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To complete the proof that \( \triangle UVX \cong \triangle WTX \), we can follow these steps, filling in the blanks for each statement:

1. **Statement 1**: \( TW \parallel UV \) (Given)
2. **Statement 2**: \( UV \cong TW \) (Given)

**Step 3**: The next logical step involves identifying corresponding angles. Since \( TW \parallel UV \) and \( X \) is a transversal line that intersects these parallel lines, we have corresponding angles:
   - **Statement 3**: \( \angle TXW \cong \angle UXV \) 
   - **Reason**: **Corresponding Angles Postulate** (because of the parallel lines)

**Step 4**: We now look at vertical angles that are congruent:
   - **Statement 4**: \( \angle W \cong \angle U \) 
   - **Reason**: **Vertical Angles Theorem** (since \( X \) is the intersection of the two lines)

**Step 5**: At this point, you have two pairs of congruent angles and a pair of congruent sides (\( UV \cong TW \)), which is enough to conclude that the triangles are congruent by the **ASA (Angle-Side-Angle) Congruence Theorem**:
   - **Statement 5**: \( \triangle UVX \cong \triangle WTX \) 
   - **Reason**: **ASA Congruence Theorem**.

Would you like further details on this proof or any clarifications?

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Here are 5 related questions to deepen your understanding:
1. What is the definition of corresponding angles in geometry?
2. How does the ASA Congruence Theorem differ from the SAS Congruence Theorem?
3. Why are vertical angles always congruent?
4. What would happen if the two lines weren’t parallel? Would the corresponding angles still be equal?
5. Can the SSA (Side-Side-Angle) condition prove triangle congruence? Why or why not?

**Tip**: Parallel lines cut by a transversal create several pairs of congruent angles, such as corresponding and alternate interior angles. These relationships are essential for many triangle proofs.

Prove that triangle UVX is congruent to triangle WTX given TW || UV and UV ≅ TW.

Learn how to prove that triangles UVX and WTX are congruent using the ASA Congruence Theorem, corresponding angles postulate, and vertical angles theorem. This geometric proof is suitable for students in Grades 8-10, covering parallel lines and triangle congruence concepts.

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