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To solve for \( TW \) in the given geometric figure, we can use the Law of Sines. Here's what we observe:

1. The triangle \( TWV \) is non-right-angled, and we know the following:
   - Angle \( V = 78^\circ \)
   - Angle \( T = 90^\circ \)
   - Angle \( U = 63^\circ \)
   
   From this, we can calculate the remaining angle in triangle \( TWV \):
   \[
   \text{Angle } W = 180^\circ - (63^\circ + 78^\circ) = 39^\circ
   \]

2. Next, to find \( TW \), we can apply the Law of Sines, which states:
   \[
   \frac{TW}{\sin(\angle V)} = \frac{WV}{\sin(\angle T)}
   \]

To complete the solution, we would need the length of \( WV \). Do you have that value, or should we proceed with a theoretical answer based on available angles?

Let me know if you want further details or clarifications!

Here are 5 related questions to think about:
1. How is the Law of Sines useful in non-right triangles?
2. Can you derive other side lengths of the triangle using the same law?
3. How do angles in a triangle relate to one another in non-right triangles?
4. What happens if we modify the triangle by changing one angle?
5. Can you apply the Law of Cosines to the same triangle?

**Tip**: Always double-check the units and angle types when working with trigonometric functions.

Learn how to solve for the length of TW in a triangle using the Law of Sines. This geometric problem involves calculating angles and applying trigonometric principles, ideal for students in Grades 10-12.

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