NCERT Solutions for Class 7th Maths Chapter 7 A Tale of Three Intersecting Lines
Updated on 2026-09-19
About this chapter
A triangle has three vertices, three sides and three angles. It is named by its vertices in any order — ∆ABC, ∆BAC and ∆CAB are the same triangle. Two arcs drawn with a compass meet at the third vertex, so a triangle whose three sidelengths are known can be constructed exactly, without trial and error. Triangle inequality: a triangle exists for three lengths exactly when each length is less than the sum of the other two. It is enough to test the longest length. A triangle can also be built from two sides and the included angle , or from two angles and the included side — the latter only when the two angles add to less than 180°. Angle sum property: the three angles of any triangle add up to 180°. The proof draws a line through one vertex parallel to the opposite side. An altitude is the pe
- Constructing a Triangle When its Sides are Given
- Are Triangles Possible for any Lengths?
- Triangle Inequality
- Visualising the construction of circles
- When do the two circles intersect?
- Conclusion
- Two Sides and the Included Angle
- Two Angles and the Included Side
- Do triangles always exist?
- Finding the third angle
Quick revision
| Idea | What it means | Example from the chapter | Key result |
|---|---|---|---|
| Triangle | Three vertices joined by three sides | ∆ABC | 3 sides, 3 angles |
| Equilateral triangle | All three sides equal | 4, 4, 4 | Each angle 60° |
| Isosceles triangle | Exactly two sides equal | 5, 5, 8 | Two equal angles |
| Scalene triangle | All three sides different | 3, 4, 5 | All angles different |
| Compass construction | Two arcs of the given radii cross at the third vertex | 4 cm, 5 cm, 6 cm | Exact, no trials |
| Triangle inequality | Each length < sum of the other two | 10, 15, 30 → 30 > 10 + 15 | No triangle |
| Shortcut test | Check only the longest length | 4, 5, 8 → 8 < 4 + 5 | Triangle exists |
| Two circles | Sum of the two smaller lengths vs the longest | <, =, > | Only > gives a triangle |
| Two sides + included angle | SAS-type construction | 5 cm, 45°, 4 cm | Always one triangle (angle < 180°) |
| Two angles + included side | ASA-type construction | 45°, 5 cm, 80° | Needs sum of angles < 180° |
| Angle sum property | ∠A + ∠B + ∠C | Line through A parallel to BC | 180° |
| Exterior angle | Angle between an extended side and the other side | ∠A = 50°, ∠B = 60° | ∠ACD = 110° = 50° + 60° |
| Altitude | Perpendicular from a vertex to the opposite side | AD, BE, CF | Height of the triangle |
| Angle-based types | By the largest angle | acute / right / obtuse | One right or obtuse angle at most |
Exercises
- Introduction · 7.1 Equilateral Triangles · 7.2 Constructing a Triangle When its Sides are Given — In-text Questions Page 146–149
- Construct — Constructing a Triangle When its Sides are Given Page 150
- Figure it Out — Constructing a Triangle When its Sides are Given Page 150–151
- Are Triangles Possible for any Lengths? — In-text Questions Page 151
- Triangle Inequality — In-text Questions Page 152–153
- Triangle Inequality — Figure it Out Page 154
- Triangle Inequality — In-text Questions Page 154
- Visualising the construction of circles — In-text Questions Page 155
- Triangle Inequality — Figure it Out Page 156
- When do the two circles intersect? — In-text Questions Page 156–159
- Conclusion — Figure it Out Page 159–160
- .3 Construction of Triangles When Some Sides and Angles are Given — In-text Questions Page 1607
- Two Sides and the Included Angle — Figure it Out Page 161
- Two Angles and the Included Side — In-text Questions Page 161
- Two Angles and the Included Side — Figure it Out Page 162
- Do triangles always exist? — In-text Questions Page 162–163
- Do triangles always exist? — Figure it Out Page 163
- Finding the third angle — In-text Questions Page 164
- Finding the third angle — Figure it Out Page 165
- Angle Sum Property · Exterior Angles — In-text Questions Page 165–167
- .4 Constructions Related to Altitudes of Triangles — In-text Questions Page 167–1697
- .5 Types of Triangles — In-text Questions Page 1707
- .5 Types of Triangles — Figure it Out Page 170–1717
- Shortest Path in a Box! — Puzzle Time Page 172