Looking at numerous examples of triangles, each with different properties, geometers develop their understanding of congruency. They use the notation of a counter-example to disprove certain conjectures and prove geometric theorems and...

Working individually and collaboratively, geometers gain a clear understanding of the Pythagorean theorem. They create, explain, and compare proofs of the theorem. Proofs involve areas of trapezoids, squares, and triangles, congruent...

MathWorld gives a discussion of the use of two-column proofs by high school and middle school students and its relevance to mathematicians.