The question is about the geometric proofs for the congruency of triangles, which involves criteria like SSS, SAS, ASA, and AAS. The Law of Sines and Law of Cosines can also play a role in proving congruency, especially when the triangles include non-right angles.
The student is asking about geometric proofs of congruency for two triangles. To prove that two triangles are congruent, we can use several criteria like Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). An understanding of the Law of Sines and Law of Cosines can also be used to establish congruency in some cases.
For example, if triangles have three pairs of sides that are equal in length (SSS), or two sides and the included angle equal (SAS), or two angles and the included side equal (ASA), or two angles and a non-included side equal (AAS), they are congruent. The proofs of congruency would involve showing that one of these criteria is met for both triangles in question.
The reference to the Pythagorean theorem is for right-angled triangles, and it complements the congruency proofs when right angles are involved. Remember, a proper proof would provide a logical sequence of statements and reasons that lead to the conclusion of congruency.