Class 7 Maths Chapter 5: Parallel and Intersecting Lines Notes

What is this chapter about?

Fold a square sheet of paper in different ways and draw along the creases: you get many lines. Some pairs meet; others do not seem to meet even if you imagine them extended beyond the paper. This chapter studies how lines on a plane surface, such as a table top, a page or the blackboard, relate to each other: lines that meet, lines that meet at right angles, and lines that never meet.

Intersecting lines and their four angles

When two lines meet at a point on a plane, they intersect, and two straight lines can meet at only one point. They make four angles, a, b, c and d. If ∠a is 120°, then ∠b is 60°, because together they form a straight angle of 180°. In the same way ∠c is 120° and ∠d is 60°. Neighbouring angles such as ∠a and ∠b are a linear pair, and a linear pair always adds up to 180°. Opposite angles such as ∠b and ∠d are vertically opposite angles. Since ∠a + ∠b = 180° and ∠a + ∠d = 180°, ∠b must equal ∠d, whatever ∠a is; similarly ∠a = ∠c. Reasoning that shows something is always true, like this, is a proof. Read linear pairs and vertically opposite angles.

Measurement and geometry

When you measure a linear pair, you might get 179° or 181°, or two vertically opposite angles that differ slightly. Protractors can be placed slightly wrongly, and drawn lines have thickness, while the ideal line of geometry has none. Geometry works with ideal shapes and finds their properties by reasoning. Measurements still come very close to what reasoning predicts, which is why geometry is so useful in physics, art, engineering and architecture.

Perpendicular lines

When the four angles at a crossing are all equal, each must be a right angle, 90°, since the four angles together make 360°. Lines that meet at right angles are perpendicular. They need not be horizontal and vertical: any two lines that meet at 90° are perpendicular.

Parallel lines

Some line segments look as if they would meet if extended, and others do not. Railway tracks, the ruled lines of a notebook and the stripes of a zebra crossing never seem to meet. Parallel lines are lines on one plane that never meet, no matter how far they are extended in either direction. Both conditions matter: lines drawn on a table and on the board may never meet, yet they lie on different planes, so they are not called parallel. Read parallel and perpendicular lines.

Paper folding and marks

Take a square sheet. Its opposite edges are parallel and its adjacent edges are perpendicular. Fold it in half horizontally: the crease is a third line parallel to the top and bottom edges. Each new fold doubles the number of strips, so after n folds you have 2ⁿ + 1 parallel lines: 3, 5, 9 and so on. A vertical fold is perpendicular to all of them. In drawings, a set of parallel lines is marked with an arrow, a second set with two arrows, and a right angle with a small square.

Transversals and corresponding angles

A line that crosses two lines is a transversal. It makes eight angles, four at each crossing, but each crossing has two pairs of equal vertically opposite angles, so there are at most four different measures. Angles in the same position at the two crossings, such as ∠1 and ∠5, are corresponding angles. When the two lines are parallel, corresponding angles are equal; and when a pair of corresponding angles is equal, the lines are parallel. If the lines are not parallel, corresponding angles can never be equal. Read transversals: corresponding and alternate angles.

Drawing parallel lines

To draw parallel lines, slide a set square along a line l and draw two lines along its edge. Both make 90° with l, so the corresponding angles are equal and the lines are parallel. To draw a line parallel to l through a point A, put an edge of the set square on l, hold a ruler against another edge, move the set square along the ruler till its edge reaches A, and draw. With paper folding, fold a crease t perpendicular to l through A, then a crease m perpendicular to t through A: m is parallel to l.

Alternate and interior angles

In a figure with parallel lines, ∠d is the alternate angle of ∠f: they lie between the lines, on opposite sides of the transversal. ∠f equals ∠b because they are corresponding angles, and ∠b equals ∠d because they are vertically opposite, so ∠f = ∠d: alternate angles are equal. Angles such as ∠3 and ∠6, between the lines on the same side of the transversal, are interior angles, and they add up to 180°. With these facts, one known angle gives all eight: if ∠6 is 135°, four angles are 135° and four are 45°.

Parallel illusions

Some pictures hide parallel lines: lines drawn over a fan of lines look bowed, and lines crossed by short slanting strokes look tilted. A ruler shows they are straight and parallel; the background lines crossing them at changing angles fool our eyes.

Before we move on

Intersecting lines make linear pairs and equal vertically opposite angles; perpendicular lines meet at 90°; parallel lines never meet; and a transversal across parallel lines makes equal corresponding and alternate angles, with interior angles adding up to 180°. Answers to every question, activity and Figure it Out are in the NCERT solutions for Chapter 5.