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To find the length of \( TW \), we can use the Law of Sines, which states that in any triangle, the ratio of a side length to the sine of its opposite angle is the same for all three sides.

Given the diagram:
- \( \angle W = 78^\circ \)
- \( \angle V = 90^\circ \)
- \( \angle T = 63^\circ \)
- \( \angle U = 63^\circ \)

Since \( \triangle TWV \) is a right triangle with \( \angle V = 90^\circ \), and we have the angle \( \angle W = 78^\circ \), we can calculate \( \angle TWV = 12^\circ \) because the sum of angles in a triangle is \( 180^\circ \).

### Steps to Solve:
1. **Determine \( \angle WTU \):**  
   - \( \angle WTU = 90^\circ - \angle W = 90^\circ - 78^\circ = 12^\circ \).

2. **Use the Law of Sines:**
   \[
   \frac{TW}{\sin 63^\circ} = \frac{WV}{\sin 12^\circ}
   \]

However, since we don't have the length of \( WV \) directly, this approach won't be effective. Instead, we'll use trigonometry in \( \triangle TWV \) with right angles.

3. **Using Trigonometry:**
   - In \( \triangle TWV \), we can use the following sine formula to solve for \( TW \):
     \[
     \sin(78^\circ) = \frac{TW}{TV}
     \]

But, since \( TV \) isn't provided, the question may involve using the properties of similar triangles or possibly another trigonometric identity.

### Clarification Needed:
To proceed accurately, I need either the length of \( TV \) or \( WV \), or confirmation that there's another method or key piece of information you might have missed in the problem.

Would you like more details or have any additional information? If not, here are some related questions you might find useful.

1. How can we find the length of \( TW \) if \( TV \) is given?
2. What role does the Law of Sines play in solving problems in non-right triangles?
3. How can the properties of similar triangles be applied here?
4. What other trigonometric identities could be useful in solving triangle problems?
5. How can the cosine rule help in solving triangles when two sides and the included angle are known?

**Tip:** Always double-check which angles and sides are given in a triangle problem before deciding on the approach to use!

Explore how to solve for the length of TW in a right triangle using the Law of Sines and trigonometric identities. This detailed solution explains how to approach problems involving non-right angles and the sum of angles in a triangle. Suitable for students in Grades 10-12.

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