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Angles with parallel lines

Geometry · level 4, lesson 4 of 6

A question to start

Over two thousand years ago, a librarian named Eratosthenes, in Alexandria in Egypt, measured the size of the whole Earth without leaving home. The story goes that at noon on midsummer's day the Sun shone straight down a deep well in the town of Syene, far to the south, while at the same moment in Alexandria a tall stick cast a short shadow. He measured the angle of that shadow: about one fiftieth of a full turn.

How could the angle of a shadow in one town tell him how much of the way round the Earth he was from another town? The secret is one fact about parallel lines: the Sun is so far away that its rays arrive as parallel lines. By the end of this lesson you will be able to follow his reasoning.

The idea

A line crossing two parallel lines

Take two parallel lines (marked with arrows) and a third line that crosses both of them. The crossing line is called a transversal ("going across"). It makes four angles at each crossing: eight angles in all.

a b c d e f g h

Look closely and you will see that the two crossings look exactly alike. The transversal meets both parallel lines at the same slant, so the four angles at the top crossing match the four at the bottom crossing: \(a = e\), \(b = f\), \(c = g\), \(d = h\).

Why the two crossings match

Imagine sliding the top parallel line straight down along the transversal, without turning it. Because the two lines are parallel (they point in exactly the same direction), it lands exactly on the bottom line. The transversal slides along itself, so it stays put. The whole top crossing lands on the bottom crossing, and every angle lands on its partner. A slide does not change any angle, so the partners are equal.

This is the key step, and it depends completely on the lines being parallel. If the bottom line were tipped even a little, the slide would not land on it, and the angles would not match.

(Careful mathematicians take one form of this, "a transversal makes equal corresponding angles with parallel lines", as a starting rule of flat geometry, closely tied to Euclid's famous parallel postulate. Everything else in this lesson is then proved from it.)

Three names for three patterns

Pairs of angles in the picture get names by where they sit.

corresponding: equal (F) alternate: equal (Z) co-interior: add to 180° (C)

Corresponding angles are in the same position at each crossing: both above-right, or both below-left, and so on. In the first picture the pairs are \(a\) and \(e\), \(b\) and \(f\), \(c\) and \(g\), \(d\) and \(h\). They make an F shape (sometimes a backwards or upside-down F). Corresponding angles are equal. That is the sliding fact.

Alternate angles are between the parallel lines, on opposite sides of the transversal. The pairs are \(c\) and \(e\), and \(d\) and \(f\). They make a Z shape (sometimes a backwards Z, like an N). Alternate angles are equal.

Reason: \(c = a\) (vertically opposite, lesson 2) and \(a = e\) (corresponding). So \(c = e\).

Co-interior angles (also called same-side interior or allied angles) are between the parallel lines, on the same side of the transversal. The pairs are \(d\) and \(e\), and \(c\) and \(f\). They make a C shape (or a U). Co-interior angles add to 180°.

Reason: \(d + c = 180°\) (angles on a straight line, lesson 1), and \(c = e\) (alternate). So \(d + e = 180°\).

So one sliding fact, plus the two facts from lessons 1 and 2, gives all three patterns. The letters F, Z and C are only memory aids; the reasons are what matter.

One angle tells you all eight

At each crossing, the four angles come in two sizes: a small one and a big one, adding to 180°. With parallel lines, both crossings have the same two sizes. So if one of the eight angles is 60°, every angle is either 60° or 120°. In the first picture, \(a\), \(c\), \(e\), \(g\) are the small ones and \(b\), \(d\), \(f\), \(h\) are the big ones.

Working backwards: testing for parallel

The facts also work the other way round. If a transversal makes equal corresponding angles (or equal alternate angles, or co-interior angles adding to 180°) with two lines, then those lines are parallel. (If they were not, one of them would be tipped compared with the other, and the angles could not match.) This is how builders and engineers check that two beams are parallel: they measure the angles a cross-piece makes with each.

Euclid's proof that a triangle's angles add to 180°

Now the promise from the last lesson. Take any triangle with angles \(a\), \(b\), \(c\). Through the top corner, draw a line parallel to the bottom side.

a b c a b draw a line through the top, parallel to the base
  1. Along the new line, at the top corner, there are three angles side by side: one on the left, \(c\) in the middle, one on the right. They make a straight line, so they add to 180° (angles on a straight line).
  2. The left one and the angle \(a\) at the bottom-left are alternate angles (a Z shape, using the left side of the triangle as the transversal). So the left one equals \(a\).
  3. In the same way, the right one equals \(b\) (alternate angles, using the right side as the transversal).
  4. So \(a + c + b = 180°\).

Every step has a reason, and nothing depends on the shape of the triangle. This is the proof in Euclid's Elements, the most famous geometry book ever written, from about 300 BCE.

A bent path between parallel lines

A favourite puzzle: a path leaves one parallel line, bends once, and meets the other. Find the angle at the bend.

40° 35° x

The trick is to draw an extra line through the bend, parallel to the other two (dashed). It splits the bend angle \(x\) into two parts. The top part is alternate with the 40° angle, so it is 40°. The bottom part is alternate with the 35° angle, so it is 35°. So \(x = 40 + 35 = 75°\).

Drawing an extra line that is not in the question is called a construction, and it is one of the most useful ideas in geometry. It turns a hard picture into easy pieces.

Back to Eratosthenes

The Sun's rays reaching Syene and Alexandria are parallel. A ray going straight down the well in Syene points towards the centre of the Earth; so does the stick in Alexandria. Those two lines meet at the Earth's centre. The ray hitting the top of the stick in Alexandria is parallel to the ray down the well, and the stick acts as a transversal. The shadow angle at the stick and the angle at the centre of the Earth are alternate angles, so they are equal: about a fiftieth of a full turn, roughly 7.2°. So the distance from Syene to Alexandria is about a fiftieth of the way round the Earth. Multiply that distance by 50 and you have the Earth's circumference. The same idea could be used on Mars, with two rovers and a sunny noon.

Worked examples

1. All eight from one. In the first picture, \(a = 60°\). Find \(b\) to \(h\).

  • \(b = 180 - 60 = 120°\) (angles on a straight line).
  • \(c = a = 60°\) (vertically opposite); \(d = b = 120°\) (vertically opposite).
  • \(e = a = 60°\), \(f = b = 120°\), \(g = c = 60°\), \(h = d = 120°\) (corresponding).

2. Naming the reason. Two parallel lines are crossed by a transversal. An angle between the lines on the left of the transversal is 72°. Find the angle between the lines on the right of the transversal at the other crossing.

They are between the lines, on opposite sides of the transversal: alternate angles, so it is 72°.

3. Co-interior with algebra. Two co-interior angles are \(3x + 15\) and \(2x - 10\). Find them.

Co-interior angles add to 180°: \(5x + 5 = 180\), so \(5x = 175\) and \(x = 35\). The angles are \(3 \times 35 + 15 = 120°\) and \(2 \times 35 - 10 = 60°\). Check: \(120 + 60 = 180\). ✓

4. Parallelograms. In a parallelogram, each pair of neighbouring angles is a pair of co-interior angles (the shared side is the transversal, and the two sides it crosses are parallel). So neighbouring angles of a parallelogram add to 180°. That is the fact noticed in level 3, now proved. If one angle is 65°, the angles are 65°, 115°, 65°, 115°.

5. Is it parallel? A transversal crosses two lines, making corresponding angles of 82° and 85°. The angles are not equal, so the lines are not parallel. They tip towards each other by 3°, and far enough along they will meet.

Try it

  1. Two parallel lines are crossed by a transversal and one angle is 110°. What are the sizes of the other seven angles?
  2. Name the pair: (a) both above the lines, on the same side of the transversal, one at each crossing; (b) both between the lines, on opposite sides of the transversal; (c) both between the lines, on the same side.
  3. Two alternate angles are \(4y - 20\) and \(2y + 30\). Find \(y\) and the angles.
  4. Two co-interior angles are in the ratio 1 : 2 (one is twice the other). Find them.
  5. A path between two parallel lines leaves the top one at 30° and meets the bottom one at 50°, with the bend pointing the same way as in the picture. Find the bend angle.
  6. In a trapezoid with two parallel sides, one angle is 70° and it sits at the end of one of the slanted sides. What is the angle at the other end of that slanted side?
  7. Explain why alternate angles are equal, using only corresponding angles and vertically opposite angles.
  8. Eratosthenes' shadow angle was about 7.2°. Why does that mean the distance from Syene to Alexandria was about one fiftieth of the Earth's circumference? (\(360 \div 7.2 = ?\))
Answers
  1. Three are 110° and four are 70° (\(180 - 110 = 70\)). At each crossing there are two of each size, so of the other seven: 110° three times and 70° four times.
  2. (a) Corresponding (the same position at each crossing). (b) Alternate. (c) Co-interior.
  3. Alternate angles are equal: \(4y - 20 = 2y + 30\), so \(2y = 50\) and \(y = 25\). Each angle is \(4 \times 25 - 20 = 80°\) (check: \(2 \times 25 + 30 = 80\) ✓).
  4. \(x + 2x = 180\) (co-interior), so \(x = 60\): the angles are 60° and 120°.
  5. Draw a line through the bend parallel to the other two. The parts of the bend are alternate with 30° and 50°, so the bend angle is \(30 + 50 =\) 80°.
  6. The slanted side is a transversal of the two parallel sides, and the two angles at its ends are co-interior. So the other angle is \(180 - 70 =\) 110°.
  7. Let \(c\) and \(e\) be alternate. The angle \(a\) vertically opposite \(c\) is equal to \(c\), and \(a\) is corresponding to \(e\), so \(a = e\). So \(c = a = e\).
  8. \(360 \div 7.2 = 50\), so 7.2° is one fiftieth of a full turn. The angle at the Earth's centre between the two towns is 7.2° (alternate angles), so the arc of the Earth's surface between them is one fiftieth of the way round.

Common mistake

Using the patterns when the lines are not parallel. The F, Z and C facts are only true when the two lines are parallel. Look for the arrow marks, or a statement that they are parallel. Without them, corresponding angles can be anything.

Mixing up "equal" and "add to 180°". Corresponding and alternate angles are equal. Co-interior angles add to 180°. A quick check: in any picture there are only two sizes, small and big. Two small or two big are equal; one of each adds to 180°. Co-interior pairs are always one small and one big.

Where it leads