Theorem 6.11: (AAS) Two triangles are congruent if and only if two angles and the side next to one of them in one triangle are congruent to two angles and the corresponding side in a second triangle, then the triangles are congruent.

Proof: If the two triangles are congruent, then when the whole triangle is moved to the other triangle, the three angles and three sides of the one triangle will be moved to the corresponding parts of the other triangle and they will all have to be congruent. As a result, any two angles and the side between them in one triangle will be congruent to the corresponding two angles and the side between them in a second triangle.

For the converse, if two angles are congruent, then by Theorem 6.6, they are the same size. Then since the angles in a triangle have to add up to 180^o by Theorem 6.3, all three corresponding angles have to be the same size, and the triangles are similar. By Theorem 6.9, the ratios of the lengths of the corresponding sides are all the same. Since one pair of corresponding sides have the same length, that ratio is 1, and all of the corresponding sides have the same length, so the triangles are congruent by Theorem 6.7.

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