Geometry and proof

Ages 13 to 17 Read alone · about 40 minutes

Read together

A proof is a chain of steps, each following from earlier facts, that shows a statement must be true.

Euclid built geometry from a few starting ideas, and people have studied his Elements for over two thousand years.

Do it

1. Prove that the angles in a triangle add to 180 degrees. Draw a line through one vertex parallel to the opposite side.
2. Use alternate angles to show the three angles make a straight line.
3. Write each step with a reason.
4. Try to find a counterexample to a false claim, such as 'all triangles have a right angle'.

Talk about it

• What does each step rely on?
• Why is one example not a proof?

For the grown-up

Proof teaches careful reasoning. Euclid's method has long been used to teach logic. A ruler and a compass are fine; use blunt compasses.

Mark as done

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Where this comes from

• Euclid, Elements, Book I (translation by T. L. Heath, Cambridge University Press, 1908)
• Common Core State Standards for Mathematics (2010), high school geometry, congruence and proof (G-CO.C)

Written with AI assistance and checked against the sources listed. Please read it before sharing it with a child.