From “it worked” to “always”

The idea

A justification that shows by general reasoning, without assuming particular values, that a statement is true in every case is called a proof.

Endless crossings

We saw ∠b = ∠d in the crossings we measured. But there are endless crossings, with ∠a of every size.

Can we be sure that ∠b = ∠d in every crossing?

• Yes: it worked each time we looked
• Only by measuring every crossing, and that is impossible
• Yes, by reasoning that works whatever ∠a is

Build the proof

Tap the tiles to build the proof that ∠b = ∠d. Watch out: some tiles are traps!

A proof

The proof never said how big ∠a is. It used only the straight angle: ∠a + ∠b = 180° and ∠a + ∠d = 180°. So ∠b and ∠d are both 180° − ∠a, and they are equal whatever ∠a is.

A justification that shows by general reasoning, without assuming particular values, that a statement is true in every case is called a proof.

• Checking some crossings | A proof
• Works for the cases we tried | Works for every case
• Uses particular sizes | Assumes no particular size

Notes

A justification that shows by general reasoning, without assuming particular values, that a statement is true in every case is called a proof.

Check yourself

Meena measures ∠b and ∠d in 20 crossings. They are equal every time. Has she proved that ∠b = ∠d in every crossing?

Rani says: “∠a is always bigger than ∠b.” It is true when ∠a = 120° and ∠b = 60°. Is her claim true in every crossing?

A proof must work for every crossing. Which statement can it not use?

The proof shows ∠b = ∠d for every crossing. In one crossing ∠a = 35°. Find ∠b + ∠d.

Answer: 290 °

∠b = 180° − 35° = 145°, and ∠d = ∠b = 145°. So ∠b + ∠d = 145° + 145° = 290°.

• Yes. Twenty is plenty.. However many we check, endless crossings are left. Only general reasoning covers all of them.
• No. Twenty crossings are only twenty cases. A proof must work for every case. — correct. Yes! Measuring can only show that it worked for the crossings she measured.
• Yes, if she measured very carefully.. Careful measuring still only covers the crossings she measured.
• Yes. It worked for the example.. It worked for that crossing only. When ∠a = 70°, ∠b = 110°, and then ∠a is smaller.
• Yes, because 120° is bigger than 60°.. That is only the one crossing she tried. When ∠a = 70°, ∠b = 110°.
• No. When ∠a = 70°, ∠b = 110°, so ∠a is not bigger. One example cannot prove a claim. — correct. Yes! One example that works proves nothing. One crossing where it fails shows the claim is false.
• ∠a = 120° — correct. Right! That is one particular size. It would only work for crossings with ∠a = 120°.
• ∠a + ∠b = 180°. This is true in every crossing. It uses the straight angle, not a particular size.
• ∠b = 180° − ∠a. This is true whatever ∠a is, so a proof can use it.