Answer:
- sines: AAS, ASA, SSA
- cosines: SAS, SSS
- ambiguous: SSA
Explanation:
You want to identify the solution process used for triangles when the information given is AAS, ASA, SSA, SAS, SSS. And you want to identify which of these cases is ambiguous.
Law of sines
The law of sines is useful when an angle and its opposite side are given, and one more of an angle or side.
In the ASA case, the angle opposite the side must be figured from the other two, but that is still solved using the law of sines.
The law of sines is used for AAS, ASA, and SSA.
Law of cosines
The law of cosines is useful for solving the triangle when two sides and the angle between are given or when all three sides are given.
The law of cosines is used for cases SAS and SSS.
Ambiguous
All of these abbreviations, except SSA, are representative of congruence postulates. The reason that SSA is not a congruence postulate is that it can be an ambiguous case. If the angle is opposite the smaller given side, then there may be two (or no) solutions to the triangle.
The ambiguous case is SSA.
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Additional comment
Essentially, HL is equivalent to SSA congruence. The reason it can be used in that case is that the angle (90°) is known to be opposite the longest side (H).
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