For the following exercises, assume ? is opposite side a, ?...

For the following exercises, assume ? is opposite side a, ? is opposite side b, and ? is opposite side c. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth. $$a=7, c=9, \quad \alpha=43^{\circ} $$

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Key Concepts

Law of Sines

The Law of Sines is a trigonometric relation in any triangle that relates the ratios of each side to the sine of its opposite angle. It is often used to solve triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA), laying the foundation for finding unknown sides or angles by setting up proportions.

Ambiguous Case in Triangle Solving

The ambiguous case arises in triangle solving when two sides and an angle not between them (SSA) are provided. Under certain conditions, this configuration may correspond to no triangle, one triangle, or two distinct triangles. Determining which scenario applies involves comparing the given side lengths with the altitude from the known angle, and understanding when the sine function can yield two possible angle measures.

Angle Sum Property

The Angle Sum Property of triangles states that the sum of the interior angles of any triangle is always 180°. This fundamental rule is used to determine the remaining angles once one or two angles have been computed, ensuring that the triangle’s angles form a valid triangle.

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