Solve each triangle. Approximate values to the nearest tenth.


<IMAGE>

Verified step by step guidance

1. Identify the given elements in the triangle: note which sides and angles are provided. Typically, you will have either two sides and an included angle (SAS), two angles and a side (AAS or ASA), or all three sides (SSS).
2. Choose the appropriate trigonometric rule based on the given information. Use the Law of Sines if you have an angle-side opposite pair, or the Law of Cosines if you have two sides and the included angle.
3. Set up the Law of Sines formula: \(\frac{a}{\sin\!A} = \frac{b}{\sin\!B} = \frac{c}{\sin\!C}\), or the Law of Cosines formula: \(c^2 = a^2 + b^2 - 2ab \cos\!C\), depending on the known values.
4. Solve for the unknown sides or angles step-by-step. For angles, use the inverse sine or cosine functions: \(A = \sin^{-1}\left(\frac{a \sin\!B}{b}\right)\) or \(C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)\).
5. Once all sides and angles are found, round each value to the nearest tenth as requested, and verify that the sum of the angles is approximately 180 degrees to confirm the solution's consistency.

Key Concepts

Triangle Classification and Properties

Understanding the types of triangles (right, acute, obtuse) and their properties is essential. This helps determine which trigonometric rules or formulas apply when solving for unknown sides or angles.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides. These ratios are fundamental for calculating unknown sides or angles in right triangles using given measurements.

Law of Sines and Law of Cosines

These laws extend trigonometric problem-solving to any triangle, not just right triangles. The Law of Sines relates sides and angles proportionally, while the Law of Cosines generalizes the Pythagorean theorem for non-right triangles.