https://file.aiquickdraw.com/mathbot/file/1729919437595rubd2whu.jpg

To complete this proof, we’ll focus on identifying the congruent angles and sides between triangles \( \triangle FED \) and \( \triangle FGH \), given in the diagram.

Here’s the step-by-step solution:

1. **In \( \triangle FED \) and \( \triangle FGH \), we compare corresponding angles and sides.**

2. **First blank:** \( \angle EDF = \angle FGH \)
   - **Reasoning:** These angles are **alternate angles** because \( DE \parallel GH \).

3. **Second blank:** \( \angle DFE = \angle GFH \)
   - **Reasoning:** These angles are **vertically opposite angles**.

4. **Third blank:** \( DF = FG \)
   - **Reasoning:** This side equality is **given**.

5. **Conclusion:** \( \triangle FED \cong \triangle FGH \) by **ASA (Angle-Side-Angle)**.

So the completed answer is:

- \( \angle EDF = \angle FGH \) (Alternate between parallel lines)
- \( \angle DFE = \angle GFH \) (Vertically opposite angles)
- \( DF = FG \) (Given)

Therefore, \( \triangle FED \cong \triangle FGH \) by **ASA**.

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Would you like further details or explanations on this topic? Here are some related questions to consider:

1. Why do alternate angles between parallel lines prove congruency?
2. How does the ASA (Angle-Side-Angle) postulate establish triangle congruency?
3. What are other congruence criteria for triangles besides ASA?
4. How do vertically opposite angles form, and why are they congruent?
5. Why is it significant to have two corresponding angles congruent in proving triangles are congruent?

**Tip:** When two parallel lines are cut by a transversal, alternate interior angles are always congruent. This is often useful in proving triangle congruency.

In this question, we will prove that triangle FED is congruent to triangle FGH.

Understand how to prove that triangle FED is congruent to triangle FGH using the ASA congruence theorem. This solution involves alternate angles between parallel lines and vertically opposite angles, making it suitable for Grades 8-10.

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