Parallel and Intersecting Lines

Why This Matters

Look at a railway track. The two rails run side by side. They never touch, no matter how far you look down the track. Now look at the corner of your notebook. Two edges meet there at a sharp point.

These are two ways lines can behave. They can run alongside each other and never meet. Or they can cross each other.

This is everywhere. The lines on ruled paper. The bars on a window grille. The roads on a city map. The threads in a woven cloth. Builders, carpenters, tailors and map-makers all care a lot about which lines are parallel and which lines cross.

In this chapter we will study these two cases carefully. When lines cross, they make angles. We will find out exactly how big those angles are, and why. The best part is this: once you know just one angle in a picture, you will be able to find every other angle in it. And you will not be guessing. You will know, because you understand the reason.

The Big Idea

When two lines cross, they make four angles, and those angles follow strict rules — angles opposite each other are always equal, and angles next to each other on a straight line always add up to 180°. When a line cuts across two parallel lines, even more pairs of equal angles appear. So from just one known angle, you can work out all the rest. Geometry is not about measuring with a protractor and hoping. It is about reasoning your way to the answer with certainty.

Let’s Break It Down

Before we start, let us refresh what an angle even is, because the whole chapter is built on angles.

Intersecting lines and the angles they make

When two lines cross each other at a point, we say they intersect. The point where they cross is called the point of intersection.

Two straight lines can only ever cross at one point. Think about it: if two straight lines met at two different points, they would have to bend to come back together, and then they would not be straight any more. So two straight lines intersect at exactly one point, or they never meet at all.

When two lines cross, they cut the space around the point into four parts. So they make four angles. Let us name them a, b, c and d, going around the point.

Figure 5.1 below shows two lines crossing, with all four angles marked. Notice the values written in.

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Look closely at Figure 5.1. Two things are happening, and each has a name.

Linear pairs — angles on a straight line add to 180°.

Look at angles a and b. They sit next to each other, and together they stretch all the way along the straight red line. A straight line is 180° (remember the recap). So a and b must add up to 180°.

A pair of angles like this — two angles next to each other that together make a straight line — is called a linear pair. A linear pair always adds up to 180°.

So in Figure 5.1: a + b = 180°. Since a = 120°, that means b = 180° − 120° = 60°. That is exactly the value shown.

The same is true for the other neighbours: b + c = 180°, c + d = 180°, and d + a = 180°.

Vertically opposite angles — opposite angles are equal.

Now look at a and c. They sit across from each other, on opposite sides of the crossing point. Angles like this are called vertically opposite angles. (“Vertically” here just means they share the same vertex, the crossing point — it has nothing to do with up and down.)

Notice in Figure 5.1 that a = 120° and c = 120°. They are equal! And b = 60° and d = 60° — also equal.

This is not a coincidence. Vertically opposite angles are always equal. Here is why, and this reasoning is worth following slowly because it works for any two crossing lines, whatever the angles are.

Figure 5.2 below walks through the reason.

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Let us say it in words, slowly:

• a and b are a linear pair, so a + b = 180°.
• b and c are a linear pair, so b + c = 180°.
• Both of these equal 180°, so a + b = b + c.
• Both sides have a b in them. Take it away from both. You are left with a = c.

We never said what a actually was. The reasoning worked no matter what. That is the power of a proof: it shows the rule holds always, not just for one picture. In maths, this kind of watertight argument is called a proof.

By the way, if all four angles turn out to be equal, each must be 90° (because 90 + 90 + 90 + 90 = 360, a full turn). Lines that cross at 90° are called perpendicular lines. The corner of your notebook is two perpendicular edges.

Parallel lines

Now the other case. Two lines that lie flat on the same surface and never meet, however far you stretch them in both directions, are called parallel lines.

The railway track is the classic example. So are the two long edges of a ruler, or the lines on ruled paper.

There is one important word in that definition: “same surface”. A line drawn on your table and a line drawn on the wall might never meet, but they are not parallel — they are not on the same flat surface. Parallel lines must be flat on the same plane, like two lines drawn on the same sheet of paper.

To show in a drawing that two lines are parallel, we put a small arrow mark (like a >) on each of them. Matching arrows mean “these are parallel”. If there is a second pair of parallel lines in the same figure, we give them double arrows so we do not mix them up.

A transversal

Here is where it gets interesting. What happens when a third line cuts across two other lines?

A line that crosses two (or more) other lines is called a transversal. (“Transversal” just means “crossing over”.)

When a transversal cuts across two lines, it crosses each of them at a point. At each crossing, four angles are made (just like before). Two crossings, four angles each — so a transversal makes eight angles in total.

Figure 5.3 below shows a transversal t cutting across two parallel lines l and m. The little arrows on l and m tell us they are parallel. The eight angles are numbered 1 to 8.

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We will use this exact picture and these exact numbers for the next three sections. The angles between the two lines (3, 4, 5 and 6) are called interior angles. The angles outside the two lines (1, 2, 7 and 8) are called exterior angles. Keep Figure 5.3 in mind as we go.

Corresponding angles

Look at the top crossing and the bottom crossing in Figure 5.3. They look like copies of each other, just slid down the transversal.

Angles that sit in the same position at each crossing are called corresponding angles. For example, angle 2 sits at the top-right of the top crossing. Angle 6 sits at the top-right of the bottom crossing. Same corner, both crossings. So 2 and 6 are corresponding angles.

Figure 5.4 below highlights this pair.

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When the two lines are parallel, corresponding angles are equal. In Figure 5.4, angle 2 equals angle 6.

Why? The crossing at the bottom is just the crossing at the top copied and shifted straight down the transversal — because the lines are parallel, nothing about the angle of the crossing changes. So every angle is copied exactly. Same shape, same angles.

This works the other way round too, and it is very useful: if the corresponding angles are equal, then the two lines must be parallel. This is how you test whether two lines are parallel — check if a transversal makes equal corresponding angles. If it does, they are parallel. If it does not, they are not.

Alternate angles

Now look at the angles between the two lines, but on opposite sides of the transversal. These are called alternate angles (more fully, alternate interior angles, because they are inside the two lines).

In Figure 5.3, angle 3 is between the lines, on the right of the transversal. Angle 5 is between the lines, on the left of the transversal. So 3 and 5 are alternate angles. They make a “Z” shape with the lines.

Figure 5.5 below highlights this pair.

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When the two lines are parallel, alternate angles are equal. So in Figure 5.5, angle 3 = angle 5.

Why is this true? We do not need a new rule — we can get it from the two rules we already have. Watch:

• Angle 3 and angle 7 are corresponding angles, so 3 = 7 (parallel lines).
• Angle 5 and angle 7 are vertically opposite angles, so 5 = 7.
• If 3 = 7 and 5 = 7, then 3 = 5.

So alternate angles being equal is just a result of the two rules we already proved. Nothing new to memorise — it follows. That is the “nothing is a given” idea: every rule here can be traced back to “a straight line is 180°”.

Co-interior angles

One more pair. Look at the two angles that are between the lines and on the same side of the transversal. These are called co-interior angles (sometimes “interior angles on the same side”).

In Figure 5.3, angle 4 is between the lines on the left, and angle 5 is between the lines on the left. Both inside, both on the left side. So 4 and 5 are co-interior angles. They make a “C” shape.

This pair behaves differently from the others. Figure 5.6 below shows it.

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When the two lines are parallel, co-interior angles add up to 180°. So in Figure 5.6, angle 4 + angle 5 = 180°.

Why do these add to 180 instead of being equal? Again, we can show it from rules we know:

• Angle 4 and angle 5… let us connect them through angle 8 (which is at the bottom-left, below line m). Try this simpler route instead:
• Angle 5 and angle 1 are corresponding angles, so 5 = 1.
• Angle 1 and angle 4 are a linear pair (they sit on line l, on the same side of t, making a straight line along l). So 1 + 4 = 180°.
• Replace 1 with 5 (since 5 = 1): 5 + 4 = 180°.

So co-interior angles add to 180°. Once more, it comes straight out of the linear-pair and corresponding-angle rules.

Two angles that add up to 180° have a special name: they are supplementary. So we can say co-interior angles are supplementary.

Here is a table that puts all three transversal pairs side by side, so you can see exactly how they differ.

The “F, Z, C” shape clues are a handy memory trick. Trace the angle pair along the lines: corresponding makes an F, alternate makes a Z, co-interior makes a C.

Finding unknown angles

Now we put it all together. The whole point of these rules is this: if you know one angle, you can find all the others. You just keep applying the rules — vertically opposite (equal), linear pair (add to 180°), corresponding (equal), alternate (equal), co-interior (add to 180°).

Let us start with the simplest case. Figure 5.7 shows two parallel lines cut by a transversal, with one angle given as 70°, and an unknown angle x at the corresponding position below.

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Here x is the corresponding angle to 70°, so x = 70°. Simple. Now let us do a harder one where we chain a few rules together.

We will lead into the worked example with a real layout: a transversal across two parallel lines, where the angle we want is not directly given, so we have to step through it.

See how we never measured anything? Each step used a rule, and each rule we have proved. That is geometry working the way it should.

Common Mistakes

These are the slips students make most often. Read each one — knowing the trap is the best way to avoid it.

Quick Check

Test yourself. Try each before checking the explanation.

Practice Problems

Try each problem yourself first. Then tap to reveal the full solution.

Easy

Medium

Challenge

Summary

Here is everything to carry away from this chapter. You should be able to explain each one in your own words.

• When two lines intersect, they make four angles, and they meet at exactly one point.
• Vertically opposite angles (the ones across from each other) are always equal.
• A linear pair (two angles next to each other on a straight line) always adds up to 180°.
• If all four angles at a crossing are equal, each is 90°, and the lines are perpendicular.
• Parallel lines lie on the same flat surface and never meet, however far you extend them. We mark them with matching arrows.
• A transversal is a line that cuts across two other lines, making eight angles in total.
• When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles add up to 180°.
• If a transversal makes equal corresponding (or alternate) angles, then the two lines are parallel — this is how we test for parallel lines.

What’s Next

You have just seen how careful reasoning lets you find angles with certainty, from a single fact about straight lines. That habit — find a rule, prove it, then use it everywhere — is the heart of all of maths.

Next, in Chapter 6 — Number Play, you will turn that same curiosity onto numbers. You will explore the playful patterns and surprises hidden inside ordinary numbers, and ask the same question we kept asking here: not just what happens, but why.