INSTRUCTION BOOKLET
Everything from the quest on one page: how to decide whether two lines are parallel, and how to prove it.
Parallel, in one sentence
Parallel lines are lines in the same plane that never meet. They point in the same direction and stay the same distance apart. We write m n. Lines that never meet but are not in one plane are skew (they only happen in 3-D). Two lines in one plane that aren't parallel intersect at exactly one point, even if that point is very far away.
How to decide: the flowchart
YES Go on.
Corresponding, alternate interior or alternate exterior: equal?
Same-side interior: add to 180°?
YES m n (that's the converse theorem). NO Not parallel: they meet somewhere.
Only angles at one crossing (a linear pair, vertical angles)? Then you can't tell yet. Use a linear pair or vertical angles to find an angle at the other crossing first.
SAME SLOPE, DIFFERENT b Parallel.
SAME SLOPE, SAME b It's the same line in disguise.
DIFFERENT SLOPES They intersect.
Both vertical (x = a number)? Parallel, even though their slope is undefined.
The angle-pair chart
| Pair | Picture | Where to look | If m n… |
|---|
Every row works both ways. “If m n, then…” is the theorem; “If …, then m n” is its converse, and both are true. Linear pairs (add to 180°) and vertical angles (equal) are true for any crossing lines, so on their own they never prove anything is parallel.
Watch out!
- Vertical angles are a trap. ∠1 = ∠4 is always true at one crossing. It says nothing about the other line.
- Same-side interior angles add to 180°; they are not equal (unless both are 90°). For the three “equal” pairs, adding to 180° is not the test.
- “Not drawn to scale.” Trust the numbers, never the look of the picture. A 1° or 2° difference is invisible.
- Same line in disguise. 2y = 4x + 2 is just y = 2x + 1. Simplify both equations before you compare.
- Perspective lies. Parallel rails look like they meet at the horizon in a photo. They don't.
Practice
1. ∠2 = 3x + 4 and ∠6 = 2x + 19 are corresponding angles. What x makes m n?
Corresponding angles must be equal: 3x + 4 = 2x + 19, so x = 15. Each angle is 49°.
2. ∠4 = 64° and ∠5 = 64°. Parallel?
∠4 and ∠5 are alternate interior angles (inside the lines, opposite sides of t). They're equal, so yes: m n.
3. ∠3 = 95° and ∠5 = 95°. Parallel?
∠3 and ∠5 are same-side interior angles. They would have to add to 180°, but 95° + 95° = 190°. Not parallel.
4. Line p goes through (1, 2) and (5, 4). Line q is y = ½x − 6. Parallel?
Slope of p = (4 − 2) ÷ (5 − 1) = 2/4 = ½. Same slope as q. Line p's y-intercept: 2 = ½(1) + b, so b = 1.5, which is not −6. Different lines with the same slope: parallel.
5. Lines a and b are both perpendicular to line t. Parallel?
Yes, in a plane. Both make 90° with t, so their corresponding angles are equal (both 90°).
Notes on the game
The rails are drawn wider than real rail heads so they stay visible from high up. 1,435 mm (4 ft 8½ in) is standard gauge, the distance between the inside faces of the two rails on most of the world's railways. Euclid's fifth postulate (about 300 BC) is about a line crossing two lines with interior angles on one side adding to less than two right angles; “through a point not on a line there is exactly one parallel” is the equivalent form popularised by John Playfair in 1795, and it's the one most school books use. Euclid proves the converses (alternate angles equal, or same-side interior angles adding to two right angles, means parallel) in Book I, Propositions 27 and 28, without the fifth postulate; Proposition 29, the “if parallel then” direction, is the first place he needs it. On a curved surface such as a sphere the postulate fails, which is where non-Euclidean geometry begins. Angle numbering (1–4 at the top crossing, 5–8 at the bottom) differs between textbooks; some say “consecutive interior” for same-side interior.
Sources
- Common Core State Standards for Mathematics, High School Geometry: G-CO.C.9 (prove theorems about lines and angles, including transversals) and G-GPE.B.5 (slope criteria for parallel and perpendicular lines). thecorestandards.org
- Euclid, Elements, Book I: Postulate 5 and Propositions 27–29, D. E. Joyce's edition, Clark University. mathcs.clarku.edu/~djoyce/elements
- John Playfair, Elements of Geometry (1795), the “one parallel through a point” axiom.
- Standard gauge, 1,435 mm: Encyclopaedia Britannica, “gauge” (railway). britannica.com
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