A Tale of Three Intersecting Lines – Class 7 Maths Chapter 7 Notes & NCERT PDF

Introduction

A triangle is the most basic closed shape: three vertices (corner points) joined by three sides. We name it by its vertices, in any order — ∆ABC, ∆YZX, ∆WUV — and write its angles ∠CAB, ∠ABC, ∠BCA simply as ∠A, ∠B and ∠C. The chapter begins with a puzzle worth keeping in mind: what happens when the three vertices lie on a straight line? (No triangle forms!)

This chapter is about how to construct triangles, when a triangle is possible at all, the famous angle sum property, altitudes, and types of triangles.

Key Ideas

7.1 Equilateral Triangles

An equilateral triangle has all three sides equal. It is the most symmetric triangle.

To construct one with sides 4 cm, a ruler alone would need many trials. A compass makes it exact — the same trick you used in Grade 6 to mark the top point of a "house" 5 cm from two points.

7.2 Constructing a Triangle When its Sides are Given

For sides 4 cm, 5 cm and 6 cm: draw base AB = 4 cm. Every point 5 cm from A lies on a circle of radius 5 cm centred at A; every point 6 cm from B lies on a circle of radius 6 cm centred at B. The third vertex C must be on both — so C is where the arcs meet.

Triangles with two equal sides are isosceles; with three different sides, scalene.

Are Triangles Possible for any Lengths?

Try 3 cm, 4 cm, 8 cm or 2 cm, 3 cm, 6 cm — the arcs never meet! The book explains why with a tent, a tree and a pole: the straight path from the tent to the tree is always shorter than going via the pole. So in any triangle, each side must be shorter than the sum of the other two.

For 10 cm, 15 cm, 30 cm: 10 < 15 + 30 = 45 and 15 < 30 + 10 = 40, but 30 > 10 + 15 = 25 — the "direct path" is longer than the "roundabout path", which is absurd. No triangle.

Triangle inequality: a set of three lengths satisfies it when each length is smaller than the sum of the other two.

Shortcut: only compare the longest length with the sum of the two smaller ones — the other two comparisons are automatically true.

Why the triangle inequality guarantees a triangle

Take the longest length as base AB and draw circles with the two smaller lengths as radii. Three cases:

For 4, 5, 8: base 8 cm, AX = 4 so BX = 4 cm, which is less than 5 — the circles cross, and a triangle exists.

Conclusion: if three lengths satisfy the triangle inequality, a triangle exists; if not, it does not.

7.3 Triangles When Some Sides and Angles are Given

When is a triangle impossible with two angles? If ∠A = 40°, the line from B fails to meet line l when it is parallel or bends away. Using co-interior angles on transversal AB, ∠B = 180° − 40° = 140° for parallel lines. So no triangle when ∠B ≥ 140°. The length AB plays no part!

Rule: a triangle exists with two given angles only when their sum is less than 180°.

The Angle Sum Property

For ∆ABC with ∠B = 50°, ∠C = 70°: draw line XY through A parallel to BC. Alternate angles give ∠XAB = 50° and ∠YAC = 70°. These three angles at A make a straight angle, so 50° + ∠BAC + 70° = 180° → ∠BAC = 60°.

Angle sum property: the three angles of any triangle add up to 180°.

This parallel-line idea is from *Euclid's The Elements* (Euclid lived around 300 BCE). You can also check it by folding a paper triangle** so the three corners meet on a line.

Exterior Angles

Extend side BC to D; ∠ACD is an exterior angle. If ∠A = 50° and ∠B = 60°, then ∠ACB = 70° and ∠ACD = 180° − 70° = 110°, which equals ∠A + ∠B.

An exterior angle equals the sum of the two opposite interior angles.

7.4 Altitudes

The altitude from A is the perpendicular segment AD from A to the opposite side BC; its length is the height of A above BC. Every triangle has three altitudes (AD, BE, CF). In some triangles the altitude falls outside — extend BC and drop the perpendicular to the extended line. Use a set square sliding along a ruler to draw altitudes accurately, or fold a paper triangle so one side lies on itself.

In a right-angled triangle, a side is itself an altitude.

7.5 Types of Triangles

Why can't "acute-angled" mean "has one acute angle"? Because every triangle has at least two acute angles!

Worked Examples

Example 1 — Construct ∆ABC with sides 4 cm, 5 cm, 6 cm.

Example 2 — Can 2, 2, 5 be sides of a triangle? Longest = 5; 2 + 2 = 4 < 5. No. What about 24, 26, 28? 24 + 26 = 50 > 28. Yes.

Example 3 — Third side for sides 1 and 100. It must be strictly between 100 − 1 = 99 and 100 + 1 = 101, e.g. 99.5, 100, 100.7.

Example 4 — Find the third angle when two angles are 75° and 45°. 180° − (75° + 45°) = 180° − 120° = 60°.

Example 5 — In ∆ABC, ∠B = ∠C and ∠A = 50°. ∠B + ∠C = 130°, so ∠B = ∠C = 65°.

Example 6 — Can 35° and 150° be two angles of a triangle? 35° + 150° = 185° ≥ 180°. No.

Figure it Out — Hints

Maths Words

Historical Connections

The proof of the angle sum property using a parallel line appears in The Elements, attributed to the Greek mathematician Euclid (around 300 BCE) — a reminder that creative ideas, like drawing one extra line, can make a hidden relationship obvious.

Common Mistakes

Maths in Daily Life

The triangle inequality is why a straight road is shorter than a detour. Carpenters and engineers use triangles because a triangle with fixed sides cannot change shape — that's why bridges, roof trusses and cycle frames are full of triangles.