In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.B = 5°, C = 125°, b = 200

Verified step by step guidance

Step 1: Use the angle sum property of triangles to find angle A. Since the sum of angles in a triangle is 180°, calculate A = 180° - B - C.

Step 2: Apply the Law of Sines to find side a. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Rearrange to solve for a: \( a = b \cdot \frac{\sin A}{\sin B} \).

Step 3: Use the Law of Sines again to find side c. The formula is \( \frac{c}{\sin C} = \frac{b}{\sin B} \). Rearrange to solve for c: \( c = b \cdot \frac{\sin C}{\sin B} \).

Step 4: Calculate the sine values for angles A, B, and C using a calculator, ensuring the calculator is set to degree mode.

Step 5: Substitute the known values into the equations from Steps 2 and 3 to find the lengths of sides a and c, rounding to the nearest tenth.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. In this problem, knowing two angles allows us to find the third angle by subtracting the sum of the known angles from 180. This is essential for solving the triangle and determining the remaining unknowns.

Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of its angles. It is expressed as a/b = sin(A)/sin(B) = c/sin(C). This law is particularly useful in non-right triangles, allowing us to find unknown side lengths or angles when given sufficient information, as in this exercise.

Rounding Rules

Rounding rules dictate how to approximate numbers to a specified degree of accuracy. In this problem, lengths must be rounded to the nearest tenth and angles to the nearest degree. Understanding these rules is crucial for providing answers that meet the problem's requirements and ensuring clarity in the final results.

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