Class 7 Mathematics Chapter 5: Parallel and Intersecting Lines

Two lines on a flat surface

Draw two straight lines on a sheet of paper. Only two things can happen. They either cross somewhere, or they never cross at all.

There is no third case. That one fact is what the whole chapter is built on.

Lines that cross

Lines that cross are called intersecting lines, and the point where they cross is their point of intersection.

Two straight lines can cross at one point only. If they crossed twice they would have to bend, and straight lines do not bend.

Lines that never cross

Lines that never cross, however far they are extended, are called parallel lines.

Parallel does not mean merely that they have not met yet. It means they never will, in either direction, for ever.

What we draw is only part of the line

A line has no ends at all. It runs on for ever in both directions. What we draw on paper is a segment, which is a finite piece of one.

So a drawing can only ever show part of the story. That is exactly why two lines looking unmet on the page proves nothing about the lines themselves.

Think of the gap

The clearest test is the gap between the lines. Parallel lines keep exactly the same gap all along.

If the gap shrinks even slightly, the lines are closing in and will meet eventually, which makes them intersecting rather than parallel.

Four angles at a crossing

When two lines cross they make four angles around the point. The four together fill a complete turn, so they add to 360 degrees.

Those four angles are not independent. Knowing one of them fixes all four.

Angles on a straight line

Two angles that sit side by side along a straight line add to 180 degrees, because a straight line is half a turn.

So at a crossing, each angle and the one next to it add to 180. That is the rule doing most of the work in this chapter.

Vertically opposite angles

The two angles facing each other across the crossing point are called vertically opposite angles, and they are always equal.

If one angle is 65 degrees, the one facing it is also 65 degrees. The other two are each 115 degrees, since 65 and 115 make 180.

Why opposite angles must be equal

Take the 65 degree angle. The angle beside it makes 180 with it, so it is 115 degrees.

Now the angle beside THAT one also makes 180 with it, so it is 65 degrees again, and it is the one facing the first. The equality is forced.

When the crossing is square

Some crossings make all four angles equal. Since the four add to 360, each one must be 90 degrees.

Lines that cross at 90 degrees are called perpendicular. The right angle is usually marked with a small square rather than written out.

Perpendicular is a special case

Every pair of perpendicular lines is a pair of intersecting lines. The reverse is not true.

So perpendicular sits inside intersecting, in the way that a square sits inside rectangles.

Where you have already seen all three

The edges of a page give you both at once. The two long edges are parallel and a long edge meets a short edge at a right angle.

Floor tiles, window frames, a chessboard and ruled paper are all built from the same two relationships.

Folding gives you both

Fold a sheet of paper once and open it. The crease is a straight line. Fold it again so the first crease lies exactly along itself.

The second crease meets the first at a right angle, because the fold matched the two halves of a straight angle.

And folding gives parallels too

Now fold the paper again so that the second crease lies along itself. The third crease is parallel to the second.

Each new crease is perpendicular to the one before it, and lines perpendicular to the same line are parallel to each other.

A rule worth keeping

Two lines perpendicular to the same line are parallel to each other.

That is why the rungs of a ladder are parallel: each one is at a right angle to both of the long sides.

Saying it precisely

Three words, three meanings. Intersecting means they cross. Perpendicular means they cross at a right angle. Parallel means they never cross.

Using the wrong word makes a correct drawing read as a wrong answer, so the word matters as much as the picture.

What carries forward

On a flat surface two straight lines either cross once or never cross. There is no other possibility.

At a crossing, neighbouring angles add to 180 degrees and opposite angles are equal. When all four are 90 the lines are perpendicular, and lines perpendicular to the same line are parallel.

Scenario:

A carpenter is making a rectangular window frame from four lengths of wood. She cuts the two long pieces to one length. She cuts the two short pieces to another. Then she assembles the frame. Checking it, she finds one corner slightly open. The long piece and the short piece there do not meet squarely. Her apprentice says the frame must still be fine. His reason is that the two long pieces have not crossed each other anywhere. The carpenter measures one angle at that corner. It is 85 degrees rather than 90. She asks the apprentice what the other three angles at that corner must be.

Analysis:

The apprentice is using unmet as a test for parallel, and the corner angles are fixed by the two rules about a crossing.

Activity:

Measure the gap between the two long edges of your desk at three different points, then check two corners with the corner of a book.

Title:

Fold, and find all three

Safety:

No safety concerns. Paper, pencil and ruler only.

What You Notice:

Each new crease is perpendicular to the one before it, so the first and third are both perpendicular to the second. That is why they come out parallel, and the three gap measurements should be identical.

Paper folding produces a right angle without any instrument, because folding a crease onto itself halves a straight angle exactly.

Perpendicular is a special case of intersecting, in the way that a square is a special case of a rectangle.

Knowing one angle at a crossing is enough to fix all four, so a single measurement settles the whole picture.

Neighbours make 180, facing angles match.

Same gap all along, or it is not parallel.

A third line across the other two

Draw two lines. Now draw a third line that cuts across both of them. That third line is called a transversal.

The word says what it does: it travels across. Nothing is claimed yet about whether the first two lines are parallel.

Eight angles, not four

The transversal crosses each of the two lines, and each crossing makes four angles. So the figure holds eight angles in all.

Four of them sit at the upper crossing and four at the lower one. Keeping track of which is which is most of the difficulty.

Naming by position

Angles are named by where they sit rather than by their size. The names describe the picture, so they work whatever the angles measure.

Two angles in matching positions at the two crossings are called corresponding angles.

What matching position means

Picture the upper crossing. Take the angle above the first line and to the left of the transversal.

Now go to the lower crossing and take the angle in the same place: above the second line and to the left of the transversal. Those two correspond.

Four corresponding pairs

There are four such pairs, one for each of the four positions around a crossing. Above-left pairs with above-left, below-right with below-right, and so on.

Sliding the upper crossing down the transversal onto the lower one would carry each angle onto its partner.

Still no claim about parallel

So far corresponding is only a position. Two corresponding angles need not be equal, and usually they are not.

If the two lines are not parallel the figure is lopsided, and corresponding angles differ.

Now make the two lines parallel

Here is the fact the chapter turns on. When the two lines are parallel, corresponding angles are equal.

If one of them is 72 degrees, its partner at the other crossing is 72 degrees too, exactly.

Why that is believable

Slide the whole upper crossing down the transversal until it sits on the lower one. If the lines are parallel the two crossings look identical.

Nothing about the picture has changed except position, so each angle arrives exactly on top of its corresponding partner.

The rule works both ways

If the lines are parallel then corresponding angles are equal. And if corresponding angles are equal then the lines are parallel.

The second direction is the useful one in practice, because it lets you test two lines by measuring two angles.

Filling in the rest of the figure

Once one angle is known, the whole figure follows. Suppose a corresponding angle is 72 degrees.

Its partner is 72. Each of their neighbours is 108, because angles on a straight line make 180. And facing angles repeat those values.

Co-interior angles

The two angles between the parallel lines, on the same side of the transversal, are called co-interior angles.

With the 72 degree angle above, the co-interior partner below is 108 degrees. Co-interior angles add to 180 rather than being equal.

Why they add instead

One of them is corresponding to the other one's neighbour. The neighbour makes 180 with it along a straight line.

So co-interior angles add to 180, and that is a different relationship from the equality of corresponding angles. Mixing the two is the commonest error here.

Equal or adding: how to tell

Corresponding angles sit in matching positions, one at each crossing, and they are equal.

Co-interior angles sit between the lines on the same side, and they add to 180. Look at the position first, then pick the relationship.

Drawing a parallel line

The rule gives a construction. To draw a line parallel to a given line through a given point, draw any transversal through that point.

Measure the angle the transversal makes with the original line. Copy that same angle at the point, in the corresponding position, and draw. The new line is parallel.

Why the construction works

The copied angle and the original are corresponding angles by position, and they were made equal by construction.

Equal corresponding angles force the two lines to be parallel, which is the reverse direction of the rule doing real work.

What carries forward

A transversal cuts two lines and makes eight angles. Corresponding angles occupy matching positions at the two crossings.

When the lines are parallel those pairs are equal, and co-interior angles add to 180 instead. The rule runs both ways, which is what lets you both check and construct a parallel.

Scenario:

A carpenter has fixed two long shelves to a wall and wants to know whether they are truly parallel. She cannot extend them to see whether they would meet. Instead she lays a straight batten across both shelves, making a transversal. At the upper shelf she measures the angle above the shelf and left of the batten, and gets 72 degrees. At the lower shelf she measures the angle in the same position and gets 75 degrees. Her apprentice says this proves nothing, because the two angles are at different crossings. He also claims the angle between the shelves on the left of the batten should be 72 degrees as well.

Analysis:

The carpenter is using the reverse of the corresponding-angles rule, which is exactly what it is for, and the apprentice has confused co-interior angles with corresponding ones.

Activity:

Lay a ruler across two lines of your exercise book, measure a corresponding pair, and then measure a co-interior pair.

Title:

Draw a parallel with one measured angle

Safety:

Take care with the protractor point and the pencil tip. No other hazards.

What You Notice:

The three gap measurements come out equal, so the new line is parallel. It had to be: the copied angle corresponds to the original and was made equal by construction, and equal corresponding angles force parallel lines.

With parallel lines the eight angles take only two values, four of each, so the whole figure is settled by one measurement.

The reverse of the rule is what makes it practical: two angle measurements test a pair of lines that cannot be extended.

The construction for a parallel line needs no set square at all, only a protractor and the converse of the rule.

Matching positions match; between and same side add to 180.

Read the position first, then choose the relationship.

A third named position

A transversal across two lines gives corresponding angles and co-interior angles. There is one more position worth a name.

Take the two angles between the lines, but on OPPOSITE sides of the transversal. Those are alternate angles.

Alternate means opposite sides

Co-interior angles sit between the lines on the same side. Alternate angles sit between the lines on opposite sides.

One word changes, and so does the relationship. That single difference is what the whole topic rests on.

Alternate angles are equal

When the two lines are parallel, alternate angles are equal.

If one of them measures 48 degrees, the alternate one across the transversal also measures 48 degrees.

Why they are equal

Take the 48 degree angle between the lines. Its neighbour along the straight line is 132 degrees.

That 132 degree angle is co-interior with the alternate angle, so the two add to 180. The alternate angle is therefore 48 degrees.

A shorter route to the same place

Alternate angles can also be reached through a corresponding angle. The 48 degree angle corresponds to an angle at the other crossing.

That corresponding angle is vertically opposite the alternate angle, so all three are 48 degrees. Either route gives the same answer.

The three named pairs together

Corresponding angles are equal. Alternate angles are equal. Co-interior angles add to 180.

Two of the three are equalities and one is a sum, and which one you have depends entirely on position.

How to read the position

Ask two questions. Are the angles between the two lines, or is one outside? And are they on the same side of the transversal, or opposite sides?

Between and opposite gives alternate. Between and same side gives co-interior. Matching positions at the two crossings gives corresponding.

Angles outside the lines too

Not every named pair lies between the lines. Two angles outside them, on opposite sides of the transversal, are also alternate.

They are equal for the same reason, and they are often called exterior alternate angles. The position rule did not change; only which side of each line we stood on did.

All three run backwards too

Each rule has a reverse. Equal alternate angles mean the lines are parallel, and co-interior angles adding to 180 means the same.

So any of the three can be used as a test. A single pair of measurements is enough to settle whether two lines are parallel.

A worked figure

Suppose a transversal cuts two parallel lines and one angle is 48 degrees. Every angle in the figure is now fixed.

The four at each crossing are 48, 132, 48 and 132. Four angles in the figure are 48 degrees and four are 132 degrees.

Checking the whole figure

The check is easy. At each crossing the four angles add to 360 degrees, and 48 with 132 makes 180.

If a figure you have filled in fails either check, an angle has been placed in the wrong position.

Lines that only look wrong

Now the illusions. Draw two genuinely parallel lines and cross them with many short slanting strokes.

The parallel lines will appear to bend or to converge. They have not moved; the strokes are tilting what your eye reports.

Why the eye is fooled

The eye judges direction partly from the shapes nearby. Slanting strokes drag the apparent direction of a line with them.

So a figure can look wrong and be right. Your eye is reporting the strokes as well as the lines.

A figure can also look right and be wrong

The illusion runs the other way too. Two lines drawn almost parallel look parallel to the eye, especially over a short length.

A lean of two or three degrees is invisible on a page, yet the lines would cross well within a room.

So appearance is not evidence

This is why the chapter has rules at all. Looking tells you what a figure seems to do, and measuring tells you what it does.

Measure a corresponding, alternate or co-interior pair. Those settle the question when the eye cannot.

Where the illusions are useful

The same effect is used on purpose. A pattern of slanting strokes makes a flat wall look textured, and tiles laid true can be made to look wavy.

Knowing why it works means you are not fooled by it, and can use it.

What carries forward

Alternate angles lie between the lines on opposite sides of the transversal and are equal when the lines are parallel.

Position decides the relationship: between and opposite are equal, between and same side add to 180. And an illusion can make a true figure look false, which is exactly why measurement settles things and appearance does not.

Scenario:

A tiler lays a floor in straight rows and checks each row against the wall before fixing it. When the floor is finished the rows appear to curve gently towards the middle of the room. The owner says the work must be redone. The tiler points out that each tile carries a pattern of short slanting lines, all leaning the same way. He lays a long straight batten along one row to show it is straight. He then measures, with a transversal across two rows, an alternate pair of angles and finds both to be 48 degrees. The owner also mentions a shelf elsewhere that looks perfectly parallel to the floor.

Analysis:

The floor is a true figure looking false, and the shelf may well be a false figure looking true. Only the angle measurements decide either.

Activity:

Draw two parallel lines with a ruler, cross them with twenty short slanting strokes, then check with the ruler that they are still straight and parallel.

Title:

Build an illusion, then measure it away

Safety:

Take care with the protractor point and the pencil tip. No other hazards.

What You Notice:

The gap is unchanged and the alternate angles are equal, so the lines are parallel. The eye said otherwise. That gap between what was seen and what was measured is the whole point of having rules.

Alternate angles can be proved equal by two different routes, through a neighbour or through a corresponding angle, and both give 48 degrees.

Co-interior angles are equal in exactly one case, when both are right angles, since 90 is the only angle equal to its own supplement.

The same illusion that spoils a tiled floor is used deliberately to make a flat wall appear textured.

Opposite sides match, same side adds.

The eye reports; the protractor decides.