list two congruence conditions for the law of sines yields a unique solution
Your question makes no grammatic sense. The law of sines is valid for any triable, and has nothing to do with conguence. Congruence conditions apply to pairs of triangles, and are:
side/side/side
side/angle/side
angle/side/angle
angle/side/side (ambiguous case)
See
http://en.wikipedia.org/wiki/Congruence_(geometry)
The Law of Sines states that for any triangle with sides a, b, and c, and angles A, B, and C respectively, the following congruence conditions must hold to yield a unique solution:
1. Side-Angle-Side (SAS) Congruence: If you know the lengths of two sides and the measure of the angle between them (ASA condition), you can use the Law of Sines to find the third side and the other two angles.
2. Angle-Side-Angle (ASA) Congruence: If you know the measures of two angles and the length of the side between them (SAS condition), you can use the Law of Sines to find the lengths of the other two sides and the remaining angle.
In both cases, the Law of Sines can be used to establish a unique solution.
To have a unique solution when using the Law of Sines, we need to satisfy two congruence conditions.
1. The Ambiguous Case:
The first condition is to avoid encountering the ambiguous case. This occurs when the given information does not uniquely determine a triangle. The ambiguous case arises when there are two possible triangles that can be formed using the given side lengths and angle measures. To ensure a unique solution, we must verify that we are not dealing with the ambiguous case. This can be done by calculating the value of the sine of the given angle and comparing it to the relative side lengths.
2. The Sine Ratio:
The second condition involves using the sine ratio to find angles or side lengths. When applying the Law of Sines, we must ensure that the ratio formed by any pair of angle-side opposite pairs is consistent throughout the triangle. In other words, the sine ratios for each angle-side opposite pair should be equal. If the ratios are not equivalent, it could indicate an error in the calculations or an invalid triangle.
By satisfying these two congruence conditions, we can use the Law of Sines to determine a unique solution for a triangle.