Side Side Side

• If the three sides of triangle A are the same length as the three sides of triangle B, these two triangles are congruent. Students may find this too obvious to mention, but teachers can illustrate the difference between triangles and other polygons by drawing a square and a non-square parallelogram on the chalkboard, with sides the same length. The two shapes are clearly not congruent, though their sides are the same length. Students can attempt to construct two triangles with the same length sides, but different internal angles. This cannot be done, illustrating this unique property of triangles.
  
  

Side Angle Side

• If triangle A has two sides and the included angle congruent to two sides and the included angle of triangle B, the two triangles are congruent. It's important to note that two triangles with two congruent sides and one congruent angle are not necessarily congruent. This is only the case if the angle is the included angle of the two sides. To illustrate this, teachers can ask the class to construct two non-congruent triangles with two congruent sides and one congruent angle. They may be surprised to find this can be done.
  
  

Hypotenuse Leg

• Two right triangles with the same hypotenuse and one side the same length are congruent. This is the only case where you can prove two triangles are congruent knowing only that two sides and one non-included angle are congruent. Students should think how this property can be derived directly from the Pythagorean theorem. If a^2+b^2=c^2, and you don't change the value of "c" or "b" (the length of the hypotenuse and one side), then the value of "a" must be unchanged as well.
  
  

Angle Side Angle (and Angle Angle Side)

• If two angles and any side of one triangle are congruent to two angles and any side of another triangle, the triangles are congruent. It doesn't matter whether or not the side is an included side or a non-included side, the triangles are congruent either way. Students should make the distinction between this and the Side Angle Side approach, which does not necessarily apply if the congruent angle is not an included angle. Students are also encouraged to try to construct any two triangles with two and only two congruent angles. They will quickly find this impossible. If a pair of triangles have two congruent angles, then the third is automatically congruent.