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The problem asks to prove that the diagonals of an isosceles trapezoid \( ABCD \) (with diagonals \( \overline{AC} \) and \( \overline{BD} \)) are congruent using a two-column proof. The hint suggests proving that triangles \( \Delta ABC \) and \( \Delta BAD \) are congruent using the SAS (Side-Angle-Side) Postulate. Let's break this down step by step.

### Plan:
1. **Given Information**: \( ABCD \) is an isosceles trapezoid. This means that \( AB \parallel CD \) and \( AB = CD \).
2. **Prove**: \( \overline{AC} \cong \overline{BD} \).
3. **Hint**: Show that \( \Delta ABC \cong \Delta BAD \) using the SAS Postulate.

### Solution:
We will fill in the two-column proof below.

#### Proof

| **Statements**                              | **Reasons**                           |
|---------------------------------------------|---------------------------------------|
| 1. \( ABCD \) is an isosceles trapezoid     | 1. Given                              |
| 2. \( AB = CD \)                            | 2. Definition of isosceles trapezoid   |
| 3. \( \angle ABC = \angle BAD \)            | 3. Alternate interior angles (since \( AB \parallel CD \)) |
| 4. \( AD = BC \)                            | 4. Definition of isosceles trapezoid   |
| 5. \( \Delta ABC \cong \Delta BAD \)        | 5. SAS Postulate (Side-Angle-Side)     |
| 6. \( \overline{AC} \cong \overline{BD} \)  | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

### Explanation:
- **Step 1**: We state the given information that \( ABCD \) is an isosceles trapezoid.
- **Step 2**: From the definition of an isosceles trapezoid, we know that the non-parallel sides are equal: \( AB = CD \).
- **Step 3**: Since \( AB \parallel CD \), the alternate interior angles \( \angle ABC \) and \( \angle BAD \) are congruent.
- **Step 4**: The non-parallel sides \( AD \) and \( BC \) are also equal, by the definition of an isosceles trapezoid.
- **Step 5**: Using the SAS Postulate, we conclude that triangles \( \Delta ABC \) and \( \Delta BAD \) are congruent because they have two sides and the included angle congruent.
- **Step 6**: Since the triangles are congruent, the diagonals \( AC \) and \( BD \) are also congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Would you like further details or clarifications on any step?

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Here are 5 related questions to further your understanding:
1. How does the SAS Postulate help in proving triangle congruence?
2. Why are alternate interior angles congruent when two lines are parallel?
3. What is the significance of CPCTC in geometric proofs?
4. Can we use other methods (e.g., SSS, ASA) to prove that \( \Delta ABC \cong \Delta BAD \)?
5. What properties define an isosceles trapezoid, and how do they aid in solving proofs?

**Tip**: Always check if you can apply triangle congruence (SAS, SSS, ASA, AAS) in geometric proofs—it’s a powerful tool for proving relationships like congruent sides or angles.

Prove that the diagonals of an isosceles trapezoid ABCD are congruent.

Learn how to prove that the diagonals of an isosceles trapezoid are congruent using the SAS Postulate and CPCTC. This problem involves geometric reasoning, triangle congruence, and the properties of trapezoids. Suitable for students in Grades 8-10.

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