In 3-14 : a. Determine the number of possible triangles for...

In $3-14 :$ a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree
$$
B C=12, A C=12 \sqrt{2}, \mathrm{m} \angle B=135
$$

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Key Concepts

Ambiguous Case (SSA Scenario)

The ambiguous case, associated with the SSA (side-side-angle) condition, occurs when two sides and a non-included angle of a triangle are known. In this configuration, depending on the measurements, there may be one unique triangle, two distinct possible triangles, or no triangle at all. This case requires careful application of the Law of Sines and consideration of the triangle’s geometric constraints to determine the number of possible solutions.

Triangle Angle Sum Property

The Triangle Angle Sum Property states that the sum of the interior angles of any triangle is always 180 degrees. This principle is critical when solving triangles, as it allows for the determination of the third angle when the other two angles are known, ensuring that all calculated angles are consistent with the geometric properties of triangles.

Law of Sines

The Law of Sines is a fundamental relation in trigonometry which states that in any triangle, the ratios of the lengths of the sides to the sine of their opposite angles are equal. It is especially useful for solving triangles when given two angles and a side (AAS or ASA) or two sides and an angle not included between them (SSA), and it often plays a central role in determining unknown sides or angles in these situations.

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