PrepShorts · Study sheet · Class 7 Mathematics · Chapter 5, Parallel and Intersecting Lines
Chapter 5 · Parallel and Intersecting Lines
Perpendicular lines as the case where all four are equal
This video could not be loaded. Reload the page to try again.
Sign in with Google9 min.
Keep your place in this chapter — sign in, it’s free.Sign in
Perpendicular is not a fact about which way two lines point. It is the one crossing where the four angles stop coming in two sizes.
The idea
Perpendicular is not a fact about which way two lines point. It is the one crossing where the four angles stop coming in two sizes and collapse into one — and the moment you demand that, the value 90° is forced on you rather than chosen. That is why the chapter opens the section with a question instead of a definition: 90° is the answer to "can a crossing be perfectly even?", not a convention someone agreed to. And the figure it answers with is drawn tilted, so that horizontal-and-vertical is exposed as one instance of perpendicularity rather than the meaning of it.
What you should be able to do
- State the condition that makes two intersecting lines perpendicular
- Derive 90° from "all four angles equal" rather than assuming it
- Show that "all four equal" and "one of them is a right angle" are the same condition, using the linear pair
- Recognise perpendicular lines when neither of them is horizontal or vertical
- Mark perpendicularity on a diagram with the square corner symbol
- Name two ways of producing a perpendicular without a protractor — a set square and a paper fold
- Explain why perpendicularity is a relation between two lines and not a property of one
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| perpendicular lines | two lines that cross at right angles | printed in bold in §5.2, p.109 |
| right angle | an angle of 90°, a quarter of a full turn | printed in §5.2, p.109 |
| linear pair | a pair of adjacent angles at a crossing, together making a straight angle | printed in §5.1, p.107 |
| square angle | the book's phrase for the small square drawn between two perpendicular lines | printed in "Notations", p.112 |
| set square | the right-angled drawing triangle from a geometry box | printed in §5.7, p.118 |
| adjacent edges | the two edges of a sheet that meet at a corner, used as the first example of perpendicularity | printed in §5.4, p.111 |
| crease | the line a fold leaves in a sheet of paper | printed in "Making Parallel Lines through Paper Folding", p.119 |
| foot of the perpendicular | the point where a perpendicular meets the line it is drawn to | the explanation's phrase; not printed in this chapter |
A note on the phrase square angle. It is the chapter's own wording for the corner mark, printed on p.112; it is not a name for a 90° angle, which the chapter calls a right angle. Keep the two apart when explaining it.
Where people slip up
- "Perpendicular means one line across and one line up." Fig. 5.4 is drawn tilted precisely to kill this. Checked against p.109. Nothing about the definition mentions the page's edges.
- "You need to measure 90° to make a perpendicular." A set square carries a right angle in its own shape (§5.7, p.118), and a fold that brings a line onto itself produces one with no instrument at all (p.119).
- "Perpendicular is something one line is." It is a relation between a pair. Say "perpendicular to", always, and never leave the second line unnamed.
- "All four equal is one way to be perpendicular; having a 90° angle is another." They are the same condition, and the linear pair is what makes them the same: if one angle is 90°, its neighbour is 180° − 90° = 90°, and the vertically opposite pair follows. Section 7 should run this both ways.
- "90° is just the number someone picked for a right angle." Within this section the number is forced: nothing else divides 180 into two equal parts. The degree itself is a convention; the 90 is not, once the degree is fixed.
- "Two perpendicular lines cross at more than the one point." They meet exactly once, like any pair of intersecting lines — the chapter asks this directly on p.107.
Ask your teacher a person
Your teacher reads this and writes back, usually within a day. For an instant answer, use Ask the video in the sidebar.
Your class sees the question and the answer. Only your teacher sees that it was you.
No questions on this topic yet.
Worked answers to this chapter’s exercises · this video explains Figure it Out · 2 Q1
Transcript1,344 words
Two lines crossing. Four angles around the point. You already know they come in two sizes, a value and its partner. So here is a question before any names get handed out. Can you draw a crossing where all four are the same? Not nearly the same. Exactly. Try it. Tilt one line a little, and two angles grow while the other two shrink. Tilt it back and they swap over.
It feels like there might be one position where they balance. And if there is, the next question is what each angle measures there. Before drawing anything, notice how little freedom you actually have. Two angles sitting side by side at a crossing lie along one straight line. So they add to a hundred and eighty. Always. That single fact does most of the work here. Call the four angles a, b, c and d going round.
a and b add to a hundred and eighty. So do b and c. You are not choosing four numbers. You are choosing one. The other three follow, whether you want them to or not. Which means the balanced crossing, if it exists, is already decided. So demand it. Say all four angles are equal, and see what that costs. Take just two of them, a and b, sitting next to each other.
They add to a hundred and eighty, because they lie along a line. And they are equal, because that is what we demanded. Two equal things adding to a hundred and eighty. Write it with one letter. x plus x is a hundred and eighty. Two x is a hundred and eighty, so x is ninety. Nothing was chosen there. Ninety is the only number that works. No other value splits a hundred and eighty into two equal halves.
Here is a second way to the same number, and this one is mine rather than the standard route. Forget the straight line for a moment. Look at the whole turn. The four angles at a crossing go once round the point. Once round is three hundred and sixty degrees. If all four are equal, then four of them make three hundred and sixty. Four x is three hundred and sixty, so x is ninety.
Same answer, and it never mentioned a straight line at all. Two independent routes landing on the same value is worth noticing. It means ninety was not a decision anybody made. It was forced. So the balanced crossing does exist, and there is exactly one of them. Every angle is ninety degrees. An angle of ninety degrees is called a right angle. It is a quarter of a full turn, which is a useful way to hold on to it.
And two lines that cross at right angles are called perpendicular. Perpendicular to each other. The phrase always needs two lines. Notice the order that just happened, because it matters. The question came first, then the number the question forced, and only then the name. Ninety is not a convention. It is an answer. Now the part almost everybody gets wrong, and it is worth slowing down for. Ask someone to draw two perpendicular lines and you will get one across and one up.
Like the corner of a page, or the corner of a window. But look again at everything the argument used. It used one fact, that neighbours add to a hundred and eighty. Nothing in it mentioned the edges of the page, or up, or across. So take this perpendicular crossing and turn it. Watch the four arcs. Keep turning. All the way round, and every angle stays at ninety. One across and one up is just one position out of all of them.
Two sentences get used for this, and they sound like different tests. All four angles are equal. Or, one of the angles is ninety. They are not two tests. They are the same one, and here is why. Suppose you are told just one angle is ninety. Only one. Its neighbour lies along a line with it, so the neighbour is a hundred and eighty take away ninety. Which is ninety. And the same argument gets the other two.
Now run it the other way. Suppose all four are equal. Then two neighbours are equal and add to a hundred and eighty, so each is ninety. One condition, two wordings. Checking either one is checking both. Because that is so useful, it gets its own mark on a drawing. A small square, drawn in the corner where the two lines meet. It sits inside one of the four angles, tucked against the crossing point.
And it says exactly one thing. This angle is ninety degrees. One mark is enough for the whole crossing, since one ninety forces the rest. Careful with the words here, because two of them look alike. The angle is a right angle. The small square is just the mark for it. Drawing the square does not make anything true. It records a thing you have already checked, so the next reader does not have to.
Which raises a fair question. How do you make one without measuring? You do not need a protractor at all, and here is the first way. A set square is the triangle in a geometry box, and it has a right angle built into its shape. You are not measuring ninety degrees. You are borrowing one that already exists. Lay a ruler along your line, and stand the set square against the ruler's edge.
Draw along the upright side. That line is perpendicular to your first one. Now slide the set square along the ruler, without lifting or turning it, and draw again. Sliding moves both ends of that edge the same distance, so its direction cannot change. Two perpendiculars to the same line, and nothing was measured. The second way needs no instrument whatsoever. Only paper. Draw a line on a sheet, anywhere, at any angle.
Now fold the sheet so the line lands exactly back on top of itself. Press the fold flat and open it out. The crease you just made is perpendicular to your line. It has to be. A fold flips one side of the crease onto the other. The only fold that lays a line back onto itself, apart from folding along it, is one that crosses it square. And now fold once more, laying the crease onto itself, to get a third line.
That third one never meets your first, no matter how far you follow either of them. One more thing, and it is about reading a question rather than answering one. You will meet figures that hand you several facts at once. Three lines running in the same direction, a fourth line cutting across all three, and one angle marked. And somewhere in the corner, a small square saying two of the lines are perpendicular.
The question asks for two of the unmarked angles. The cutting line meets all three in exactly the same way, so the marked angle repeats. And its neighbour is a hundred and eighty take away it. Both answers, and the small square in the corner never came into it once. A figure can hand you more than the answer needs, and spotting which fact did the work is half of reading one.
So, two things the word does not mean. It is not something a single line can be. A line on its own is not perpendicular, any more than a person on their own is a neighbour. It is a relation between two, so always say perpendicular to something. And it does not mean steep enough, or close enough. Here are two crossings, both tilted, and they look much the same.
In the first, the four angles are ninety, ninety, ninety and ninety. In the second, they are eighty five and ninety five, twice each. Five degrees out. Your eye cannot tell those apart, and the four angles can, which is exactly why the condition is worth stating.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- The four angles at a crossing: vertically opposite and linear pairsClass 7 · Ch 5, Parallel and Intersecting Lines
Comes up again in
- What "parallel" actually claims, and why it is hard to checkClass 7 · Ch 5, Parallel and Intersecting Lines
- Drawing a parallel line using the corresponding-angle testClass 7 · Ch 5, Parallel and Intersecting Lines
- What an altitude is, and how to construct oneClass 7 · Ch 7, A Tale of Three Intersecting Lines