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To solve for angle \( A \) and the sides \( b \) and \( c \) of the triangle, we can use the following steps:

### Step 1: Calculate Angle \( A \)
The sum of angles in a triangle is always \( 180^\circ \). Therefore, we can find angle \( A \) using:

\[
A = 180^\circ - 80^\circ - 53^\circ
\]

\[
A = 47^\circ
\]

### Step 2: Use the Law of Sines to Find Sides \( b \) and \( c \)
The Law of Sines states:

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

Given:
- \( a = 2 \)
- \( A = 47^\circ \)
- \( B = 80^\circ \)
- \( C = 53^\circ \)

We can now calculate \( b \) and \( c \) using:

\[
\frac{b}{\sin(80^\circ)} = \frac{2}{\sin(47^\circ)}
\]

\[
b = 2 \times \frac{\sin(80^\circ)}{\sin(47^\circ)}
\]

\[
\frac{c}{\sin(53^\circ)} = \frac{2}{\sin(47^\circ)}
\]

\[
c = 2 \times \frac{\sin(53^\circ)}{\sin(47^\circ)}
\]

### Calculating Numerically:

Let's compute the values:

\[
b = 2 \times \frac{\sin(80^\circ)}{\sin(47^\circ)}
\]
\[
b \approx 2 \times \frac{0.9848}{0.7314} \approx 2 \times 1.347 \approx 2.7
\]

\[
c = 2 \times \frac{\sin(53^\circ)}{\sin(47^\circ)}
\]
\[
c \approx 2 \times \frac{0.7986}{0.7314} \approx 2 \times 1.0917 \approx 2.2
\]

### Summary of Results:
- Angle \( A \) = \( 47^\circ \)
- Side \( b \) = \( 2.7 \) (rounded to the nearest tenth)
- Side \( c \) = \( 2.2 \) (rounded to the nearest tenth)

Would you like more details on any of the steps? 

Here are some related questions:

1. How does the Law of Sines apply to non-right triangles?
2. Can you explain why the sum of angles in a triangle is always \( 180^\circ \)?
3. What are some scenarios where the Law of Cosines would be more appropriate than the Law of Sines?
4. How do rounding errors impact the precision of trigonometric calculations?
5. Can you describe how to find the area of a triangle given two sides and the included angle?

**Tip:** When solving triangle problems, always double-check your angle calculations to ensure they sum up to \( 180^\circ \). This can help catch any potential mistakes early on.

Learn how to find side A, angle B, and angle C in a triangle using trigonometric principles and the Law of Sines. This problem involves calculating angles and sides based on given angle measurements and the Law of Sines formula. Suitable for high school students studying trigonometry.

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