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Accepted Answer

To solve this matching problem, we recall the properties of parallelograms, rectangles, rhombuses, and squares:

### 1. "Show that one pair of opposite sides of the quadrilateral is both congruent and parallel."
- Property: A quadrilateral with one pair of opposite sides congruent and parallel is a **Parallelogram**.

### 2. "Show that a diagonal of a parallelogram bisects a pair of opposite angles."
- Property: A parallelogram with a diagonal bisecting opposite angles is a **Rhombus** (since rhombus diagonals bisect angles).

### 3. "Show that the parallelogram is both a rhombus and a rectangle."
- Property: A figure that is both a rhombus (all sides equal) and a rectangle (all angles right) is a **Square**.

### 4. "Show that the diagonals of the quadrilateral bisect each other."
- Property: A quadrilateral with diagonals bisecting each other is a **Parallelogram**.

### 5. "Show that both pairs of opposite angles of the quadrilateral are congruent."
- Property: A quadrilateral with both pairs of opposite angles congruent is a **Parallelogram**.

### 6. "Show that the diagonals of the parallelogram are congruent."
- Property: A parallelogram with congruent diagonals is a **Rectangle**.

### 7. "Show that an angle of the quadrilateral is supplementary to both of its consecutive angles."
- Property: If an angle is supplementary to its consecutive angles, the opposite sides are parallel (since consecutive angles are supplementary in parallelograms), so it is a **Parallelogram**.

### 8. "Show that the diagonals of the parallelogram are perpendicular."
- Property: A parallelogram with perpendicular diagonals is a **Rhombus**.

### 9. "Show that both pairs of opposite sides of the quadrilateral are congruent."
- Property: A quadrilateral with both pairs of opposite sides congruent is a **Parallelogram**.

### 10. "Show that all four angles of the parallelogram are right angles."
- Property: A parallelogram with all angles right is a **Rectangle**.

### 11. "Show that both pairs of opposite sides of the quadrilateral are parallel."
- Property: A quadrilateral with both pairs of opposite sides parallel is a **Parallelogram**.

### 12. "Show that all four sides of the parallelogram are congruent."
- Property: A parallelogram with all sides congruent is a **Rhombus**.

### 13. "Show that the diagonals of the parallelogram are congruent." (Repeated, matches "Rectangle" as in Step 6)
- Matches **Rectangle**.

### 14. "Show that the diagonals of the parallelogram are perpendicular." (Repeated, matches "Rhombus" as in Step 8)
- Matches **Rhombus**.

### 15. "Show that all four angles of the parallelogram are right angles." (Repeated, matches "Rectangle" as in Step 10)
- Matches **Rectangle**.

### Final Matches:
| Left Column (Property) | Right Column (Figure) |
|------------------------|-----------------------|
| 1. Both congruent + parallel | Parallelogram |
| 2. Diagonal bisects angles | Rhombus |
| 3. Both rhombus + rectangle | Square |
| 4. Diagonals bisect each other | Parallelogram |
| 5. Both pairs opposite angles congruent | Parallelogram |
| 6. Diagonals congruent | Rectangle |
| 7. Angle supplementary to consecutive angles | Parallelogram |
| 8. Diagonals perpendicular | Rhombus |
| 9. Both pairs opposite sides congruent | Parallelogram |
| 10. All angles right | Rectangle |
| 11. Both pairs opposite sides parallel | Parallelogram |
| 12. All sides congruent | Rhombus |

(Note: The pre-filled "Square", "Rectangle", and "Rhombus" in the right column align with the properties above.)

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Question

Explanation:

1. "Show that one pair of opposite sides of the quadrilateral is both congruent and parallel."

2. "Show that a diagonal of a parallelogram bisects a pair of opposite angles."

3. "Show that the parallelogram is both a rhombus and a rectangle."

4. "Show that the diagonals of the quadrilateral bisect each other."

5. "Show that both pairs of opposite angles of the quadrilateral are congruent."

6. "Show that the diagonals of the parallelogram are congruent."

7. "Show that an angle of the quadrilateral is supplementary to both of its consecutive angles."

8. "Show that the diagonals of the parallelogram are perpendicular."

9. "Show that both pairs of opposite sides of the quadrilateral are congruent."

10. "Show that all four angles of the parallelogram are right angles."

11. "Show that both pairs of opposite sides of the quadrilateral are parallel."

12. "Show that all four sides of the parallelogram are congruent."

13. "Show that the diagonals of the parallelogram are congruent." (Repeated, matches "Rectangle" as in Step 6)

14. "Show that the diagonals of the parallelogram are perpendicular." (Repeated, matches "Rhombus" as in Step 8)

15. "Show that all four angles of the parallelogram are right angles." (Repeated, matches "Rectangle" as in Step 10)

Final Matches:

Answer: