PrepShorts · Study sheet · Class 7 Mathematics · Chapter 5, Parallel and Intersecting Lines
Chapter 5 · Parallel and Intersecting Lines
Alternate angles, and interior angles that add to 180°
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Nothing new is assumed in this topic, and that is the whole video. Every result is manufactured out of two facts you already own.
The idea
Nothing new is being assumed here. Every result in this section is manufactured out of two facts you already own — corresponding angles across parallels, and vertically opposite angles at a single crossing — by doing them one after the other. Alternate angles are "corresponding, then flip"; same-side interior angles are "corresponding, then linear pair". That is why one composition gives equality and the other gives a total of 180°: the second step is different, so the answer is different. Learning the section as two more rules to memorise hides exactly the thing that makes them provable.
What you should be able to do
- Identify the alternate angle of a named angle in a transversal figure
- Find an alternate angle in two steps — corresponding, then vertically opposite — and say which fact each step used
- State the condition under which alternate angles are equal, and not state it without the condition
- Identify the interior angles on one side of the transversal
- Show that same-side interior angles total 180°, by composing a corresponding angle with a linear pair
- Given one of the eight angles at a pair of parallel lines, find the other seven
- Use unequal corresponding angles to establish that two lines are not parallel
- Handle a figure with two independent pairs of parallel lines, applying the interior-angle result twice
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| alternate angle | the angle on the opposite side of the transversal and between the two lines | printed in bold in §5.8, p.120 |
| interior angles | the pair lying between the two lines, one at each crossing | printed in bold in §5.8, p.122 |
| corresponding angles | two angles, one at each crossing, in matching positions | printed in §5.6, p.115 |
| vertically opposite angles | the two angles of a crossing that face each other across the crossing point | printed in §5.1, p.108 |
| linear pair | a pair of adjacent angles at a crossing, together making a straight angle | printed in §5.1, p.107 |
| transversal | a line cutting across two other lines | printed in §5.5, p.115 |
| justified | the chapter's word for having given a reason rather than a measurement | printed in §5.8, p.120 |
| co-interior angles | the common examination name for the same-side interior pair | an added term; not printed in this chapter, which names the pair simply as interior angles |
| alternate interior angles | the fuller name used in later classes for the pair this section calls alternate angles | an added term; not printed in this chapter |
Which name is examinable. In this book the examinable words are alternate angles and interior angles. Later classes and many question banks say alternate interior angles and co-interior angles for the same two pairs. The explanation may mention the longer names once, as an aside, but must answer in the book's words.
Where people slip up
- "Alternate angles are equal." Only across parallel lines. This is the chapter's most dangerous half-sentence, and the danger is real here: the printed wording inside Activity 6 says the equality holds irrespective of the measure of the starting angle, which is true, but it holds because the corresponding-angle step is available, and that step needs parallelism. The boxed statement on p.120 states the condition properly.
- "Interior angles are equal, like the other pairs." They total 180°. They are equal only in the special case where the transversal meets the parallels at 90°, and then both are 90°. Say that case out loud; a student who has met only perpendicular examples will generalise the wrong way.
- "Alternate means the two angles are next to each other." They sit on opposite sides of the transversal, one at each crossing, both between the two lines. Show a wrong pick beside the right one on Fig. 5.25's lettering.
- "There are four separate rules to memorise here." There are two facts and two compositions. Section 4 and section 9 should be built as the same move with a different second step.
- "Alternate angles are a new fact, so they need a new proof." The chapter derives them and says so — it points out that no measurement was used. That sentence is the point of the section, not a flourish.
- "If two angles at a transversal are supplementary, the lines must be parallel." Only for the interior pair on one side. Any linear pair at either crossing is already supplementary whatever the lines do.
- "Example 4 needs a rule about parallelograms." It does not, and the chapter does not use one. It applies the interior-angle result twice, once to each pair of parallel sides.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 4 Q1, Figure it Out · 4 Q2, Figure it Out · 4 Q3, Figure it Out · 4 Q4, Figure it Out · 4 Q5, Figure it Out · 4 Q6
Transcript1,359 words
Here is a promise about this one, and I intend to keep it. Nothing new is going to be assumed. Not one thing. Everything that follows is built out of two facts you already have. The first: corresponding angles agree, across a pair of parallel lines. The second: at any single crossing, the two angles facing each other are equal. That is the whole toolkit, and both tools are already yours.
What is new is doing them one after the other, in a particular order. Two facts and two ways of chaining them. Learn it as four rules and you have hidden the good part. So. Two lines, and a third one cutting across both of them. That gives eight angles, and I am going to letter them. a, b, c and d at the upper crossing, going round. e, f, g and h at the lower one, the same way round.
One more thing before we start, and I want to be open about it. I am marking these two lines as parallel, with the little arrows. Everything that follows leans on that, so it belongs on the board where you can see it. Now look at which of the eight sit between the two lines. c and d up here, e and f down there. Those four in the middle are the interior angles, and they are where the action is.
Among them there is a pairing worth naming. Take one from each crossing, on opposite sides of the cutting line. d up here, and f down there. Those two are alternate angles. The other alternate pair is c with e. Same rule, other side. Now, alternate does not mean next to each other, and that trips people constantly. c and d are next to each other, and they are not an alternate pair at all.
They share a crossing, and an alternate pair never does. Here is the question this whole section is answering. You know one of the eight. How do you reach its alternate angle? Not by looking it up in a list. By walking there, in two hops. Start at f. Hop one goes to the angle that corresponds to it. That is b, at the other crossing, in the matching position.
And corresponding angles agree across parallels, so b is f. Hop two goes from b to the angle facing it across its own crossing. That is d. Angles facing each other are equal, so d is b. Let us put a number in and watch it travel. Say f is a hundred and twenty degrees. Hop one. b corresponds to f, so b is a hundred and twenty. Hop two. d faces b across the crossing, so d is a hundred and twenty as well.
And d was the alternate angle we set out to find. Now notice what did not happen anywhere in that. Nobody put a protractor on d and read it off. Nothing was measured. The number arrived by two steps, and each step was a fact you already had. So take the number away and run the same walk again. Whatever f happens to be, b is the same, because they correspond.
Whatever b is, d is the same, because they face each other. So d is f. Always. For any starting value at all. Which is a result, and it needs saying carefully. Alternate angles are equal when the two lines are parallel. Not the first half on its own. The condition travels with it, every time. Because hop one is the step that needed parallel lines. Take that away and the walk has no first step.
Here is what that buys you, and it is a lot. One angle decides all eight of them. New figure, numbered this time. One to four at the upper crossing, five to eight below. And you are told exactly one thing. Angle six is a hundred and thirty five degrees. Its corresponding angle at the other crossing is a hundred and thirty five. The angle facing it is a hundred and thirty five. So is its alternate angle.
That is four of them settled, all at a hundred and thirty five. The other four are what is left of a straight line. Forty five each, and the two values add to a hundred and eighty. Now run the whole thing backwards, which is where it really earns its keep. Two lines and a cutting line, and this time nothing tells you they are parallel. You are told two measures. This one is a hundred and twenty, and this one is seventy.
The hundred and twenty sits along a straight line with its neighbour. So the neighbour is sixty. And that neighbour is the corresponding partner of the seventy. So if these two lines were parallel, the seventy would have to be a sixty. It is not. So they are not parallel, and they are ten degrees off it. You never measured the gap between the lines. You caught a contradiction instead.
Now keep the first hop and change the second one, and watch the answer change with it. Same picture, parallel again. This angle down here is fifty degrees. Its neighbour along the line makes a straight angle with it, so the neighbour is a hundred and thirty. And that neighbour has a corresponding partner at the other crossing. So that partner is a hundred and thirty as well. Now put the fifty and the hundred and thirty side by side, and hold on to them.
Try the same walk starting from seventy five. You end up at a hundred and five. Once more, starting from a hundred and ten. You end up at seventy. Notice anything? Every one of those pairs adds to a hundred and eighty. And in each case the two angles are both between the lines, on the same side of the cutting line. That is the other pairing this section names, and that is what it does.
Interior angles on the same side add to a hundred and eighty. Now look at how we got there, because it is almost the same walk as before. Same first move. A different second move. A different answer. Facing across a crossing gave equality. Lying along a straight line gives a hundred and eighty. So do not expect this pair to be equal. They only ever are when the cutting line comes in square, and then they are ninety and ninety.
One last figure, because it shows the result doing real work. A four sided shape with corners A, B, C and D, and a line drawn from A across to C. Two pairs of parallel sides here, both marked. And you are given two angles. The corner at D is sixty degrees, and that diagonal cuts off sixty five at A. Now, the side from A to D is a cutting line across one parallel pair.
So D and the whole corner at A are interior angles on the same side. They add to a hundred and eighty. That puts the corner at A at a hundred and twenty, which leaves fifty five beside the diagonal. Use the same result on the other parallel pair and C is a hundred and twenty, B is sixty, and the four corners close at three hundred and sixty. So here is the map, and it is smaller than it looked.
Two facts at the bottom. Corresponding angles agree across a parallel pair. And angles facing each other at one crossing are always equal, parallel or not. Chain the first with a flip across the crossing, and you get alternate angles. Equal. Chain it with a straight line instead, and you get the same side interior pair. A hundred and eighty. Same start, different second step, different answer. That is the entire section.
You will meet longer names for those two pairs later on. They are these same pairs. And one last thing, because it is the most dangerous half sentence in the topic. Alternate angles are equal across parallel lines. Never let that second half travel on its own.
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
- Corresponding angles are equal exactly when the lines are parallelClass 7 · Ch 5, Parallel and Intersecting Lines
- The four angles at a crossing: vertically opposite and linear pairsClass 7 · Ch 5, Parallel and Intersecting Lines
- A transversal creates two matching sets of four anglesClass 7 · Ch 5, Parallel and Intersecting Lines
Comes up again in
- Constructing from two angles and the side between themClass 7 · Ch 7, A Tale of Three Intersecting Lines
Either side of this one
- Drawing a parallel line using the corresponding-angle testClass 7 · Ch 5, Parallel and Intersecting Lines
- Parallel illusions: why the eye is not a proofClass 7 · Ch 5, Parallel and Intersecting Lines