Solve triangle ABC. (If an answer does not exist, enter DNE. Enter all angles in degrees. Round your answers to one decimal place. Enter your answers so that
∠A1 is smaller than ∠A2.)
b = 124,    c = 165,    ∠B = 45°

∠A1=

∠C1=

a1=

∠A2=

∠C2=

a2=

We are given a triangle \(ABC\) with side lengths \(b = 124\), \(c = 165\), and angle \( \angle B = 45^\circ \). We need to solve for the remaining angles and sides.

This is a **non-right triangle** situation, so we can apply the **Law of Sines**:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]

1. **Find \( \angle A1 \) and \( \angle C1 \):**

We'll first calculate the unknowns for the first solution. Start by applying the Law of Sines to find angle \(A\):

\[
\frac{b}{\sin(B)} = \frac{a}{\sin(A)} \implies \frac{124}{\sin(45^\circ)} = \frac{165}{\sin(C)}
\]

Let me solve this to get the angles.For the first solution:

\[
\angle A1 = 32.1^\circ, \quad \angle C1 = 102.9^\circ, \quad a1 = 93.2
\]

2. **Find the second set of possible values (A2, C2, a2):**

In a triangle with two solutions, \( \angle A2 \) is the supplement of \( \angle A1 \) since there could be another valid configuration.

\[
\angle A2 = 180^\circ - \angle A1
\]

Let me calculate the second set of values now.For the second solution, no valid values exist because \( \angle A2 \) results in an impossible configuration. Therefore:

\[
\angle A2 = \text{DNE}, \quad \angle C2 = \text{DNE}, \quad a2 = \text{DNE}
\]

### Summary of the solution:

1. \( \angle A1 = 32.1^\circ \)
2. \( \angle C1 = 102.9^\circ \)
3. \( a1 = 93.2 \)
4. \( \angle A2 = \text{DNE} \)
5. \( \angle C2 = \text{DNE} \)
6. \( a2 = \text{DNE} \)

Would you like more details or further explanation?

Here are 5 related questions:
1. How is the Law of Sines applied to non-right triangles?
2. What conditions can cause two possible solutions in a triangle?
3. What happens if the sum of angles exceeds 180 degrees?
4. How do you use the Law of Cosines when angles are not given?
5. Can you explain the difference between acute and obtuse triangles?

**Tip:** Always check if the sum of angles stays within 180 degrees when solving triangles with multiple solutions.

Solve triangle ABC. (If an answer does not exist, enter DNE. Enter all angles in degrees. Round your answers to one decimal place. Enter your answers so that ∠A1 is smaller than ∠A2.) b = 124, c = 165, ∠B = 45° ∠A1= ∠C1= a1= ∠A2= ∠C2= a2=

Learn how to solve triangle ABC with side lengths b = 124, c = 165, and angle ∠B = 45° using the Law of Sines. Find the remaining angles and sides with step-by-step calculations, suitable for students in Grades 10-12. Understand multiple solutions and impossible configurations.

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