Students will infer the triangle inequality by playing with AngLegs. Remove all pieces except those having a few specific lengths, such as 2, 5, and 6. Have students make two columns on a sheet of paper, possible and impossible. The possible column will contain triples of side lengths that can form triangles, such as 2, 5, 6, while the impossible column will have triples such as 2, 2, 5. Lead them to the triangle inequality by asking for a rule that determines when a triangle is possible. Hopefully students will find that a triangle is possible only when the two shorter sides add up to be longer than the longest side. There is no need to make the triangle inequality formal at this point. Conclude by giving your students the following challenge:
Problem: The ancient Egyptians were said to make right-angled triangles using ropes knotted at unit intervals. If you have a rope knotted like this, with 12 equal sections, what triangles can you make? For example, one triangle has side lengths 3, 4, and 5. This happens to be a right triangle. Not all the triangles you find have to be right triangles, but there must be a knot at each vertex. What rectangles can you make? regular polygons?