Statement, converse and proof
Split a theorem into Voraussetzung and Behauptung, form the Kehrsatz and refute it with a Gegenbeispiel, and prove and use the Satz des Pythagoras.
- France: grade 7
- Germany: grade 7
- Syria: grade 7
- Palestine: grade 8
- Ukraine: grade 7
Guide
A theorem has the form "If … then …". The part after "if" is the Voraussetzung, the part after "then" is the Behauptung. A proof leads from the Voraussetzung to the Behauptung.
If you swap both parts, you get the Kehrsatz. The Kehrsatz is not automatically true: it is either proved or refuted by a Gegenbeispiel.
The Satz des Pythagoras can be proved by completing a square: (a + b)² = c² + 4 · ½ab. Its Umkehrung is true as well and decides whether a triangle is right-angled.