Solve each triangle ABC.
A = 68.41°, B = 54.23°, a = 12.75 ft

Verified step by step guidance

1. Identify the given information: angle A = 68.41°, angle B = 54.23°, and side a = 12.75 ft. Since two angles and one side are given, this is an ASA (Angle-Side-Angle) case.
2. Find the third angle C using the fact that the sum of angles in a triangle is 180°. Use the formula: \(C = 180^\circ - A - B\).
3. Use the Law of Sines to find side b. The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearrange to solve for \(b\): \(b = \frac{a \sin B}{\sin A}\).
4. Similarly, use the Law of Sines to find side c: \(\frac{a}{\sin A} = \frac{c}{\sin C}\), so \(c = \frac{a \sin C}{\sin A}\).
5. After calculating sides b and c, summarize the triangle with all sides and angles known.

Key Concepts

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third by subtracting their sum from 180°, which is essential for solving the triangle completely.

Law of Sines

The Law of Sines relates the sides and angles of a triangle: (a/sin A) = (b/sin B) = (c/sin C). It is used to find unknown sides or angles when given some combination of sides and angles, especially in non-right triangles.

Solving Triangles

Solving a triangle means finding all unknown sides and angles. Using known angles and sides, along with the Law of Sines and angle sum property, you can systematically determine all missing measurements.