To solve the proof in the image, we need to fill in the reasons for each step that leads to the conclusion that $GD \parallel EF$. Here's how we can approach the problem:

Problem Recap:

• Given:
  • $GBD$ and $CDE$ are straight lines.
  • $AB \parallel CE$.
• To Prove: $GD \parallel EF$.

Steps and Reasons:

1. Step 1: $\angle GDC = \angle GBA = 73^\circ$
   Reason: Corresponding Angles Postulate (When a transversal intersects two parallel lines, corresponding angles are equal).

2. Step 2: $\angle GDE + \angle GDC = 180^\circ$
   Reason: Linear Pair Postulate (The sum of the angles in a linear pair is 180°).

3. Step 3: $\angle GDE + 73^\circ = 180^\circ$
   Reason: Substitution (Substituting the value of $\angle GDC$ from Step 1).

4. Step 4: $\angle GDE = 107^\circ$
   Reason: Subtraction (Subtracting 73° from both sides).

5. Step 5: $\angle DEF = 107^\circ = \angle GDE$
   Reason: Vertical Angles Theorem (Vertical angles are equal).

6. Step 6: $GD \parallel EF$
   Reason: Converse of Corresponding Angles Postulate (If corresponding angles are equal, the lines are parallel).

Filled Proof:

• Step 1: $\angle GDC = \angle GBA = 73^\circ$
  Reason: Corresponding Angles Postulate

• Step 2: $\angle GDE + \angle GDC = 180^\circ$
  Reason: Linear Pair Postulate

• Step 3: $\angle GDE + 73^\circ = 180^\circ$
  Reason: Substitution

• Step 4: $\angle GDE = 107^\circ$
  Reason: Subtraction

• Step 5: $\angle DEF = 107^\circ = \angle GDE$
  Reason: Vertical Angles Theorem

• Step 6: $GD \parallel EF$
  Reason: Converse of Corresponding Angles Postulate

Would you like further details on any of these steps? Here are some related questions:

1. What is the Corresponding Angles Postulate, and how is it used in proofs?
2. Can you explain the Linear Pair Postulate with an example?
3. What is the Vertical Angles Theorem, and why is it important?
4. How do you identify corresponding angles in parallel lines cut by a transversal?
5. Can two lines be parallel if their corresponding angles are not equal?
6. How would you prove two lines are parallel using alternate interior angles?
7. Can we use the same proof strategy if the angles given were different?
8. How does this proof change if the lines $AB$ and $CE$ were not parallel?

Tip: Always double-check your reasons in a geometric proof to ensure that each logical step is justified by an appropriate theorem or postulate.