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To find the measure of \( FG \) in the triangle \( \triangle FBG \), we can use the Law of Sines. The triangle has angles \( \angle B \) and \( \angle G \) with measures of 18° and 32° respectively. Let \( \angle F \) be the third angle.

### Step 1: Calculate \( \angle F \)
The sum of angles in a triangle is always 180°:
\[
\angle F = 180^\circ - \angle B - \angle G = 180^\circ - 18^\circ - 32^\circ = 130^\circ.
\]

### Step 2: Apply the Law of Sines
The Law of Sines states:
\[
\frac{FG}{\sin(\angle B)} = \frac{BF}{\sin(\angle G)} = \frac{BG}{\sin(\angle F)}.
\]

Using \( BF = 18 \) and \( \angle G = 32^\circ \):
\[
\frac{FG}{\sin(18^\circ)} = \frac{18}{\sin(32^\circ)}.
\]

### Step 3: Solve for \( FG \)
Now, solve for \( FG \):
\[
FG = 18 \cdot \frac{\sin(18^\circ)}{\sin(32^\circ)}.
\]

Let's calculate \( FG \):

\[
FG \approx 18 \cdot \frac{0.3090}{0.5299} \approx 10.50.
\]

Therefore, the measure of \( FG \) is approximately 10.50 units.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you calculate the length of a side in a triangle using the Law of Cosines?
2. What other methods can be used to find a side length in a non-right triangle?
3. How can the Law of Sines be used to find an angle in a triangle?
4. What are some common mistakes when using the Law of Sines?
5. How do you determine if a triangle is possible with given angle measures?

**Tip:** Always ensure the angle sum in a triangle equals 180° when checking for consistency in your calculations!

Find the measure of FG in the triangle with given angles B = 18° and G = 32°.

Learn how to find the side length FG in a triangle with angles B = 18° and G = 32° using the Law of Sines. This trigonometry problem is ideal for high school students in Grades 10-12.

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