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
$$
a=9, c=10, \mathrm{m} \angle C=150
$$

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Key Concepts

Law of Sines

The Law of Sines states that in any triangle, the ratios of each side length to the sine of its opposite angle are equal. This law is especially useful in solving triangles when given two sides and a non-included angle (SSA condition). It provides a way to determine unknown angles or sides, and it is key to identifying potential multiple solutions in ambiguous cases.

Ambiguous Case (SSA Condition)

The ambiguous case arises in triangle problems when two sides and an angle not included between them are known. Under this condition, using the Law of Sines can sometimes yield two different valid solutions for a triangle, one solution, or even no solution at all depending on how the given measurements relate to the triangle's height. Recognizing the ambiguous case is crucial for determining all possible triangles consistent with the provided measures.

Triangle Existence and Angle Sum Property

For any set of measurements to form a valid triangle, the resulting angles must add up to 180 degrees, and all computed angles must be positive and less than 180 degrees. This requirement ensures that after applying trigonometric relations like the Law of Sines, the identified angles yield a geometrically possible triangle. It also helps in verifying the validity of multiple solutions derived from the ambiguous case.

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