In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 95, c = 125, A = 49°

Verified step by step guidance

1. Identify the given elements: side \(a = 95\), side \(c = 125\), and angle \(A = 49^\circ\). Note that angle \(A\) is opposite side \(a\).
2. Use the Law of Sines to find angle \(C\) or to check the number of possible triangles. The Law of Sines states: \(\frac{a}{\sin A} = \frac{c}{\sin C}\).
3. Rearrange the Law of Sines to solve for \(\sin C\): \(\sin C = \frac{c \cdot \sin A}{a}\).
4. Calculate \(\sin C\) using the given values (do not find the final value yet). Then analyze the value of \(\sin C\) to determine the number of possible triangles: if \(\sin C > 1\), no triangle; if \(\sin C = 1\), one right triangle; if \(0 < \sin C < 1\), two possible triangles (since \(\sin\) is positive in two quadrants).
5. If one or two triangles exist, find angle \(C\) by taking \(\arcsin(\sin C)\) for the first solution, and for the second solution use \(180^\circ - C\). Then find angle \(B\) using the triangle angle sum \(B = 180^\circ - A - C\), and finally find side \(b\) using the Law of Sines: \(\frac{b}{\sin B} = \frac{a}{\sin A}\).

Key Concepts

Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is essential for solving triangles when given two sides and a non-included angle (SSA), as it helps find unknown angles or sides.

Ambiguous Case of SSA Triangles

The SSA configuration can produce zero, one, or two possible triangles, known as the ambiguous case. This depends on the given side lengths and angle, particularly the height relative to the known side. Understanding this helps determine how many triangles satisfy the given measurements.

Triangle Solution and Rounding

After determining the number of triangles, solving involves calculating missing sides and angles using trigonometric laws. Final answers should be rounded appropriately—angles to the nearest degree and sides to the nearest tenth—to provide clear, practical results consistent with the problem's instructions.