Class 7 Maths Chapter 7: A Tale of Three Intersecting Lines Notes

What is this chapter about?

A triangle is the simplest closed shape: three vertices joined by three sides, with an angle at each corner. In ∆ABC the angles are ∠CAB, ∠ABC and ∠BCA, written simply ∠A, ∠B and ∠C, and the vertices can be named in any order. If the three points lie on one straight line, no triangle forms at all. The chapter asks how to construct triangles accurately, when a triangle can exist, and what is always true of its angles.

Constructing a triangle from three sides

To draw an equilateral triangle of side 4 cm with only a ruler, we would mark the third point by trial and error. A compass does it in one go: draw the base AB = 4 cm, draw an arc of radius 4 cm from A and another from B, and join their meeting point C to A and B. C lies on both arcs, so it is 4 cm from each end. The same method builds any triangle from its three sides, such as 4 cm, 5 cm and 6 cm: draw one side as the base, then arcs with the other two lengths from its ends. Read constructing triangles.

When can three lengths make a triangle?

Try 3 cm, 4 cm and 8 cm: the arcs never meet. The reason is simple. Walking straight from a tent to a tree is shorter than going via a pole, so in any triangle the direct side between two vertices is shorter than the roundabout path through the third vertex. For 10, 15 and 30 this fails, since 30 is more than 10 + 15, so no such triangle exists. Three lengths satisfy the triangle inequality when each is less than the sum of the other two, and it is enough to check the longest. Picturing the two circles from the ends of the longest side shows why the rule is enough: they touch when the two smaller lengths add up exactly to the longest, miss each other when they add up to less, and cross, giving a triangle, when they add up to more. Read triangle inequality and angle sum property.

Two sides and an angle, or two angles and a side

Given two sides and the included angle, draw one side, draw the angle at its end, and mark the second side along the new arm; a triangle exists for any included angle less than 180°. Given two angles and the included side, draw the side and both angles at its ends; the arms meet at the third vertex. This works only when the two angles add up to less than 180°. With 40° at A, an arm from B at 140° runs parallel to the first arm and never meets it, however long AB is.

The angle sum property

Fix two angles, say 60° and 70°, and the third angle comes out the same whatever the base length. To find it without measuring, draw a line XY through the top vertex parallel to the base. The angles it makes with the two sides are alternate angles, equal to the two base angles, and together with the top angle they form a straight angle. So the three angles of every triangle add up to 180°. This clever idea appears in Euclid’s Elements, written around 300 BCE. Folding the corners of a paper triangle to meet on its base shows the same thing. Closely linked, an exterior angle, made by extending one side, equals the sum of the two interior angles opposite it.

Altitudes

The height of a vertex is measured straight down to the opposite side. The perpendicular segment from a vertex to the opposite side is an altitude, and every triangle has three. In an obtuse triangle the base must be extended and the altitude falls outside. A paper fold that lays the base onto itself through the top vertex makes the altitude, and a ruler with a set square draws it accurately. In a right-angled triangle, the sides at the right angle are themselves altitudes.

Types of triangles

By sides, a triangle is equilateral (three equal sides), isosceles (two equal sides) or scalene (all sides different). By angles, it is acute-angled (all three angles acute), right-angled (one right angle) or obtuse-angled (one obtuse angle). An acute-angled triangle needs all three angles acute, because every triangle already has at least two acute angles. Read types of triangles.

In a nutshell

A compass turns triangle construction into a few clean steps; the triangle inequality and the 180° rule tell us in advance whether a triangle can exist; parallel lines prove the angle sum property; and altitudes and types complete the picture. The chapter ends with a puzzle about the shortest path for a spider walking over a box. Answers to every question are in the NCERT solutions for Chapter 7.