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
$$
A B=14, B C=21, \mathrm{m} \angle C=75
$$

Key Concepts

Ambiguous Case in Triangles

This concept refers to the scenario encountered when two sides and an angle not enclosed by them (SSA) are given. In this situation, there can be no triangle, one unique triangle, or two distinct possible triangles, depending on the given measurements. The ambiguity arises because the sine function has the same value for an acute angle and its supplement, potentially leading to two solutions for one of the angles.

Law of Sines

The Law of Sines is a trigonometric relationship which states that the ratios of each side length to the sine of its opposite angle are equal in any triangle. This law is particularly useful when solving for unknown sides or angles in cases where standard congruence conditions (such as SSA) are applied, and it is instrumental in identifying and resolving the ambiguous case scenarios.

Triangle Existence Conditions

When determining the number of possible triangles that can be constructed from given measurements (especially in SSA cases), it is essential to use triangle existence conditions. This involves comparing the provided side lengths and specified angle to determine if the given measures result in a valid triangle and, if so, whether one or two triangles are possible. These conditions help in identifying instances of no solution, a single unique solution, or the ambiguous case with two potential solutions.

Angle Sum Property of Triangles

The Angle Sum Property states that the sum of the interior angles in any triangle is always 180 degrees. This fundamental property is used to calculate the remaining unknown angle(s) once one or two angles have been determined, providing a necessary check and ensuring the consistency of the triangle's internal angle measurements.

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