Reading the protractor and the compass constructions

In Discover you met the geometry box, measured a few angles and made a 60° angle with a compass. This layer slows everything down and asks how each method works and why it gives the right answer, so you can do it confidently every time, including the tricky cases: the wrong scale, short arms, angles bigger than 180°, and constructions that go wrong because the compass slipped.

There are two families of methods in this topic:

1. Measuring and drawing with a protractor. Fast and flexible: any whole number of degrees. Accuracy depends on your eye and your pencil.
2. Constructing with a ruler and compass only. Slower, and only certain angles are possible (60°, 120°, 90°, 30°, 45° and their relatives), but the result is exact in principle, because it comes from reasoning rather than reading a scale.

An angle is formed by two rays, called its arms, that start from the same point, its vertex. We name an angle with three letters, the vertex in the middle: ∠AOB has vertex O and arms OA and OB.

The size of an angle is the amount of turn needed to rotate one arm onto the other, about the vertex. We measure turn in degrees: one complete turn is 360°, so 1° is 1/360 of a full turn.

Notice what is not in that definition: the length of the arms. The arms of an angle are rays, which go on for ever. When we draw them, we draw only a piece, and the piece can be short or long. Drawing the arms longer does not change the amount of turn, so it does not change the angle.

A standard school protractor is a semicircle of clear plastic, graduated from 0° to 180°. Four features matter:

• The centre point (also called the origin or reference point) is the centre of the semicircle. It is marked with a small hole, a cross or a short line on the straight edge. Every angle is measured from here.
• The base line (the zero line) is the straight line through the centre point joining the two 0° marks. It is usually a few millimetres above the plastic edge.
• The two scales run round the curved edge in opposite directions: one reads 0° at the right end and 180° at the left, the other reads 0° at the left end and 180° at the right. Which of them is printed on the inside and which on the outside varies from protractor to protractor, so the rule below never mentions inner or outer. It only asks where the 0 is.
• The graduations: long marks every 10°, medium marks every 5°, short marks every 1°.

The most common protractor mistake in the world is reading the wrong scale. It is easy to make, because both numbers are printed right next to each other where the second arm crosses, and they are both perfectly real numbers.

Here is why there are two. Suppose the arm you lined up points to the right. You must count the turn starting from that arm, so you need a scale that starts at 0 on the right. If the arm you lined up points to the left, you need a scale that starts at 0 on the left. A single protractor carries both, so it works either way round.

The rule: read the scale whose 0 lies on the arm you lined up with the base line. Then count upwards along that scale to the second arm.

To draw an angle of a given measure is to reverse the measuring process. You create one arm, then use the protractor to find where the other arm must go.

The same rule decides the scale: count from the 0 on the arm you have already drawn. The same safety net catches mistakes: an estimate of what the angle should look like.

A reflex angle is more than 180° and less than 360°. A half-circle protractor only reaches 180°, so it cannot measure a reflex angle in one go. There are two simple methods.

Method 1: measure the other angle and subtract from 360°. The two angles at a vertex add up to a full turn. Measure the ordinary (non-reflex) angle, x, then the reflex angle is 360° − x.

Method 2: split at a straight line. Extend one arm backwards through the vertex to make a straight line. The reflex angle is then 180° plus the extra piece beyond the straight line, which you can measure normally.

In geometry, to construct means to draw a figure using only two tools:

• a straightedge (a ruler used only for drawing straight lines through two points, never for reading its markings), and
• a compass (for drawing circles and arcs with a chosen centre and radius, and for copying a length).

Why bother, when a protractor exists? Because a construction is exact in principle. A protractor reading is only as good as your eyesight. A construction's correctness comes from a reason, such as "these three lengths are equal, so this triangle is equilateral". If your drawing is careful, the result is as accurate as your pencil line allows.

A perpendicular bisector of a segment AB is a line that does two jobs at once: it cuts AB into two equal halves (it passes through the midpoint) and it crosses AB at a right angle. It is one of the most useful constructions, because it gives you both a midpoint and a 90° angle.

There are two standard ways to make a 90° angle at a point O on a ray OA:

Method A: between 60° and 120°. Make the 60° and 120° marks, Q and R, on the big arc as in Chapter 8. Then bisect the angle between them: with centres Q and R and equal radii, draw arcs that meet at S. The ray OS makes 90° with OA, because it sits exactly halfway between 60° and 120°: (60° + 120°) ÷ 2 = 90°.

Method B: perpendicular at a point on a line. Extend OA backwards through O to make a straight line. With centre O, draw an arc cutting the line on both sides of O, at X and Y. Now draw the perpendicular bisector of XY: it passes through O (because OX = OY) and makes 90° with the line.

Always check a construction with a protractor. A careful construction should read within ± 1° of the target. If it is further off, the usual culprits are: the compass width changed between arcs; the compass point slipped off the exact point; arcs that meet at a very shallow crossing, so the crossing point is blurry; or a blunt pencil. Keep your construction arcs on the page: they are your working and they let a teacher see where a slip happened.