PrepShorts · Study sheet · Class 7 Mathematics · Chapter 5, Parallel and Intersecting Lines
Chapter 5 · Parallel and Intersecting Lines
The four angles at a crossing: vertically opposite and linear pairs
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Two lines cross and four angles open out. How many different sizes can those four be? Four is the obvious answer and it is wrong.
The idea
Two crossing lines make four angles, but they can only ever make two different sizes — and you can know that without owning a protractor. The section's real move is to swap measuring for arguing: because each neighbour pair lies along a straight line, one angle fixes the other three, and the same three lines of reasoning work whatever that first angle is. That step from "I checked four drawings" to "it cannot come out otherwise" is what the chapter calls a proof, and the page that follows explains why the protractor will keep disagreeing with it anyway.
What you should be able to do
- State how many angles two intersecting lines form, and how many different measures those angles can have
- Given one of the four angles at a crossing, work out the other three by reasoning rather than by measuring
- Identify every linear pair and every pair of vertically opposite angles in a labelled crossing
- Explain why linear pairs add to 180°, from the straight angle
- Reconstruct the argument that vertically opposite angles are equal, using letters rather than a chosen value
- Say what makes that argument a proof and a set of protractor readings not one
- Give two reasons a measured linear pair may fail to total 180° on the page
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| intersect | to meet at a point, said of two lines on one flat surface | printed in bold in §5.1, p.106 |
| plane surface | a flat surface such as a table top or a sheet of paper | printed in bold in §5.1, p.106 |
| angle | the opening between two rays from a shared point, measured in degrees | printed throughout §5.1, pp.106–108 |
| straight angle | the 180° angle a straight line makes at a point on it | printed in §5.1, p.107 |
| adjacent angles | two angles at a crossing that share an arm and sit next to each other | printed in §5.1, p.107 |
| linear pair | a pair of adjacent angles at a crossing; the two together make a straight angle | printed in bold in §5.1, p.107 |
| vertically opposite angles | the two angles of a crossing that face each other across the vertex | printed in bold in §5.1, p.108 |
| proof | a justification that settles a claim by reasoning instead of by measuring | printed in bold in §5.1, p.108 |
| protractor | the instrument for measuring an angle in degrees | printed in §5.1, p.107 and again in "Measurements and Geometry", p.108 |
| measurement errors | readings that are off because the instrument was used badly | printed in "Measurements and Geometry", p.108 |
| vertex | the shared point where the two lines cross | the explanation's word for the crossing point; not printed in this chapter |
Two cautions: First, the chapter never abbreviates vertically opposite angles; keep the full phrase, because the short form invites the wrong picture (see Misconceptions). Second, the chapter names proof here, on p.108, for a three-sentence argument — do not present proof as something reserved for a later class.
Where people slip up
- "Vertically opposite means one is above the other." Fig. 5.3 breaks this on the same page the term is introduced: its vertically opposite pair ∠a and ∠c sit to the left and right of the vertex, level with each other. Checked against p.108. The word points at the vertex, not at the vertical direction — say so out loud when the term first appears.
- "Four angles, so four different sizes." Only two sizes can appear, and they are forced as soon as one of them is fixed. Section 5 should show the fourth value arriving with no new information used.
- "The pattern is true because I measured four pairs and it worked." Activity 1 produces evidence, not a reason. The chapter deliberately puts the letters argument after the measuring, and then names the difference. An explanation that stops at the measuring has taught the opposite of the section.
- "My protractor said 179°, so the rule is only roughly true." The chapter reverses this: the reasoning is exact and the drawing is approximate. The ideal, thickness-free line is a decision about what is being reasoned about, not a fudge.
- "If two of these four are equal, the lines must be perpendicular." Every crossing already has two equal pairs. Equal neighbours is the extra condition, and that is the next topic.
- "Linear pairs add to 180° because the book says so." They add to 180° because their two outer arms together make one straight line, and a straight line makes a straight angle. Show the two arms sweeping into a line.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 1 Q1
Transcript1,308 words
Fold a square sheet in half, and in half again, and then corner to corner. Open it out and look at what the folds left behind. Creases running across the sheet, crossing each other everywhere. Some of them never meet, however far you follow them. Others cross, and where they cross, something happens worth looking at closely. So pick one crossing. Just one. Two lines meeting at a point, and four angles opening out around it.
The question is how those four are related, and whether you need to measure them to find out. You do not, and the reason why is the good part. First, a little care about what is being said. Two lines that meet at a point are said to intersect. And they meet at exactly one point, never at two. Because two straight lines sharing two points would have to be the same line.
There is one more condition, and it is easy to skip past. Both lines have to lie on the same flat surface. A tabletop, a sheet of paper. Two lines in open space can miss each other completely without being parallel. So: one flat surface, and one shared point. That is a crossing. Now count what it makes. Two lines through one point cut the space around it into four.
Four angles, and they go all the way round. Label them going clockwise from the top left. a, b, c and d. Each one is an amount of turning, measured at the point where the lines cross. Together they go once round, so they add to three hundred and sixty. And now the question worth asking. Four angles. How many different sizes can they possibly be? Four looks like the obvious answer, and it is wrong.
It is never four. It is never even three. Draw four crossings, all different, and measure every angle in every one. Sixteen readings. First crossing. Thirty five, a hundred and forty five, thirty five, a hundred and forty five. Second. Sixty two, a hundred and eighteen, and then those two again. Third and fourth. The same two pairs, the other way round. Two values in every crossing, each of them appearing twice.
And in each one, the two that face each other across the point are the equal ones. That is a pattern, and sixteen readings is decent evidence for it. But evidence is not a reason, and the reason is the thing worth having. Take a single crossing, and suppose somebody tells you one angle. Say a is a hundred and twenty degrees. Now look at a and b together. They sit side by side along one straight line.
A straight line makes a straight angle, and a straight angle is a hundred and eighty. So b is a hundred and eighty take away a hundred and twenty. Sixty. Now b and c. Those sit along the other line, so they add to a hundred and eighty too. Which makes c a hundred and twenty. And c with d, along the first line again. d is sixty. All four angles, and only one of them was ever measured.
But look at what that argument actually used, because it is less than you think. It used the hundred and twenty exactly once, right at the start. After that, every single step was subtraction from a hundred and eighty. So run it again, without choosing a number at all. Call the first angle a. Not a value. Just a name for whatever it happens to be. a and b lie along a line, so a plus b is a hundred and eighty.
a and d lie along the other line, so a plus d is a hundred and eighty. Both b and d are a hundred and eighty minus a. So b and d are equal. No number chosen. No angle measured. It cannot come out otherwise. Those neighbouring pairs deserve a name, because they do all the work. Two angles sitting next to each other at a crossing, sharing an arm, are called a linear pair.
There are four of them here. a with b, b with c, c with d, and d with a. And every linear pair adds to a hundred and eighty. Not because somebody decided that it should. Look at what the two outer arms of a linear pair actually are. They point in opposite directions from the crossing, so together they make one straight line. And the turning from one end of a straight line to the other is a hundred and eighty degrees.
That is the whole reason, and it holds at every crossing you will ever draw. Now the other pairing. The two that face each other across the point. a with c. And b with d. These are called vertically opposite angles, and they are always equal. Here is why, in three lines and with nothing measured. b and a make a linear pair, so a is a hundred and eighty minus b.
b and c make a linear pair as well, so c is a hundred and eighty minus b. a and c are both a hundred and eighty minus the very same thing. So a equals c. Whatever b happens to be. Two angles you never touched, forced equal by the angle sitting between them. Stop for a moment, because something changed and it is worth naming. Measuring sixteen angles told you the pattern held sixteen times.
The three lines you just wrote tell you it cannot fail. An argument that settles a claim by reasoning instead of by checking cases is called a proof. And that was one. Three sentences long, using nothing but the straight angle. It also answers the question from earlier. How many different sizes? One angle fixes the other three, so the four are a value and its partner. Two sizes. Unless those two happen to be the same, which is exactly one crossing: the one at ninety degrees.
So the honest statement is at most two, and never, ever three. Now a warning about the name, because it misleads almost everybody. Vertically opposite does not mean one is above the other. Here is the same crossing, drawn as a narrow X, and lettered differently. a is on the left. b is on top. c is on the right. d is at the bottom. a and c are the vertically opposite pair, and they are side by side, level with each other.
The word points at the vertex, the point the lines turn about. It is not about up and down. So take this crossing and list them. Four linear pairs. a with b, b with c, c with d, d with a. And two facing pairs. a with c, and b with d. Four and two, out of the six ways of choosing two from four. Every pair is one or the other.
One last thing, and it is the honest one. Measure a linear pair carefully and you may well get a hundred and seventy nine. That does not make the reasoning roughly true. The reasoning is exact. It is the drawing that is approximate, and there are two reasons for it. One is the instrument. A protractor sitting slightly off the crossing reads slightly wrong. The other is that a drawn line has thickness, and the line in the argument has none.
A ballpoint stroke is about seven tenths of a millimetre wide. Sighted along a fifty millimetre arm, that is eight tenths of a degree. An angle has two arms, so a reading can be out by more than a whole degree. Which is precisely what you saw. So the ideal line is not a fudge. It is a decision about what is being reasoned about, and it stays close enough to build with.
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
- A letter-number stands for any number, not one numberClass 7 · Ch 4, Expressions using Letter-Numbers
Comes up again in
- Perpendicular lines as the case where all four are equalClass 7 · Ch 5, Parallel and Intersecting Lines
- What "parallel" actually claims, and why it is hard to checkClass 7 · Ch 5, Parallel and Intersecting Lines
- A transversal creates two matching sets of four anglesClass 7 · Ch 5, Parallel and Intersecting Lines
- Corresponding angles are equal exactly when the lines are parallelClass 7 · Ch 5, Parallel and Intersecting Lines
- Alternate angles, and interior angles that add to 180°Class 7 · Ch 5, Parallel and Intersecting Lines
- Parallel illusions: why the eye is not a proofClass 7 · Ch 5, Parallel and Intersecting Lines
Either side of this one
- Why a formula says in one line what words take a paragraph to sayClass 7 · Ch 4, Expressions using Letter-Numbers