Angles and parallel lines

A straight road crosses a railway at a slant. The two rails are parallel, the road cuts both of them, and eight angles appear at the crossings. Does an engineer have to measure all eight? No — just one. The other seven follow from the rules in this lesson. We start with the angle itself, then study the angles made by intersecting and parallel lines. In the next lesson, “Triangles”, these facts will prove that the angles of a triangle add up to 180°.

The angle: measuring and classifying

Angles are measured in degrees. An angle whose sides form a straight line is a straight angle; split it into 180 equal parts and each part is 1°. A full turn is 360°. For more precision a degree is split into 60 minutes: 1° = 60′, so 37.5° = 37°30′. To measure an angle with a protractor:

1. 1
   Centre on the vertex

   Put the centre mark of the protractor on the vertex of the angle.

2. 2
   Zero line along a side

   Line up the zero line of the protractor with one side of the angle.

3. 3
   The right scale

   Read the number where the other side crosses the scale, using the scale that starts at 0 on the first side.

4. 4
   Check by eye

   If the angle looks acute, the answer must be less than 90°: reading 130° instead of 50° is the most common mistake.

Adjacent and vertical angles. Perpendicular lines

Why? Each of ∠1 and ∠3 is adjacent to ∠2. So ∠1 = 180° − ∠2 and ∠3 = 180° − ∠2, which means ∠1 = ∠3. In the same way ∠2 = ∠4. So knowing one of the four angles at a crossing is enough: its vertical angle is equal to it, and the other two are adjacent to it.

Why are all four angles right? The angle next to a 90° angle is 180° − 90° = 90°, and vertical angles are equal. Through any point of the plane there is exactly one line perpendicular to a given line. The length of the perpendicular dropped from a point to a line is the distance from the point to the line — it is the shortest of all segments joining the point to points of the line.

Parallel lines and a transversal

A line that crosses two other lines is called a transversal. When the transversal c crosses lines a and b, four angles appear at each crossing — eight in total. Angles 3, 4, 5 and 6, which lie between a and b, are interior; angles 1, 2, 7 and 8 are exterior. The names of the angle pairs describe where the angles are:

• Corresponding angles are in the same position at each crossing: 1 and 5, 2 and 6, 3 and 7, 4 and 8.
• Alternate interior angles are interior and on opposite sides of the transversal: 3 and 5, 4 and 6 (the alternate exterior angles are 1 and 7, 2 and 8).
• Co-interior angles (same-side interior angles) are interior and on the same side of the transversal: 3 and 6, 4 and 5.

Why? Parallel lines have the same direction: if you slide line b along c until it lies on a, the “cross” at the lower crossing becomes an exact copy of the upper one — that is why corresponding angles are equal. (In a geometry course this fact is proved strictly from the parallel postulate.) The other two properties follow from it: ∠3 = ∠7 (corresponding) and ∠7 = ∠5 (vertical), so ∠3 = ∠5. And ∠6 is adjacent to ∠5, so ∠3 + ∠6 = ∠5 + ∠6 = 180°.

Tests for parallel lines

The reverse question matters too: how can we tell from the angles that two lines are parallel? After all, we cannot extend the lines forever and check that they never meet. The properties also work in reverse — these are the tests for parallel lines.

Proof (for alternate angles). Let ∠3 = ∠5. Through the point where a meets c draw the line a′ parallel to b; by the parallel postulate there is only one such line. Since a′ ∥ b, the alternate interior angle between a′ and c equals ∠5, that is ∠3. So a and a′ leave that point on the same side of c at the same angle — they coincide. Hence a = a′, i.e. a ∥ b. The other two tests reduce to this one: equal corresponding angles give equal alternate angles through vertical angles, and co-interior angles adding up to 180° give the same through adjacent angles.

• Two lines perpendicular to the same line are parallel: the co-interior angles are 90° + 90° = 180°.
• Two lines parallel to a third line are parallel to each other: otherwise two parallels to the third line would pass through their common point, which contradicts the postulate.

The angle between the hands of a clock

Clock problems are good practice with angles. A clock face is divided into 12 equal parts of 360° ÷ 12 = 30° each. The minute hand makes a full turn in 60 minutes, so it turns 360° ÷ 60 = 6° per minute. The hour hand turns 30° per hour, which is only 30° ÷ 60 = 0.5° per minute. At time h:m the minute hand has turned 6m degrees from 12, and the hour hand 30h + 0.5m degrees. The angle between the hands is the difference of these two numbers.