https://r2.erweima.ai/mathbot/image/1726698430773cqoj29vu.jpg

Let's work through this matching problem based on the information given in the image.

### Statements and Reasons:
You need to match the statements to the correct reasoning, so here's a breakdown:

#### 1. \( ABCD \) is a rhombus.
- **Reason:**
   - Given.

#### 2. \( BC \cong AD \) and \( AB \cong CD \)
- **Reason:**
   - Definition of a rhombus (all sides of a rhombus are congruent).

#### 3. \( AC \cong AC \)
- **Reason:**
   - Reflexive property of congruence (an element is always congruent to itself).

#### 4. \( \triangle ABC \cong \triangle CDA \)
- **Reason:**
   - **SAS (Side-Angle-Side):** You have two pairs of congruent sides and the included angles.

#### 5. \( \angle 1 \cong \angle 4 \) and \( \angle 2 \cong \angle 3 \)
- **Reason:**
   - CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

#### 6. \( BC \parallel AD \) and \( AB \parallel CD \)
- **Reason:**
   - The **converse of the alternate interior angles theorem** (since the angles are congruent, the lines must be parallel).

#### 7. \( ABCD \) is a parallelogram.
- **Reason:**
   - Definition of a parallelogram (a quadrilateral where opposite sides are parallel).

### Final Matching:
- **Statement 1** → Reason 1: Given.
- **Statement 2** → Reason 2: Definition of rhombus.
- **Statement 3** → Reason b: Reflexive property.
- **Statement 4** → Reason f: SAS.
- **Statement 5** → Reason 5: CPCTC.
- **Statement 6** → Reason a: Converse of the alternate interior angles theorem.
- **Statement 7** → Reason 7: Definition of parallelogram.

Do you have any questions or want more details?

### Related Questions:
1. Why does the definition of a rhombus imply that all sides are congruent?
2. How is the reflexive property used in proving triangle congruence?
3. What conditions are needed for two triangles to be congruent by SAS?
4. How does the CPCTC principle help in geometric proofs?
5. What are the conditions for the converse of the alternate interior angles theorem?

### Tip:
In geometric proofs, always look for congruent triangles to help establish relationships between angles and sides.

Match the correct answers. Given: ABCD is a rhombus. Prove: ABCD is a parallelogram.

Learn how to prove that the quadrilateral ABCD is a parallelogram, given that ABCD is a rhombus. This problem covers geometric concepts like SAS, CPCTC, and the converse of alternate interior angles theorem. Suitable for Grades 9-10.

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