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* = 24,    *c* = 45,    ∠*A* = 128°

∠*B* =

°

∠*C* =

°

*b* =

We can solve this triangle using the Law of Sines. The Law of Sines is given by:

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

### Step 1: Solve for ∠C
We know the angles in a triangle sum up to 180°. Therefore:

\[
\angle C = 180^\circ - \angle A - \angle B
\]

But first, we need to find ∠B.

### Step 2: Use the Law of Sines to find ∠B

From the Law of Sines, we can write:

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

Substitute the known values (\(a = 24\), \(c = 45\), \(\angle A = 128^\circ\)):

\[
\frac{24}{\sin(128^\circ)} = \frac{45}{\sin C}
\]

Solve for \(\sin C\):

\[
\sin C = \frac{45 \times \sin(128^\circ)}{24}
\]

Calculating \(\sin(128^\circ)\):

\[
\sin(128^\circ) \approx 0.7818
\]

Now substitute this value:

\[
\sin C = \frac{45 \times 0.7818}{24} \approx 1.4666
\]

Since \(\sin C\) cannot be greater than 1, this means there is no possible solution for angle \(\angle C\), and thus the triangle does not exist.

### Final Answer:

\[
\boxed{\text{DNE (Does Not Exist)}}
\]

Would you like further clarification on any steps? Below are some related questions for practice:

---

1. How does the Law of Cosines differ from the Law of Sines in solving triangles?
2. What does it mean when \(\sin C\) exceeds 1 in a triangle problem?
3. What conditions ensure that a triangle exists given side lengths and angles?
4. How would this problem change if angle \(A\) were smaller than 90°?
5. Can you explain the ambiguous case in the Law of Sines?

**Tip:** Always check if your sine value is physically possible for a given triangle before proceeding.

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* = 24, *c* = 45, ∠*A* = 128° ∠*B* = ° ∠*C* = ° *b* = 

Solve a triangle using the Law of Sines with given values: side a = 24, side c = 45, and angle A = 128°. The solution involves calculating the angles and sides, determining if a solution exists. Appropriate for students in Grades 10-12.

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