.
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°.
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.
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
Master the World of Triangles!
Inequality • Constructions • Angle Sum • Exterior Angles • Altitudes • Types
START QUIZ NOW
← Previous
Next →
Submit Quiz
As an Amazon Associate, ProSyllabus earns from qualifying purchases. Prices shown are subject to change.
Group Discussions No forum posts available.
Easily Share with Your Tribe
