Is it always true? Testing angle ideas

Mathematicians do not just learn facts; they test them. They ask "what happens if…?", make a guess, try examples, and then ask the most important question of all: is it always true?

In this layer you are the investigator. Every chapter starts with a question. You will predict, test with labs and numbers, look for patterns and counterexamples, and write down what you found. Some findings will be always true, some only sometimes, and some never. Learning to tell these apart is the heart of mathematical thinking.

Keep a notebook open. For each investigation write: my prediction, what I tried, what I found, always / sometimes / never?

What type of angle do you get if you add two angles of known types? Try these investigations before reading on.

Acute + acute. 20° + 30° = 50° (acute). 50° + 60° = 110° (obtuse). 45° + 45° = 90° (right). So the sum of two acute angles is sometimes acute, sometimes right, sometimes obtuse. It is never straight or bigger, because each is less than 90°, so the sum is less than 180°.

Acute + right. 10° + 90° = 100°. 89° + 90° = 179°. The sum is always more than 90° and less than 180°, so it is always obtuse.

Obtuse + obtuse. 91° + 91° = 182°. 179° + 179° = 358°. Each is more than 90°, so the sum is more than 180°; each is less than 180°, so the sum is less than 360°. The sum is always reflex.

Choose any acute angle. Write down its complement and its supplement. Now subtract: supplement − complement. Try a few before reading on.

Every row gives 90°. Is that always true, or did we just pick lucky angles? Here is a reason that works for every acute angle x:

• supplement − complement = (180° − x) − (90° − x)
• = 180° − x − 90° + x
• = 90°, because the −x and +x cancel.

So it is always true. The reason is more convincing than a thousand examples, because it covers every possible x at once. In words: the supplement is always exactly one right angle bigger than the complement.

Here is a beautiful investigation. Take a linear pair, say 50° and 130°. Cut each angle exactly in half with a ray (a ray that halves an angle is called its bisector). What is the angle between the two bisectors?

• Half of 50° is 25°; half of 130° is 65°. The angle between the bisectors is 25° + 65° = 90°.
• Try 20° and 160°: 10° + 80° = 90°.
• Try 90° and 90°: 45° + 45° = 90°.

It seems to be always 90°. Why? The two angles add to 180°, so their halves add to half of 180°, which is 90°. So the bisectors of a linear pair are always perpendicular. You can construct bisectors with a compass in the constructing angles topic and check this on paper.

When two lines cross, four angles appear. How many of them do you need to be told before you can work out all four?

Try it: if one angle is 65°, the angle opposite is 65° (vertically opposite), and the two others are each 180° − 65° = 115° (linear pairs). One angle is enough.

What if the lines cross at a right angle? Then all four angles are 90°. That is the only way to have four equal angles at a crossing: four equal angles adding to 360° must each be 90°.

The table shows a pattern: n lines through a point make 2n angles, arranged in n pairs of vertically opposite angles. The n angles on one side of any of the lines lie along that line, so they add to 180°, which is why you only need to be told n − 1 of them. For two lines, one clue is enough; for three lines, two clues; for six lines, five clues.

Now make it harder. What if three lines pass through the same point? Draw it: you get six angles around the point. Predict before reading on: how many angles do you need to know to find all six?

Label them a, b, c, d, e, f going round. Each line gives a pair of vertically opposite angles, so d = a, e = b, f = c. The six angles add up to 360°, so 2a + 2b + 2c = 360°, which means a + b + c = 180°. (You can also see this directly: a, b and c sit side by side along one straight line.)

So you need to know two of a, b, c; the third comes from the straight line, and the other three come from vertically opposite angles.

Look down the last column. The angle starts at 90°, shrinks to almost nothing near 3:16 (the hands overlap), then grows again, passes 90° between 3:30 and 3:35, reaches almost 180° near 3:49, and then shrinks towards 120° at 4:00.

The minute hand gains on the hour hand by 6° − 0.5° = 5.5° every minute. That single number explains everything:

• From 3:00 the gap of 90° closes at 5.5° per minute, so the hands overlap after 90 ÷ 5.5 = 16 4⁄11 minutes, at about 3:16.
• The next right angle comes when the minute hand is 90° ahead: that needs a total gain of 180°, which takes 180 ÷ 5.5 = 32 8⁄11 minutes, at about 3:32 and 44 seconds.

Imagine a tiny robot that can only do two things: walk forward, and turn on the spot. It walks around a square: forward, turn, forward, turn, forward, turn, forward, turn, and it is back where it started, facing the same way.

How much did it turn altogether? Facing the same way again means it made exactly one full turn: 360°. There were 4 equal turns, so each was 360° ÷ 4 = 90°.

What about walking around an equilateral triangle? Again it ends facing the way it started, so the total turning is 360°, and each of the 3 equal turns is 360° ÷ 3 = 120°. Notice: the robot turns 120° at each corner, but the angle inside each corner of the triangle is 180° − 120° = 60°. The turn and the inside angle form a linear pair.

Treat clockwise turns as positive and anticlockwise turns as negative. Then a whole sequence of turns can be added up like ordinary numbers, and the net turn tells you where you end up facing.

For example, +90° (right turn), +90°, −45°, +180° gives a net turn of 90 + 90 − 45 + 180 = 315° clockwise. Turning 315° clockwise ends in the same direction as turning 45° anticlockwise, because 315° + 45° = 360°.

Investigation question: does the order of the turns matter? Try +90°, then −45°, then +180°, and compare with +180°, then +90°, then −45°. Predict first.

Almost everyone finds the same thing: the three corners of any triangle fit together to make a straight line, 180°. The four corners of any four-sided shape fit together all the way round a point, 360°.

An experiment is powerful evidence, but it is not a proof. Paper tears roughly, and we could never test every triangle in the universe. Is it always true? In the Deepen layer you will prove it using parallel lines, so that nobody ever needs to tear another triangle.

A good detective does not guess; they reason from clues. A good missing-angle solver does the same:

1. Mark everything you are told on a sketch.
2. Look for straight lines (180°), right angles (90°), crossing lines (equal opposite angles) and points with angles all round (360°).
3. Find one new angle, write its reason, and mark it.
4. Repeat until you reach the angle you want.
5. Check with a different route or a total (360° round a point).