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The image shows a triangle where you're tasked to find the lengths \( x \) and \( y \). The information provided includes:

- Two angles: \( 21^\circ \) and \( 37^\circ \).
- A side of length 100 units opposite the angle \( 21^\circ \).

We can use the **Law of Sines** to solve for \( x \) and \( y \). The Law of Sines states that:

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

Where \( a, b, c \) are the sides opposite to angles \( A, B, C \), respectively.

### Step 1: Find the third angle of the triangle
Using the sum of angles in a triangle, the third angle is:
\[
C = 180^\circ - 21^\circ - 37^\circ = 122^\circ
\]

### Step 2: Apply Law of Sines
We will solve for \( x \) and \( y \).

\[
\frac{100}{\sin(122^\circ)} = \frac{x}{\sin(37^\circ)} = \frac{y}{\sin(21^\circ)}
\]

Now, let me compute the values for \( x \) and \( y \).The values for \( x \) and \( y \) are approximately:

- \( x \approx 70.96 \) units
- \( y \approx 42.26 \) units

Would you like more details or have any questions about the process? Here are five related questions for further exploration:

1. How would the Law of Cosines be applied in this problem if angles were not given?
2. How does the Law of Sines differ from the Law of Cosines?
3. How do you determine when to use the Law of Sines versus the Law of Cosines?
4. What is the importance of significant figures in geometric problems like this?
5. Can you solve for the area of the triangle using the given sides and angles?

**Tip:** Always check if you can calculate all the necessary angles using the sum of angles in a triangle before applying any trigonometric rules.

Find x and y in a triangle with given angles 21°, 37°, and side of 100 units opposite to the 21° angle.

Learn how to find the lengths of sides x and y in a triangle using the Law of Sines. The given angles are 21° and 37°, and the side opposite the 21° angle is 100 units. This detailed solution is suitable for high school students in Grades 10-12.

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