Naming, sorting and pairing angles

In Discover you met angles as turns: doors, clock hands, scissors and compass directions. Now it is time to be precise, the way a mathematician is.

This layer answers four questions:

1. Exactly what is an angle, and how do we name one so that everyone knows which angle we mean?
2. How big is it? Degrees, landmarks, estimation and the seven types, with exact boundaries.
3. How do angles team up? Adjacent angles, complementary and supplementary pairs, linear pairs, vertically opposite angles and angles around a point.
4. How do we use all this to find an angle we cannot measure, and give a reason for every step?

By the end you will be able to look at a diagram of crossing lines, know one angle, and work out all the others, with reasons, like a detective.

Here is the precise definition used in NCERT textbooks and around the world:

An angle is the figure formed by two rays with a common end point.

• The common end point is the vertex of the angle.
• The two rays are the arms (or sides) of the angle.
• The measure (size) of the angle is the amount of rotation needed to turn one arm onto the other, about the vertex.

Both ideas from Discover are inside this definition. The two rays describe what an angle looks like; the amount of rotation describes how big it is. Because rays go on forever, the arms have no length, so an angle's size can only depend on the turn.

Every angle divides the plane (the flat surface it is drawn on) into three parts:

• the interior of the angle: all the points between the arms (on the side of the smaller opening);
• the exterior: all the points outside the arms;
• the angle itself: the points on the two arms, including the vertex.

A point on an arm is on the angle, neither inside nor outside. This matters later, because when we say two angles are adjacent, we will need their interiors not to overlap.

There are three ways to name an angle. Suppose the vertex is B, one arm passes through point A and the other arm passes through point C.

1. Three letters, vertex in the middle: ∠ABC or ∠CBA. The sign ∠ means angle. The middle letter must always be the vertex. ∠ABC and ∠CBA are the same angle (you can read the arms in either order).
2. Just the vertex: ∠B. This is fine when only one angle has its vertex at B.
3. A number or small letter written inside the angle, near the vertex: ∠1, ∠2, or ∠x. Handy in diagrams with many angles.

We write the measure of the angle as ∠ABC = 50° (some books write m∠ABC = 50°).

A degree is one 360th of a full turn: 1° = 1⁄360 of a full turn. For finer work, each degree is split into 60 minutes (60′) and each minute into 60 seconds (60″). Map-makers and astronomers still use these, which is why a place's position is written like 28° 36′ N for Delhi.

You will learn to measure angles exactly with a protractor in the measuring and constructing angles topic. For now the goal is a good eye: to be able to glance at an angle and say "about 60°" or "a bit more than a right angle". Estimating first is also the best way to catch protractor mistakes, such as reading the wrong scale.

Two special words go with right angles:

• Two lines that meet at a right angle are called perpendicular, written with the sign ⊥. If line AB is perpendicular to line CD, we write AB ⊥ CD.
• The four angles where two perpendicular lines cross are all right angles.

And two special facts about types:

• A straight angle is two right angles: 90° + 90° = 180°.
• A complete angle is four right angles or two straight angles: 4 × 90° = 2 × 180° = 360°.

Two numbers let you find the angle between clock hands at almost any time.

• The minute hand goes round once (360°) in 60 minutes, so it turns 360 ÷ 60 = 6° every minute.
• The hour hand goes from one number to the next (30°) in 60 minutes, so it turns 30 ÷ 60 = 0.5° every minute, or half a degree.

On the hour, the minute hand is at 12 and the hour hand is exactly on a number, so the angle is simply gaps × 30°. But at half past, the hour hand has moved half-way to the next number, 15° further on. That is why the hands at 3:30 do not make a right angle, even though they point "at 3 and 6".

A compass rose has four cardinal directions 90° apart: N, E, S, W. Half-way between each pair are the four intercardinal (in-between) directions: NE, SE, SW, NW. Each is 45° from its neighbours, because 360° ÷ 8 = 45°.

Turns have a size and a direction: clockwise (the way clock hands move: N → E → S → W) or anticlockwise (N → W → S → E). A clockwise turn of 90° from north takes you to east; an anticlockwise turn of 90° from north takes you to west.

Here is a handy fact: turning x° clockwise ends in the same place as turning (360 − x)° anticlockwise. For example, 90° clockwise and 270° anticlockwise both take you from north to east.

Now we start working with pairs of angles. The first idea is about where two angles sit.

Two angles are adjacent (neighbours) when all three of these are true:

1. They have a common vertex.
2. They have a common arm.
3. Their interiors do not overlap: the common arm lies between the other two arms.

The door and the frame, the two parts of a pizza cut by one extra slice, the angles on either side of a clock's hour hand between the other two hands: these are adjacent angles.

Now two ideas about the sum of two angles, whether or not they are adjacent.

• Two angles are complementary if their sum is 90°. Each is the complement of the other. The complement of an angle x is 90° − x.
• Two angles are supplementary if their sum is 180°. Each is the supplement of the other. The supplement of an angle x is 180° − x.

For example, 30° and 60° are complementary (30 + 60 = 90). 110° and 70° are supplementary (110 + 70 = 180). They do not have to touch: one could be at a corner of your book and the other at a corner of the blackboard.

Stand a ray on a straight line, like a lane meeting a main road. It makes two adjacent angles whose outer arms point in opposite directions, forming a straight line. Such a pair is called a linear pair.

The angles of a linear pair are supplementary: they add up to 180°. Reason: together they make the straight angle, which is 180°.

So a linear pair is adjacent and supplementary at the same time. Be careful with the reverse: two supplementary angles are not a linear pair unless they are also adjacent with outer arms on a line. 120° at one corner of your desk and 60° on the blackboard are supplementary but not a linear pair.

Now cross two straight lines, like the two blades of open scissors or an X. Four angles form around the crossing point. The angles directly across from each other, which share only the vertex, are called vertically opposite angles. (Vertical here means at the vertex, not up and down.)

Vertically opposite angles are always equal. Here is why. Call the four angles ∠1, ∠2, ∠3 and ∠4, going round, so ∠1 and ∠3 are opposite each other, and ∠2 and ∠4 are opposite each other.

• ∠1 and ∠2 form a linear pair on one line, so ∠1 + ∠2 = 180°.
• ∠2 and ∠3 form a linear pair on the other line, so ∠2 + ∠3 = 180°.
• Both ∠1 and ∠3 are "180° minus ∠2". So ∠1 = ∠3.

The same argument with ∠1 as the shared partner shows ∠2 = ∠4. This is why the angle between the handles of scissors always matches the angle between the blades.

If several angles fill all the space around one point, with no gaps and no overlaps, they make a full turn. So:

Angles around a point add up to 360°.

Together with the facts from earlier chapters, you now have a toolkit. The skill in geometry is choosing the right tool and saying which one you used.