Sure, let's prove that [tex]\(WXYZ\)[/tex] is a rectangle step by step, using geometric properties and theorems.
### Given:
- [tex]\(WXYZ\)[/tex] is a parallelogram.
- [tex]\(\overline{ZX} \cong \overline{WY}\)[/tex]
### To Prove:
- [tex]\(WXYZ\)[/tex] is a rectangle.
### Proof:
| Statements | Reasons |
|----------------|--------------|
| 1. [tex]\(WXYZ\)[/tex] is a parallelogram | 1. Given |
| 2. [tex]\(\overline{ZX} \cong \overline{WY}\)[/tex] | 2. Given |
| 3. [tex]\(\overline{ZW} \cong \overline{ZW}\)[/tex] and [tex]\(\overline{XY} \cong \overline{XY}\)[/tex] | 3. Reflexive property of congruence |
| 4. [tex]\(\overline{WX} \cong \overline{YZ}\)[/tex] | 4. Opposite sides of a parallelogram are congruent |
| 5. [tex]\(\angle W + \angle X = 180^\circ\)[/tex] | 5. Consecutive angles of a parallelogram are supplementary |
| 6. [tex]\(\angle X + \angle Y = 180^\circ\)[/tex] | 5. Consecutive angles of a parallelogram are supplementary (Alternate angle pairs) |
| 7. If one angle is [tex]\(90^\circ\)[/tex], then all angles are [tex]\(90^\circ\)[/tex] because consecutive angles are supplementary | 7. Property of supplementary angles in a parallelogram |
| 8. [tex]\(WXYZ\)[/tex] has at least one right angle | 8. Given [tex]\(\overline{ ZX} \cong \overline{ WY }\)[/tex] forming a pair of congruent triangles which implies right angles |
| 9. Thus, [tex]\(WXYZ\)[/tex] has four right angles, proving it is a rectangle | 9. Definition of a rectangle |
Therefore, [tex]\(WXYZ\)[/tex] is indeed a rectangle.
