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Proving When a Parallelogram Is a Rectangle

Given: [tex]\( WXYZ \)[/tex] is a parallelogram.
[tex]\[ \overline{ZX} \cong \overline{WY} \][/tex]

Prove: [tex]\( WXYZ \)[/tex] is a rectangle.

Statements | Reasons
--- | ---
1. [tex]\( WXYZ \)[/tex] is a parallelogram | Given
2. [tex]\( \overline{ZX} \cong \overline{WY} \)[/tex] | Given
3. Opposite sides of a parallelogram are congruent | By definition of a parallelogram
4. Consecutive angles of a parallelogram are supplementary | By definition of a parallelogram
5. [tex]\( WXYZ \)[/tex] has one right angle | If one angle is right in a parallelogram, all are right angles
6. [tex]\( WXYZ \)[/tex] is a rectangle | A parallelogram with one right angle and congruent diagonals is a rectangle

Sagot :

GinnyAnsweridn
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.
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