Parallel lines
Two straight avenues run side by side across a town map. How can you be sure they will never meet, however far the map goes?
Where you are 8 in Angle types and pairs
Why this exists
Peadar is drawing a map of his town. The avenues run side by side from one end of town to the other. Clodagh writes on lined paper, where the printed lines run across the page side by side. Cian copies pictures onto grid paper.
All three keep meeting the same kind of pair: two straight lines that run side by side and never cross. The rails of a straight railroad track are like that, and so are the top and bottom edges of a door.
The last two lessons were about lines that cross: kinds of angles and perpendicular lines, which cross at a square corner. This lesson is about lines that never cross at all.
The lesson takes two days. Part 1 is the meaning, the symbol and the marks, on a grid and in everyday things. Part 2 is telling parallel from not parallel: by the gap, by grid steps, and in rectangles, segments and rays.
Definition
Part 1: parallel lines. Think of a flat surface, like a sheet of paper, that goes on forever in every direction. Two lines on the same flat surface are parallel when they never meet, however far they go.
In symbols, \ell \parallel m, read “\ell is parallel to m”. The symbol is two upright strokes side by side, a picture of the idea. With a slash, \ell \nparallel m means “\ell is not parallel to m”.
On a drawing, parallel lines carry small matching arrow marks partway along, as in Figure 1. Lines with one arrow mark are parallel to each other. A second set of parallel lines gets two arrow marks each, so the sets cannot be mixed up. The arrowheads at the very ends still mean what they meant in points, lines, rays and segments: the line keeps going.
Segments and rays are parallel when the lines they lie on are parallel. So \overline{AB} \parallel \overline{CD} says that the line through A and B and the line through C and D never meet.
Part 2: two tests. The gap between two lines at a place is measured straight across, along a segment that meets the bottom line at a square corner, like the dashed segments in Figure 1. On a square grid, a line’s steps are a move along the grid from one point on the line to another, such as “3 right and 2 up”.
Intuition
Picture a railroad track. A train’s wheels sit on both rails at once, and the axle between them has one fixed length. So the rails must stay the same distance apart the whole way, and that is what keeps them from meeting. Parallel lines are the same distance apart everywhere.
Now picture two lines a little closer at the left than at the right. Walk left along them and the gap shrinks, the same amount for every step, because both lines are straight. Sooner or later it shrinks to nothing, and there the lines cross, perhaps far off the page. So two straight lines are either the same distance apart all the way, or they meet somewhere.
On grid paper there is an easy way to keep the gap the same: make both lines take the same steps. If both go 3 right and 2 up, then each time both have moved 3 squares right, both have climbed 2. Neither one gains on the other.
Results
Same distance apart. Two parallel lines are the same distance apart everywhere. Two lines that are not parallel get closer and closer in one direction, and cross.
Same steps on a grid. On a square grid, two different lines that take the same steps are parallel. The same moves made bigger count as the same steps: “6 right and 4 up” is “3 right and 2 up” done twice. So do the same moves backward: “3 left and 2 down”. Two lines with different steps cross somewhere.
Two square corners. On a flat surface, two lines that are both perpendicular to the same line are parallel to each other.
Meet once, or never. On one flat surface, two different lines either cross at exactly one point or are parallel. There is no third choice.
Why it is true
Same steps. Take two lines that both go “3 right and 2 up”, and look at the gap straight up the grid from one to the other. Each time you move 3 squares right, both lines climb 2, so the gap does not change. Moving left, both drop 2, and again it does not change. A gap that never changes never reaches zero, so the lines never meet.
With different steps, one line climbs faster. The gap changes by the same amount for every move, growing one way and shrinking the other, and going the shrinking way it reaches zero.
Two square corners. This is an explanation more than a full proof, and it needs one flat surface. Lines a and b both make a square corner with line \ell. Slide a along \ell, without turning it, until it reaches b. It still makes a square corner with \ell, and only one line can do that at that point (perpendicular lines showed why), so a lands exactly on b. Sliding without turning moves every point of a the same distance, so every point of a was the same distance from b. That gap is never zero, so a and b never meet.
Meet once, or never. Only one line goes through two points, so two different lines cannot cross twice. They meet once or not at all, and “not at all” on one flat surface is what parallel means.
Worked examples
Examples 1 to 3 go with Part 1, and Examples 4 to 7 with Part 2.
Lines on writing paper
Clodagh’s notebook page (Figure 2) has three printed lines a, b and c running across it, and a margin line m running down it, with a square corner marked at each crossing. Name every pair of the four lines that is parallel and every pair that is perpendicular, using \parallel and \perp.
The printed lines carry matching arrow marks, and even if the page went on forever they would never meet. So each pair of them is parallel. Pair them in order so none is missed: a with b and with c, then b with c.
That is 2 + 1 = 3 parallel pairs.
The margin line crosses every printed line, so it is parallel to none of them: m \nparallel a, for example. The square corners say it crosses each one at a right angle:
That is 3 perpendicular pairs.
Check. Four lines make 3 + 2 + 1 = 6 pairs in all, and 3 parallel plus 3 perpendicular is 6. Every pair has been sorted.
Reading the marks on a map
Peadar’s town map (Figure 3) shows three avenues f, g and h, two streets s and t, and a road r. Which lines does the map say are parallel? Does it say that the avenues are perpendicular to the streets?
f, g and h each carry one arrow mark, so f \parallel g, f \parallel h and g \parallel h. s and t each carry two arrow marks, so s \parallel t.
The two kinds of mark are different on purpose. They do not say f \parallel s, and the map shows f and s crossing, so f \nparallel s. Road r has no marks, and it crosses every avenue and both streets, so it is parallel to none of them.
The avenues look square to the streets, but the map has no square corner mark. Arrow marks say “never meet”; only a square corner mark says “perpendicular”. So the map does not tell us f \perp s.
Drawing a parallel line on a grid
On Cian’s grid paper (Figure 4), line k passes through grid points A and B, where B is 3 right and 2 up from A. Point P is 3 squares straight up from A. Draw the line through P parallel to k, and show that it stays the same distance from k.
The new line has to take the same steps as k. So from P, count 3 right and 2 up, mark that grid point Q, and draw the line n through P and Q.
Check the gap straight up the grid. At A’s column it is 3 squares, since that is where P was put. At B’s column, B is 2 squares above A’s row and Q is 3 + 2 = 5 above it, so the gap is
again. Every further 3 right adds 2 to both lines, and every 3 left takes 2 from both, so the gap stays 3. It never reaches zero, so n \parallel k.
Measuring the gap
Clodagh draws two pairs of straight lines. She makes three marks 5 centimeters apart along the bottom line of each pair, and at each mark measures the gap straight up, at a square corner, to the top line.
| first mark | second mark | third mark | |
|---|---|---|---|
| pair 1 | 3 cm | 3 cm | 3 cm |
| pair 2 | 4 cm | 3 cm | 2 cm |
Which pair is parallel? For the other pair, where do the lines meet?
SolutionPair 1 is the same distance apart at every mark, so its lines are parallel.
Pair 2 is not. Going right, its gap shrinks by 1 cm for every 5 cm along the bottom line, and because both lines are straight it keeps shrinking at that rate. At the third mark the gap is 2 cm, so it takes 2 more shrinks of 1 cm to reach zero:
The lines of pair 2 meet 10 cm to the right of the third mark. Going left their gap grows instead: 5 cm to the left of the first mark it would be 5 cm.
Same steps or different steps
In Figure 5, line p passes through O with steps of 2 right and 1 up. Line q passes through B, 2 squares straight up from O, with steps of 1 right and 1 up. Line r passes through C, 2 squares straight down from O, with steps of 2 right and 1 up. Which line is parallel to p? Where does the other one meet p?
Line r takes the same steps as p. At O’s column the gap from r up to p is 2 squares, and every 2 right adds 1 to both, so the gap stays 2. So r \parallel p.
Line q takes different steps. For every 2 squares right, q climbs 2 but p climbs only 1, so the gap from p up to q grows by 1. Going left it shrinks by 1 for every 2 squares. At O’s column it is 2, so it is gone after
Check both lines there. From O, two steps of 2 left and 1 down reach the point 4 left and 2 down from O. From B, four steps of 1 left and 1 down drop 4 squares; B started 2 above O’s row, so q ends 2 below it, at the same point M. So q meets p at M, and q \nparallel p.
A block on the map
One block on Peadar’s map is the rectangle ABCD in Figure 6, with a square corner at each of its four corners. Explain why \overline{AB} \parallel \overline{DC} and \overline{AD} \parallel \overline{BC}. Then name every pair of sides that is perpendicular.
Sides \overline{AD} and \overline{BC} both make a square corner with \overline{AB}, at A and at B. Two lines perpendicular to the same line are parallel, so \overline{AD} \parallel \overline{BC}. In the same way, \overline{AB} and \overline{DC} both make a square corner with \overline{BC}, so \overline{AB} \parallel \overline{DC}.
Neighboring sides meet at square corners, so there are four perpendicular pairs: \overline{AB} \perp \overline{BC}, \overline{BC} \perp \overline{CD}, \overline{CD} \perp \overline{DA} and \overline{DA} \perp \overline{AB}.
Check. Four sides make 3 + 2 + 1 = 6 pairs, and 2 parallel plus 4 perpendicular is 6.
Segments that do not touch
On Cian’s grid (Figure 7), segment \overline{EF} goes from E, 3 right and 1 up. Segment \overline{GH} starts 3 squares straight up from E and goes 3 right and 0 up. Ray \overrightarrow{JK} starts at J, 3 right and 2 down from E, and passes through K, 3 right and 1 up from J. None of the three touch. Is \overline{EF} \parallel \overline{GH}? Is \overrightarrow{JK} \parallel \overline{EF}?
Being parallel is about the lines, not the pieces drawn, so look at the lines through them.
The line through G and H goes straight across, and the line through E and F climbs 1 for every 3 right. At E’s column the gap between them is 3 squares; at F’s column it is 3 - 1 = 2. It shrinks by 1 for every 3 squares right, so it reaches zero after
at the point M, 9 right and 3 up from E. Extended, as the dashed lines show, the two meet there. So \overline{EF} \nparallel \overline{GH}, even though the segments never touch.
The ray takes the same steps as \overline{EF}. J is in F’s column, 2 squares below E’s row, and F is 1 above it, so the gap from the ray up to the line through E and F is 2 + 1 = 3 squares, and the same steps keep it there. So \overrightarrow{JK} \parallel \overline{EF}.
Boundaries
One flat surface. Parallel is about lines on the same flat surface. In space there is a third kind of pair. At a highway overpass, a straight road on the bridge runs east and a straight road underneath runs north. They never meet, and they are not parallel either: they point different ways, and no flat surface holds both. Such lines are called skew; lines and planes in space takes them up.
Not touching is not enough for pieces. Two segments that do not touch can lie on lines that cross, as in Example 7. Only the whole lines decide.
Two different lines. A line drawn on top of itself meets itself everywhere, so this lesson does not count it as parallel to itself. Some books choose the other way and do count it; both choices are in use.
A ruler cannot prove it. A ruler reads to about a millimeter. Two lines 3 cm apart at one end and a hair less at the other are not parallel; they meet very far away. Lines are known to be parallel only when it is given, marked, or shown from something exact, such as the same grid steps or two square corners.
Where this goes next
Four-sided shapes are sorted by their parallel sides: parallelograms, rectangles, rhombuses and squares says what each kind guarantees. When a third line crosses two parallel lines, the angles it makes match up in pairs, in parallel lines and transversals; using angles to show two lines are parallel is proving lines parallel.
The “same steps” test later becomes a test with numbers on a coordinate grid, in writing a parallel or perpendicular line through a point. And the fact that through a point off a line there is exactly one parallel line turns out to be one of the starting rules of geometry, the parallel postulate.
Exercises
Part 1: the symbol and the marks
- Write “line p is parallel to line q” in symbols. Then write “line p is not parallel to line s”.
- Clodagh’s next page has 5 printed lines running across it, all parallel. How many pairs of parallel lines is that?
- Which of these are models of parallel lines? (a) the two rails of a straight railroad track, (b) the two hands of a clock at 3 o’clock, (c) the top and bottom edges of a door, (d) the two long edges of a ruler.
- In Figure 3, name two lines that are not parallel, and say how you can tell.
- A drawing marks \overline{AB} and \overline{CD} with one arrow mark each, and \overline{EF} and \overline{GH} with two arrow marks each. Which pairs does the drawing say are parallel? Does it say \overline{AB} \parallel \overline{EF}?
Part 1: on a grid and in letters
- On a grid, line k passes through A and B, where B is 4 right and 1 up from A. Point P is not on k. Which of these moves from P reach a point Q with \overleftrightarrow{PQ} \parallel k? (a) 4 right and 1 up, (b) 1 right and 4 up, (c) 4 left and 1 down, (d) 8 right and 2 up.
- In Figure 4, point S is 2 squares straight down from A. From S, move 3 right and 2 up to a point T. Where is T compared with B? Is \overleftrightarrow{ST} \parallel k?
- In Figure 8, which letters have a pair of parallel strokes? How many pairs does each have?
- In Figure 8, which letters have a pair of perpendicular strokes? How many pairs does each have?
Part 2: measuring the gap
In Exercises 10 to 12, the gap between two straight lines is measured straight up from the bottom line, at marks along it, from left to right.
- The marks are 4 cm apart and the gaps are 5 cm, 4 cm and 3 cm. Are the lines parallel? If not, how far to the right of the last mark do they meet?
- The marks are 2 cm apart and the gaps are 2 cm, 3 cm and 4 cm. Which way from the marks do the lines meet, and how far from the first mark?
- The gap is 7 cm at one mark and 7 cm at another mark 10 cm away. Are the lines parallel? Explain.
Part 2: steps on a grid
- Line p takes steps of 2 right and 3 up. Lines q, r and s are three other lines, none of them on top of p, with steps of 2 right and 3 up, 3 right and 2 up, and 4 right and 6 up. Which of them are parallel to p?
- Line p passes through O with steps of 1 right and 2 up. Line q passes through the point 3 squares straight up from O, with steps of 1 right and 1 up. Do they meet? If so, where?
- Line p passes through O with steps of 3 right and 1 up. Line q passes through the point 2 squares straight up from O, with steps of 3 right and 1 down. Where do they meet?
Part 2: shapes, segments and rays
- Cian’s garden is an L shape with 6 straight sides and a square corner at every corner. Three sides run east to west and three run north to south. How many pairs of sides are parallel? How many pairs lie on perpendicular lines, whether or not they touch?
- On a grid, \overline{UV} goes from U, 2 right and 1 up. Point W is 5 squares to the right of U, and \overline{WX} goes from W, 2 right and 1 up. Is \overline{UV} \parallel \overline{WX}?
- On a grid, \overline{OA} goes from O, 4 right and 0 up. \overline{BC} starts at B, 2 squares straight up from O, and goes 4 right and 1 up. The segments do not touch. Are they parallel? If not, where do their lines meet?
Reasoning
- Explain why two different lines on a flat surface cannot meet at two points.
- Peadar says that two segments that do not touch must be parallel. Show that he is wrong.
- Two lines both take steps of 5 right and 1 up, and one starts 4 squares straight above the other. Explain why they never meet.
- On a shoebox, take the edge along the top of the front face and the edge along the bottom of the right face. Do the lines through them meet? Are they parallel? Why does this not break the rule “meet once, or never”?
- Line a is 1 cm above line b everywhere on Clodagh’s page, and line c is 2 cm below line b everywhere. How far apart are a and c? Are they parallel?
- Can two lines be parallel and perpendicular at the same time? Explain.
- In Example 4, the lines of pair 2 meet 10 cm to the right of the third mark. Clodagh’s paper ends 8 cm to the right of it, so on her paper the lines never touch. Are they parallel?
Answers
- p \parallel q and p \nparallel s.
- 10 pairs. The top line pairs with the 4 below it, the next with 3, and so on: 4 + 3 + 2 + 1 = 10.
- (a), (c) and (d): each pair stays the same distance apart all along. The clock hands meet at the center, so they are not parallel.
- For example g and t: they cross on the map, and lines that meet are not parallel. Any avenue with any street, or r with any other line, also works.
- \overline{AB} \parallel \overline{CD} and \overline{EF} \parallel \overline{GH}. No: the two kinds of mark say nothing about \overline{AB} and \overline{EF} together.
- (a), (c) and (d). (a) is the same steps, (c) is the same steps backward, and (d) is 4 right and 1 up done twice. Move (b) climbs 4 for every 1 right, so its line would cross k.
- T is 2 squares straight below B, and yes, \overleftrightarrow{ST} \parallel k. The same steps that take A to B take S to T, so the gap is 2 squares at both columns and stays there.
- H has 1 pair (its uprights), E has 3 (its three arms, taken two at a time), Z has 1 (top and bottom) and N has 1 (its uprights). In V and T the strokes meet, so there are none.
- H has 2 pairs (the crossbar with each upright), E has 3 (the upright with each arm) and T has 1. Z, N and V have none: their slanted strokes meet nothing at a square corner.
- Not parallel; they meet 12 cm to the right of the last mark. The gap shrinks 1 cm for every 4 cm, and 3 cm is left: 3 \times 4 = 12.
- To the left, 4 cm from the first mark. Going left the gap shrinks by 1 cm for every 2 cm, and at the first mark it is 2 cm: 2 \times 2 = 4.
- Yes. Between straight lines the gap changes by the same amount for every centimeter along. Here it changed by 0 over 10 cm, so it changes by 0 everywhere and never reaches zero.
- q and s. Line q takes the same steps, and s’s steps are p’s done twice. Line r goes 3 right for 2 up instead of 2 right for 3 up, so it crosses p.
- Yes, at the point 3 right and 6 up from O. Line q starts 3 above p. For each square right, p climbs 2 and q climbs 1, so the gap shrinks by 1 and is gone after 3 squares. There p has climbed 3 \times 2 = 6, and q is at 3 + 3 = 6.
- At the point 3 right and 1 up from O. For each 3 squares right, p climbs 1 and q drops 1, so the gap of 2 is gone after one move. There p is 1 up, and q is 2 - 1 = 1 up.
- 6 parallel pairs and 9 perpendicular pairs. The three east to west sides make 2 + 1 = 3 parallel pairs, and the three north to south sides another 3. Every east to west side is perpendicular to every north to south side: 3 \times 3 = 9. Check: 6 sides make 5 + 4 + 3 + 2 + 1 = 15 pairs, and 6 + 9 = 15.
- Yes. Both take steps of 2 right and 1 up, and they are different lines: by W’s column the line through U and V has climbed more than 2 squares, but W is in U’s row.
- Not parallel; their lines meet 8 squares to the left of O. The line through B and C starts 2 above the line through O and A and, going left, drops 1 for every 4 squares. Dropping 2 takes 2 \times 4 = 8 squares.
- Two lines meeting at two points would both go through those two points. Only one line does, so they would be the same line.
- For example, on a grid, one segment goes from O 2 right, and another starts 3 squares right of O and goes 1 right and 1 up. They never touch, but the line through the first passes through the start of the second, so the lines meet there and are not parallel. Example 7 is another case.
- Every 5 squares right both lines climb 1, and every 5 left both drop 1, so the gap straight up stays 4 squares. It is never zero, so they never meet.
- They never meet (one is at the top of the box, the other at the bottom), and they are not parallel, since one runs left to right and the other front to back. The rule is about two lines on one flat surface, and no flat surface holds both of these. They are skew lines.
- 3 cm apart everywhere, so a \parallel c. Wherever you measure, a is 1 cm above b and c is 2 cm below it, so the gap is 1 + 2 = 3 cm, which is never zero.
- No. Perpendicular lines cross at a square corner, so they meet; parallel lines never meet.
- No. Parallel is about the whole lines, and lines go on past the edge of the paper. These meet just off the paper, so they are not parallel.