Which one of the following sets of data does not determine a unique triangle?
a. A = 50°, b = 21, a = 19
b. A = 45°, b = 10, a = 12
c. A = 130°, b = 4, a = 7
d. A = 30°, b = 8, a = 4

Verified step by step guidance

1. Identify the given information for each case: angle A, side b, and side a. We want to check if these data sets determine a unique triangle.
2. Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). We can use this to find angle B for each set by rearranging to \(\sin B = \frac{b \sin A}{a}\).
3. Calculate \(\sin B\) for each set using the given values (without final numeric evaluation): \(\sin B = \frac{b \sin A}{a}\). Then analyze the possible values of angle B.
4. Check the range of \(\sin B\). If \(\sin B > 1\), no triangle exists. If \(0 < \sin B < 1\), there can be one or two possible angles for B (since \(\sin \theta = \sin (180^\circ - \theta)\)), which affects uniqueness.
5. Determine for each set whether the data leads to zero, one, or two possible triangles. The set that leads to two possible triangles does not determine a unique triangle.

Key Concepts

Law of Sines

The Law of Sines relates the ratios of sides to the sines of their opposite angles in a triangle: (a/sin A) = (b/sin B) = (c/sin C). It is essential for solving triangles when given two sides and an angle not included between them (SSA), helping determine possible triangle configurations.

Ambiguous Case of SSA Triangles

The SSA condition can produce zero, one, or two possible triangles depending on the given measurements. This ambiguity arises because the given side opposite the known angle may or may not form a valid triangle, leading to no solution, a unique triangle, or two distinct triangles.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. This property helps verify the validity of computed angles and ensures that the solutions derived from the Law of Sines or other methods correspond to a real triangle.