Unit 10: Angles

An angle measures turning

Stand facing a wall and spin all the way round back to where you started. That whole turn is 360°.

A quarter turn is 90°, and we call it a right angle. Half a turn is 180°, which points you the opposite way.

Why 360? The Babylonians liked it, because it divides by almost everything.

Angles on a straight line

A straight line is half a turn, so any angles sitting on one side of it must add to 180°.

If one is 130°, the other has to be 50°. There is nothing to work out beyond subtracting.

Around a point — a full turn — they add to 360° instead.

Opposite angles are equal

When two straight lines cross, they make four angles. The ones facing each other are always equal.

You can see why: each of them is 180° minus the same neighbour, so they have no choice.

These are called vertically opposite angles, and spotting them is the first move in most angle puzzles.

Parallel lines and a crossing line

Parallel lines never meet. When a third line cuts across both, something useful happens: the pattern of angles at the top is repeated exactly at the bottom.

So angles in matching positions are equal (corresponding), and the ones in a Z shape between the lines are equal too (alternate).

Every angle in the picture is one of just two values.

Three letters name an angle exactly

Saying "angle B" is fine until three lines meet at B. Then it is ambiguous.

So we use three letters, with the corner in the middle: angle ABC is the angle at B, between the arms BA and BC.

The outer two letters say which arms. The middle one says where you are standing.

The four sizes

Acute: less than 90°.
Right: exactly 90°, marked with a little square rather than an arc.
Obtuse: between 90° and 180°.
Reflex: more than 180° — the big way round.

Every angle has a reflex partner: if one way round is 110°, the other way is 250°, and together they make a full turn.

A direction, written as an angle

Sailors and pilots need directions with no room for doubt, so they use bearings.

Always measured from north, always clockwise, always written with three digits.

So north is 000°, east is 090°, south is 180°, west is 270°. A bearing of 45° is written 045°.

Coming back the other way

If a ship sails from A to B on a bearing of 070°, what bearing takes it home?

Turn right round — add 180°: 250°.

And if that would take you past 360°, subtract 360 instead. A bearing of 200° has a back bearing of 200 + 180 - 360 = 020°.

Two hands, two speeds

The minute hand goes all the way round in an hour: 360/60 = 6° per minute.

The hour hand takes twelve hours: 360/12 = 30° per hour, which is only 0.5° per minute.

Almost everybody forgets that second one. The hour hand is not sitting still while the minutes tick by.

Half past is not straight

At half past three, the minute hand points at 6 — that is 180° from the top.

The hour hand is not at the 3. It is halfway to the 4, so it is at 3.5 × 30 = 105°.

So the angle between them is 180 - 105 = 75°, not 90°. That drifting hour hand is the whole puzzle.

Write the rule down as an equation

Angle problems become algebra the moment you name the unknown.

Two angles on a straight line are x and 2x. They must add to 180, so 3x = 180 and x = 60.

The geometry gives you the equation. Unit 3 solves it. That is the whole method for the rest of this track.

Two names worth knowing

Two angles adding to 90° are complementary. Two adding to 180° are supplementary.

They come up constantly, so it is worth being able to say which is which. The trick most people use: C comes before S in the alphabet, and 90 comes before 180.