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.)

*a* = 28,    *c* = 44,    ∠*A* = 129°

∠*B* =

 °

∠*C* =

  °

*b* =


To solve this triangle using the Law of Sines, we will follow these steps.

### Given:
- \( a = 28 \)
- \( c = 44 \)
- \( \angle A = 129^\circ \)

The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

First, we can use the Law of Sines to find \( \sin C \) (which allows us to solve for \( \angle C \)):

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Substitute the known values:
\[
\frac{28}{\sin 129^\circ} = \frac{44}{\sin C}
\]

### Step 1: Calculate \( \sin C \)
First, find \( \sin 129^\circ \):
\[
\sin 129^\circ \approx 0.7818
\]

Now, substitute this value into the equation:
\[
\frac{28}{0.7818} = \frac{44}{\sin C}
\]
\[
\sin C = \frac{44 \times 0.7818}{28} \approx 1.228
\]

### Step 2: Check the value of \( \sin C \)
Since \( \sin C \) cannot be greater than 1, this tells us that no such triangle exists under these conditions. Therefore, there is no possible solution.

### Conclusion:
- \( \angle B = \text{DNE} \)
- \( \angle C = \text{DNE} \)
- \( b = \text{DNE} \)

There is no possible triangle that satisfies the given conditions.

---

Would you like more details on this, or do you have any questions? Here are some related problems you might find useful:

1. Solve a triangle where \( a = 28 \), \( c = 44 \), and \( \angle A = 90^\circ \).
2. Solve a triangle with two sides and an angle given: \( a = 30 \), \( b = 40 \), and \( \angle A = 45^\circ \).
3. Use the Law of Cosines to solve a triangle with \( a = 35 \), \( b = 50 \), and \( c = 60 \).
4. Solve a triangle where \( a = 25 \), \( b = 35 \), and \( \angle B = 60^\circ \) using the Law of Sines.
5. Solve for the area of a triangle with \( a = 40 \), \( b = 50 \), and \( \angle C = 45^\circ \).

**Tip:** Always check if the sine of an angle exceeds 1 when using the Law of Sines. If it does, the triangle is not possible.

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.) a = 28, c = 44, ∠A = 129°. ∠B = ° ∠C = ° b =

This problem uses the Law of Sines to solve a triangle where a = 28, c = 44, and ∠A = 129°. Follow a step-by-step process to determine the angles and sides of the triangle, or conclude that the triangle is impossible. Suitable for Grades 10-12.

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