Theorem 6.10: (ASA) Two triangles are congruent if and only if two angles and the side between them in one triangle are congruent to two angles and the side between them in a second triangle, then the triangles are congruent.

Proof: If two angles in one triangle are congruent to two corresponding angles in a second triangle, then, since the angles in a triangles add up to 180o, by Theorem 6.3, it follows that the third angles are also congruent, the triangles are similar, and, by Theorem 6.9 the ratios of the lengths of corresponding sides are the same. Since one pair of corresponding sides have the same length, all of the corresponding sides have the same length, so by Theorem 6.5, all pairs of corresponding sides will be congruent, and the triangles will be congruent by Theorem 6.7 (SSS).

For the converse, if the triangles are congruent then all corresponding sides and angles will be congruent.

next theorem (6.11)