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 = 10, b = 30, A = 150°

Verified step by step guidance

1. Step 1: Identify the given information. You have side a = 10, side b = 30, and angle A = 150°. This is an SSA (Side-Side-Angle) scenario.
2. Step 2: Use the Law of Sines to find angle B. The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the known values: \( \frac{10}{\sin 150°} = \frac{30}{\sin B} \).
3. Step 3: Solve for \( \sin B \). Rearrange the equation to find \( \sin B = \frac{30 \cdot \sin 150°}{10} \). Calculate \( \sin 150° \) using known trigonometric values.
4. Step 4: Determine the possible values for angle B. Since \( \sin B \) can have two possible angles (B and 180° - B) in the range of 0° to 180°, check both possibilities to see if they form valid triangles.
5. Step 5: Check the sum of angles. For each possible angle B, calculate angle C using \( C = 180° - A - B \). Ensure that the sum of angles in the triangle is 180° to confirm valid triangles.

Key Concepts

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides and angles in a triangle. This law is particularly useful in SSA (Side-Side-Angle) cases, allowing us to determine unknown angles or sides when two sides and a non-included angle are known.

Ambiguous Case of SSA

The SSA condition can lead to an ambiguous situation where two different triangles may be formed, one triangle, or no triangle at all. This ambiguity arises because the given angle and side lengths may not uniquely determine a triangle. Understanding how to analyze the given measurements is crucial to identifying whether one, two, or no triangles can be constructed.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential when solving triangles, as it allows us to find unknown angles once we have determined some angles using the Law of Sines. It ensures that the angles calculated from the given sides and angles adhere to the fundamental properties of triangles.