Theorem 6.7: (SSS) Two triangles are congruent if and only if their corresponding sides all have the same lengths.
Proof: If they are congruent, then it is possible to move one of the triangles until it coincides with the other by an invertible isometry. Since, by definition, an isometry preserves distances, the sides that will coincide after the movement will have to all have the same lengths by Theorem 6.5.
For the converse, assume that for the three sides in triangle ABC and triangle A'B'C',
Translate A' to A, and rotate the plane about A until B' coincides with B, as in the proof of Theorem 6.5. Then C' will have been moved to a point whose distance from A is |AC| and whose distance from B will be |BC|. That is, it will be on the circle centered at A of radius |AC| and on the circle centered at B of radius |BC|.
By the triangle inequality, Theorem 3.5, and Theorem 3.13, there are two such points, and by Theorem 4.12, they are reflections of each other about the line determined by AB. So either C' coincides with C or it will after a reflection, and the triangles are congruent.
next theorem (6.8)