Question

use the given information to complete the proof of the following theorem.
if one pair of opposite sides of a quadrilateral is both parallel and congruent, then it is a parallelogram.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
use this definition in your proof.
given: $overline{wx} \cong \overline{yz}$
$overline{wx} \parallel \overline{yz}$
prove: $wxyz$ is a parallelogram

1. $overline{wx} \cong \overline{yz}$ given
2. $overline{wx} \parallel \overline{yz}$ given
3. $\angle 1 \cong \angle 4$ reason?
4. $\overline{\quad} \cong \overline{\quad}$ reflexive property
5. $\triangle \quad \cong \triangle \quad$ reason?
6. $\angle 2 \cong \angle 3$ reason?
7. $\overline{\quad} \parallel \overline{\quad}$ if alt. int. $\angle s \cong$, then lines $\parallel$
8. $wxyz$ is a parallelogram reason?

Explanation:

Step1: Analyze ∠1 ≅ ∠4

Since \( \overline{WX} \parallel \overline{YZ} \), \( \angle 1 \) and \( \angle 4 \) are alternate interior angles. By the Alternate Interior Angles Theorem, alternate interior angles formed by a transversal cutting two parallel lines are congruent. So \( \angle 1 \cong \angle 4 \) because of Alternate Interior Angles Theorem.

Step2: Identify the reflexive side

The reflexive property states that a segment is congruent to itself. Here, \( \overline{XZ} \cong \overline{XZ} \) (since \( XZ \) is a common side to both \( \triangle WXZ \) and \( \triangle YZX \)).

Step3: Prove triangle congruence

We have \( \overline{WX} \cong \overline{YZ} \) (given), \( \angle 1 \cong \angle 4 \) (from step 1), and \( \overline{XZ} \cong \overline{XZ} \) (reflexive). By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle WXZ \cong \triangle YZX \).

Step4: Analyze ∠2 ≅ ∠3

Since \( \triangle WXZ \cong \triangle YZX \), corresponding parts of congruent triangles are congruent (CPCTC). So \( \angle 2 \) and \( \angle 3 \) are corresponding angles of the congruent triangles, hence \( \angle 2 \cong \angle 3 \) by CPCTC.

Step5: Prove \( \overline{WZ} \parallel \overline{XY} \)

Since \( \angle 2 \cong \angle 3 \), and these are alternate interior angles formed by transversal \( XZ \) cutting \( \overline{WZ} \) and \( \overline{XY} \). By the Alternate Interior Angles Converse (if alternate interior angles are congruent, then the lines are parallel), \( \overline{WZ} \parallel \overline{XY} \).

Step6: Conclude WXYZ is a parallelogram

We know \( \overline{WX} \parallel \overline{YZ} \) (given) and we just proved \( \overline{WZ} \parallel \overline{XY} \). By the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), WXYZ is a parallelogram.

Filling the table:

Statement	Reason
2. \( \overline{WX} \parallel \overline{YZ} \)	Given
3. \( \angle 1 \cong \angle 4 \)	Alternate Interior Angles Theorem
4. \( \overline{XZ} \cong \overline{XZ} \)	Reflexive Property
5. \( \triangle WXZ \cong \triangle YZX \)	SAS Congruence Postulate
6. \( \angle 2 \cong \angle 3 \)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
7. \( \overline{WZ} \parallel \overline{XY} \)	Alternate Interior Angles Converse
8. WXYZ is a parallelogram	Definition of a parallelogram (both pairs of opposite sides parallel)

Answer:

The reasons are as follows:

• For statement 3: Alternate Interior Angles Theorem.
• For statement 4: \( \overline{XZ} \cong \overline{XZ} \)
• For statement 5: SAS Congruence Postulate ( \( \triangle WXZ \cong \triangle YZX \))
• For statement 6: CPCTC
• For statement 7: \( \overline{WZ} \parallel \overline{XY} \) (by Alternate Interior Angles Converse)
• For statement 8: Definition of a parallelogram (both pairs of opposite sides parallel)