The image shows two triangles, $\triangle EDC$ and $\triangle GHF$, with known angles and sides. Our goal is to solve for $x$ in triangle $\triangle GHF$.

Triangle $\triangle GHF$ information:

• Angle $G = 79^\circ$
• Angle $H = 48^\circ$
• Angle $F = 53^\circ$
• Side $HF = 66$
• Side $GH = x$

We can use the Law of Sines to find $x$. The Law of Sines states:

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

Where:

• $A, B, C$ are the angles
• $a, b, c$ are the sides opposite to the respective angles.

For $\triangle GHF$:

• Side $GH = x$ is opposite to angle $F = 53^\circ$,
• Side $HF = 66$ is opposite to angle $G = 79^\circ$.

We apply the Law of Sines:

$\frac{x}{\sin 53^\circ} = \frac{66}{\sin 79^\circ}$

Solving for $x$:

$x = \frac{66 \cdot \sin 53^\circ}{\sin 79^\circ}$

Now, calculate the sine values and substitute them:

$x = \frac{66 \cdot 0.7986}{0.9816} \approx \frac{52.89}{0.9816} \approx 53.9$

Thus, $x \approx 53.9$ units.

Would you like further details or explanations? Here are some additional questions you could explore:

1. How would the Law of Cosines be applied to solve this triangle?
2. What if the triangle was not a non-right triangle? How would that affect the solution?
3. Can you verify this result by calculating the sum of the angles of the triangle?
4. How does the accuracy of sine values affect the result when solving for $x$?
5. What are other uses of the Law of Sines in different types of triangle problems?

Tip: Always ensure that the angles you are using for trigonometric calculations are in degrees (or radians as appropriate).