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
$$
c=18, b=10, \mathrm{m} \angle C=120
$$

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Key Concepts

Ambiguous Case in Triangles (SSA Condition)

This concept deals with situations where two sides and an angle not included between them are given, which can potentially lead to no triangle, one triangle, or two distinct triangles due to how the angle can be positioned relative to the given side lengths. Analyzing this case requires careful use of trigonometric functions and consideration of possible geometric configurations.

Law of Sines

The Law of Sines is a fundamental relation in any triangle that relates the lengths of its sides to the sines of its opposite angles. It is particularly useful in solving for unknown angles or sides in non-right triangles, especially in ambiguous SSA cases where multiple triangle orientations must be analyzed.

Triangle Angle Sum Theorem

In any triangle, the sum of the three interior angles must equal 180 degrees. This theorem is crucial for determining the remaining angle once two angles are known, and it validates whether a computed set of angles can constitute a valid triangle.

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