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Problem Type Theorem
SSS
Congruence: Two triangles with equal corresponding side lengths must be congruent.
SAS
Congruence: Two triangles with two equal corresponding side lengths and congruent angle included between the sides must be congruent.
SAA
Congruence: Two triangles with one equal side length and two congruent angles must be congruent.
ASA
Congruence: Two triangles with two congruent angles and equal included side length must be congruent.
AAA
Similarity: Two triangles with congruent corresponding angles must be similar but not necessairly congruent.
This similarity theorem is often called "AA" since two congruent angles imply that the third is also congruent.
SSA
Neither: In this case, similarity is not implied.
Congruence and Similarity Theorems
Problem Type Theorem
SSS
Congruence: Two triangles with equal corresponding side lengths must be congruent.
Solve for any of the angles using the law of cosines. For example:
SAS
Congruence: Two triangles with two equal corresponding side lengths and congruent angle included between the sides must be congruent.
Solve for the length of the third side using the law of cosines, then you can find any of the remaining angles using the law of cosines as in an SSS-type problem.
Solving for the third side:
SAA
Congruence: Two triangles with one equal side length and two congruent angles must be congruent.
Whenever you are given two angles and one side, you can find the third angle using the fact that the angles must sum to . Then, use the law of sines to solve for any of the remaining sides:
ASA
Congruence: Two triangles with two congruent angles and equal included side length must be congruent.
Use the law of sines as in the SSA case.
AAA
Similarity: Two triangles with congruent corresponding angles must be similar but not necessairly congruent.
This similarity theorem is often called "AA" since two congruent angles imply that the third is also congruent.
SSA
Neither: In this case, similarity is not implied.
This is the ambiguous case for the law of sines, which may yield zero, one, or two solutions. You won't be asked to solve these types of problems in this course.
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