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°

We are given the following conditions for the triangle:

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

### Step 1: Use the Law of Sines

The Law of Sines states:

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

We can first use this formula to solve for angle \( \angle C \), using sides \( a \) and \( c \) along with \( \angle A \).

### Step 2: Solve for \( \angle C \)

Rearranging the Law of Sines for \( \sin C \), we get:

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

Substitute the known values:

\[
\frac{28}{\sin 129^\circ} = \frac{44}{\sin C}
\]

First, calculate \( \sin 129^\circ \):

\[
\sin 129^\circ \approx 0.781
\]

Now, solve for \( \sin C \):

\[
\frac{28}{0.781} = \frac{44}{\sin C}
\]

\[
\sin C = \frac{44 \times 0.781}{28}
\]

\[
\sin C \approx 1.227
\]

### Step 3: Check if \( \sin C \) is possible

Since \( \sin C \) cannot exceed 1 (as sine values are between -1 and 1), this result is impossible. Therefore, no triangle can be formed with the given conditions.

Thus, the answer is:

\[
\boxed{\text{DNE}}
\]

### Would you like more details, or do you have any questions?

Here are five related questions to explore further:
1. What if \( \sin C \) were a valid value? How would we proceed to find the angles and side \( b \)?
2. How does the Law of Sines differ from the Law of Cosines?
3. Can we always use the Law of Sines when two sides and an angle are given?
4. How does an obtuse angle like \( \angle A = 129^\circ \) affect the solution of triangles?
5. What happens if \( \angle A \) was smaller, like \( 30^\circ \)? How would the solution change?

**Tip:** Always check the sine value when using the Law of Sines. If it's greater than 1, no triangle can exist.

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°

Solve a triangle using the Law of Sines with sides a = 28, c = 44, and angle A = 129°. This detailed solution checks the sine values and explains why no triangle can exist for the given conditions. Suitable for high school students in Grades 10-12.

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