Mathematics · Plane geometry · Grade 7

Parallel lines

The street crossing the rails makes eight angles — why are four of them equal?

The situation

However far railway rails run, they never meet — they are parallel. A street crossing them makes eight angles, and these take only two different values. A draughtsman, with a ruler and a set square, draws a parallel line through any point in a single movement. Why can we tell that two lines are parallel from their angles, and what angles do parallel lines make with a transversal?

Topic

Parallel lines: criteria, angles formed by a transversal. Railway tracks and a street crossing them: corresponding, alternate and co-interior angles

Key ideas

Tests for parallel lines
Transversal and angles
Corresponding and alternate angles

Formulas

a ∥ b

Two straight lines in one plane that do not meet are parallel. Axiom: through a point outside a line there passes only one straight line parallel to it.

∠ 3 = ∠ 5 \Rightarrow a ∥ b

Test for parallel lines: if alternate interior angles are equal, the lines are parallel (so too if corresponding angles are equal, or if the co-interior angles add up to $180^{\circ}$ ).

a ∥ b \Rightarrow ∠ 1 = ∠ 5, ∠ 3 = ∠ 5

Property (converse theorem): parallel lines make equal corresponding angles and equal alternate interior angles with a transversal.

∠ 4 + ∠ 5 = 180^{\circ}

For parallel lines, the co-interior angles add up to $180^{\circ}$ .

α + β + γ = 180^{\circ}

Application: if we draw a line through a vertex of a triangle parallel to the base, the three angles fill a straight angle.

Theory

Definition and axiom

Two straight lines in a plane either meet at one point or do not meet at all. Those that do not meet are called parallel: $a ∥ b$ . A line turned even slightly — even by $1^{\circ}$ — will certainly meet, only far away (read off the distance in the first step). The parallel axiom: through a point outside a straight line there passes only one straight line parallel to it. This statement is not proved — it is laid down as a foundation of geometry.

The transversal and eight angles

A third line crossing two lines — a transversal — makes eight angles. Those between the lines are interior ( $∠ 3, ∠ 4, ∠ 5, ∠ 6$ ), those outside are exterior. The pairs: corresponding angles are in the same position ( $∠ 1$ and $∠ 5$ ); alternate interior angles are interior and on different sides of the transversal ( $∠ 3$ and $∠ 5$ ); co-interior angles are interior and on the same side ( $∠ 4$ and $∠ 5$ ). In the second step, rotate the transversal: the eight angles take only two different values.

Tests for parallel lines

If alternate interior angles are equal, the lines are parallel. Proof: the reflection in the midpoint $O$ of $P Q$ sends $P$ to $Q$ , the transversal to itself, and $a$ to the line through $Q$ that makes the same angle with the transversal — that is, to $b$ . If $a$ and $b$ met at some point $X$ , they would also meet at the symmetric point $X^{'}$ — but two straight lines cannot meet at two points. Hence $a ∥ b$ . The same holds when corresponding angles are equal ( $∠ 1 = ∠ 3$ as vertically opposite angles), or when the co-interior angles add up to $180^{\circ}$ ( $∠ 3 = 180^{\circ} - ∠ 4$ ) — the alternate angles then turn out to be equal.

Properties of parallel lines

The converse theorem: if $a ∥ b$ , the alternate interior angles are equal. Proof by contradiction: suppose $∠ 3 \neq ∠ 5$ ; through $P$ we draw a line $a^{'}$ making an angle equal to $∠ 5$ with the transversal — by the test, $a^{'} ∥ b$ . But $a ∥ b$ as well: two parallels to $b$ through $P$ — which contradicts the axiom. Hence $a^{'} = a$ and $∠ 3 = ∠ 5$ . Consequences: corresponding angles are equal, and co-interior angles add up to $180^{\circ}$ . If a line is perpendicular to one of two parallel lines, it is perpendicular to the other too.

Drawing a parallel line

The draughtsman's method: we place one side of a set square along $a$ , rest a ruler against its other side, and slide the set square along the ruler to the point $M$ — the side stays parallel to $a$ (corresponding angles are equal). With a compass: through $M$ we draw any transversal and copy the angle it makes with $a$ to $M$ (equal arcs and an equal chord) — the corresponding angles are equal, so the new line is parallel to $a$ . In the fifth step, slide the set square yourself.

Applications and the model

Parallel lines are the key to many theorems: if we draw a line through a vertex of a triangle parallel to the base, the alternate angles gather the three angles into one straight angle — $α + β + γ = 180^{\circ}$ . The treads of a staircase are parallel and the sloping support is a transversal: it makes the same (corresponding) angle with every tread. Limits of the model: angles are set in $5^{\circ}$ steps, lengths are shown rounded; the “rails” are taken as ideal straight lines; lines are infinite, but on the screen they are drawn only to the edge.

Did you know?

For two thousand years people tried to prove Euclid's fifth postulate (the parallel axiom). In 1829 N. Lobachevsky and in 1832 J. Bolyai gave up the attempt and built a new, consistent geometry — non-Euclidean geometry.
On Uzbekistan's railways the distance between the rails (the gauge) is 1520 mm; in many European countries it is 1435 mm. If the rails were not parallel, the train wheels would leave the track.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.