Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B1 is smaller than ∠B2.)


To solve a triangle using the Law of Sines, we need specific information about the triangle such as two angles and one side, or two sides and one angle.

Since you haven't provided specific values, let's outline how to solve the problem step by step and identify any missing information:

### **Steps for solving triangles using the Law of Sines:**

1. **Write down the Law of Sines formula**:
   \[
   \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
   \]
   Here, \( a \), \( b \), and \( c \) are the sides of the triangle, and \( A \), \( B \), and \( C \) are the opposite angles.

2. **Identify the given information**:
   - Are you given two angles and one side (AAS or ASA)?
   - Or, are you given two sides and an angle (SSA)?

3. **Solve for missing angles or sides**:
   - **Case 1**: If you know two angles, you can find the third angle using the fact that the sum of the angles in a triangle is always 180°.
   - **Case 2**: If you know two sides and an angle opposite one of them (SSA case), you may get two possible solutions due to the ambiguous case in the Law of Sines.

4. **Solve for multiple solutions (Ambiguous case)**:
   In the SSA case, the Law of Sines can yield two possible triangles (one acute, one obtuse). If the sine of angle \( B \) gives you two values (one acute, one obtuse), you'll have two possible solutions:
   - A triangle where \( \angle B_1 \) is the smaller angle.
   - A triangle where \( \angle B_2 \) is the larger angle.

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Please provide the specific triangle data (such as angles or sides) so I can apply the Law of Sines to solve it accurately. 

Would you like more details on how the ambiguous case works? Here are 5 questions for further exploration:

1. How does the Law of Sines help solve triangles with two angles and one side?
2. Can the Law of Sines be applied to all triangles?
3. What is the ambiguous case, and when does it arise?
4. How can we determine if a triangle has one solution or two using the Law of Sines?
5. In what situations does the Law of Sines yield no solution?

**Tip**: When dealing with the ambiguous case (SSA), always check if both angles are possible (one acute, one obtuse). If the sum of the obtuse angle and the known angle exceeds 180°, then the obtuse triangle is impossible!

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B1 is smaller than ∠B2.)

Learn how to solve triangles using the Law of Sines, including handling the ambiguous case where two possible triangles exist. This guide explains step-by-step how to calculate multiple angles (∠B1 and ∠B2) and identify whether the triangle has one or two solutions. Suitable for Grades 10-12.

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