https://file.aiquickdraw.com/mathbot/file/17291119790442dov562l.jpg

To solve the problem, let's analyze the given information and the diagram:

### Given:
- \( EF \parallel GH \)
- \( AB \parallel CD \)

### To Prove:
- \( \angle GJI \cong \angle LKF \)

### Proof Strategy:
Since we have pairs of parallel lines and transversal lines, we will use properties of corresponding angles and alternate interior angles to establish the congruence between the angles.

### Proof:
1. **Identify the transversal lines:**
   - \( AB \) and \( CD \) are parallel, and \( JI \) and \( LK \) act as transversal lines.
   - \( EF \) and \( GH \) are parallel, and \( IK \) and \( JL \) act as transversal lines.

2. **Use Alternate Interior Angles Theorem:**
   - Since \( EF \parallel GH \) and \( IK \) is a transversal, \( \angle GJI \) and \( \angle IKL \) are alternate interior angles, so:
     \[
     \angle GJI \cong \angle IKL
     \]

3. **Use Corresponding Angles Postulate:**
   - Since \( AB \parallel CD \) and \( EF \) is a transversal, \( \angle IKL \) and \( \angle LKF \) are corresponding angles, so:
     \[
     \angle IKL \cong \angle LKF
     \]

4. **Conclude the proof using transitivity:**
   - From steps 2 and 3:
     \[
     \angle GJI \cong \angle IKL \cong \angle LKF
     \]
   - Therefore, \( \angle GJI \cong \angle LKF \).

Thus, the proof is complete.

Would you like more details or have any questions on any of the steps?

### Related Questions:
1. What is the Alternate Interior Angles Theorem and how is it used in proofs?
2. How does the Corresponding Angles Postulate help in proving angle congruence?
3. What role do parallel lines play in determining angle relationships?
4. How can we prove that two lines are parallel using angle congruence?
5. What is the Transitive Property in geometry, and when is it applied?

### Tip:
When working with parallel lines and transversals, always look for corresponding angles, alternate interior angles, and alternate exterior angles—they are often key to solving problems!

Prove that angle GJI is congruent to angle LKF given EF || GH and AB || CD.

Learn how to prove that angle GJI is congruent to angle LKF using the properties of parallel lines and transversals. This geometry problem involves applying the Alternate Interior Angles Theorem, Corresponding Angles Postulate, and the Transitive Property. Suitable for students in Grades 9-10.

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