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To solve for angle \( C \), side \( a \), and side \( b \) of the triangle, we can use the Law of Sines and Law of Cosines. Let's go through the steps:

### Given:
- Angle \( A = 45^\circ \)
- Angle \( B = 100^\circ \)
- Side \( AB = 15 \) (opposite angle \( C \))

### 1. Find Angle \( C \)
Since the sum of angles in a triangle is always \( 180^\circ \), we can find angle \( C \):

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

Substituting the values:

\[
C = 180^\circ - 45^\circ - 100^\circ = 35^\circ
\]

So, angle \( C \) is \( 35^\circ \).

### 2. Find Side \( a \) and Side \( b \)
Now, we'll use the Law of Sines to find sides \( a \) and \( b \). The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{15}{\sin C}
\]

#### Solving for side \( a \):
\[
a = \frac{15 \times \sin A}{\sin C}
\]

Substitute the values:
\[
a = \frac{15 \times \sin 45^\circ}{\sin 35^\circ}
\]

#### Solving for side \( b \):
\[
b = \frac{15 \times \sin B}{\sin C}
\]

Substitute the values:
\[
b = \frac{15 \times \sin 100^\circ}{\sin 35^\circ}
\]

Let me calculate these values.### Final Results:
1. **Angle \( C \)**: \( 35^\circ \)
2. **Side \( a \)**: \( 18.5 \) units
3. **Side \( b \)**: \( 25.8 \) units

Would you like further details on any of the steps? If so, feel free to ask!

Here are 5 related questions for further exploration:
1. How would you use the Law of Cosines instead of the Law of Sines to find side lengths?
2. What are the conditions under which the Law of Sines can produce ambiguous results?
3. How do you verify that the triangle is valid given the calculated side lengths?
4. Can you derive the area of this triangle using trigonometric methods?
5. How would the calculations change if the given angle measurements were in radians instead of degrees?

**Tip**: Always double-check your angle calculations when using trigonometric functions, especially if you switch between radians and degrees!

Learn how to find angle C, side a, and side b in a triangle using the Law of Sines and angles in triangles theorem. This problem involves solving for unknown angles and sides given specific angle measures and a side length. Suitable for high school students.

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