Definition: The SAS criterion states that if two sides and the included angle of one triangle match those of another, the triangles are congruent.
What does included mean?
The angle must sit between the two sides. If you know sides AB and AC, the included angle is angle A. A matching angle at a different vertex does not qualify.
How is SAS used in a proof?
Show two pairs of congruent sides and the congruent angle between them. Often the angle comes from vertical angles, a shared angle, or parallel lines.
Example. Segments AC and BD intersect at M, which is the midpoint of both. Then AM = MC and BM = MD. The angles AMB and CMD are vertical angles, so they are congruent. Triangle AMB is congruent to triangle CMD by SAS, and so AB = CD.
What do students mistake?
Using SSA, two sides and a non-included angle, which does not guarantee congruence. A triangle with sides 8 and 11 and a 30 degree angle opposite the 8 can be drawn in two different ways. Another error is forgetting to state the reason for each congruent part in a proof.