Theorem 6.6: Two angles are congruent if and only if they have the same size.

Proof: If the angles are congruent, then it is possible to move one onto the other by an invertible isometry. Since such isometries preserve the size of angles by Theorem 5.6, the sizes of the angles will both be the same.

For the converse, suppose that   / ABC   has the same size as   / A'B'C'.   Translate   A'   to   A.   Rotate the plane about   A   until   B'   is on the ray from   A   through   B.   By Theorem 5.13, there are two angles having the ray from   A   through   B   as one arm, and they are both reflections of each other about the line determined by   AB.   So if   B'   is not on the ray from   A   through   B,   then it will be after reflecting about the line determined by   AB.   In either case it will be possible to move   / A'B'C'   to   / ABC   by a translation, rotation, and perhaps a reflection, so the angles will be congruent.

next theorem (6.7)