Challenge 22: Congruent Sides and Angles of Triangles
- Gerry Stahl
Two triangles are called “congruent” if all their corresponding angles and sides are equal size. However, you can constrain two triangles to be congruent by just constraining 3 of their corresponding parts to be equal – for certain combinations of 3 parts. Dynamic geometry helps you to visualize, to understand and to remember these different combinations. What constraints of sides and angles are necessary and sufficient to constrain the size and shape of a triangle? Two congruent triangles have 6 corresponding parts equal (3 sides and 3 angles), but you do not have to constrain all 6 parts to be equal in order to make sure the triangles are congruent. For instance, two triangles with their corresponding 3 angles equal are called “similar” but they may not be “congruent.” You can drag one of them to be larger that the other one. They will still have the same shape, but the corresponding side lengths of one will all be larger than the side lengths of the other triangle. If two triangles have their corresponding angles constrained to be equal and then you constrain two corresponding sides to be equal length, will the triangles necessarily be congruent? Suppose you only constrained one of the corresponding sides to be equal? Explore different combinations of 4 or 5 of the 6 corresponding triangle parts being constrained to be equal. Which combinations guarantee that the triangles are congruent?