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In this tutorial, the instructor explains the concept of proving triangle congruence using the side-angle-side (SAS) criterion. They emphasize the importance of having an included side, which is when an angle is between two sides of a triangle. By identifying the side-angle-side pattern in two triangles, it is possible to prove their congruence. However, if the angle is not between the two sides, the triangles are no longer congruent by the SAS criterion. Understanding the criteria for congruence, specifically the role of an included side, is crucial in proving triangles congruent.