Using the information given about a triangle, which law must you use to solve the triangle? Law of Sines, Law of Cosines, or Neither.

ASA
SSS
SAS
AAA
SSA
AAS

To determine which law to use to solve a triangle, we can use the following guidelines:

1. ASA (Angle-Side-Angle):
Given two angles and the included side, we can use the Law of Sines or Law of Cosines to solve the triangle.

2. SSS (Side-Side-Side):
Given all three sides, we can use the Law of Cosines to solve the triangle.

3. SAS (Side-Angle-Side):
Given two sides and the included angle, we can use the Law of Cosines or Law of Sines to solve the triangle.

4. AAA (Angle-Angle-Angle):
Given all three angles, we cannot solve the triangle because the side lengths cannot be determined.

5. SSA (Side-Side-Angle):
Given two sides and a non-included angle, we cannot use the Law of Sines or Law of Cosines to solve the triangle because it may result in ambiguous cases.

6. AAS (Angle-Angle-Side):
Given two angles and a non-included side, we can use the Law of Sines to solve the triangle.

Therefore, the appropriate law to use for each case is:

- ASA: Law of Sines or Law of Cosines
- SSS: Law of Cosines
- SAS: Law of Cosines or Law of Sines
- AAA: Neither
- SSA: Neither
- AAS: Law of Sines

To determine which law to use to solve a triangle based on the given information, you can use the following guidelines:

1. ASA (Angle-Side-Angle): The ASA condition provides information about two angles and the included side of the triangle. To solve the triangle, you would typically use the Law of Sines or Law of Cosines.

2. SSS (Side-Side-Side): The SSS condition provides information about the lengths of all three sides of the triangle. To solve the triangle, you would typically use the Law of Cosines.

3. SAS (Side-Angle-Side): The SAS condition provides information about two sides and the included angle of the triangle. To solve the triangle, you can use the Law of Cosines or Law of Sines.

4. AAA (Angle-Angle-Angle): The AAA condition provides information about all three angles of the triangle. However, this information alone is insufficient to determine the lengths of the sides. Therefore, you cannot solve the triangle with this information alone.

5. SSA (Side-Side-Angle): The SSA condition provides information about two sides and a non-included angle of the triangle. In most cases, this information is insufficient to determine a unique triangle. It may result in the possibility of one, two, or no triangle solutions.

6. AAS (Angle-Angle-Side): The AAS condition provides information about two angles and a non-included side of the triangle. To solve the triangle, you would typically use the Law of Sines or Law of Cosines.

In summary, you would use the Law of Sines or Law of Cosines for ASA, SAS, or AAS conditions, and the Law of Cosines for SSS conditions. Neither the Law of Sines nor the Law of Cosines can be used for solving triangles with the AAA condition, and SSA conditions may not provide enough information for a unique solution.