In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Verified step by step guidance

1. Step 1: Identify the type of triangle (right, acute, or obtuse) based on the given information.
2. Step 2: Use the appropriate trigonometric ratios (sine, cosine, tangent) or the Pythagorean theorem to find missing side lengths or angles.
3. Step 3: If necessary, apply the Law of Sines or the Law of Cosines to solve for unknown sides or angles.
4. Step 4: Round the calculated side lengths to the nearest tenth and angle measures to the nearest degree as instructed.
5. Step 5: Double-check your calculations to ensure all sides and angles are consistent with the properties of the triangle.

Key Concepts

Triangle Types

Understanding the types of triangles—such as scalene, isosceles, and equilateral—is essential for solving triangle problems. Each type has unique properties that affect how angles and sides relate to one another. For instance, in an isosceles triangle, two sides are equal, which can simplify calculations when finding unknown angles or lengths.

Law of Sines

The Law of Sines is a fundamental principle used to solve triangles when given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA). It states that the ratio of a side length to the sine of its opposite angle is constant across the triangle. This law is particularly useful for finding unknown angles or sides in non-right triangles.

Law of Cosines

The Law of Cosines is another critical tool for solving triangles, especially when dealing with three sides (SSS) or two sides and the included angle (SAS). It relates the lengths of the sides of a triangle to the cosine of one of its angles, allowing for the calculation of unknown sides or angles. This law is particularly useful when the triangle is not a right triangle.