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

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

(reason options: reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, definition of midpoint, definition of segment bisector, definition of perpendicular)

Explanation:

Step1: Identify Alternate Interior Angles

Since \( \overline{WX} \parallel \overline{YZ} \) and \( \overline{XZ} \) is a transversal, the alternate interior angles \( \angle 1 \) and \( \angle 3 \) are congruent. So statement 3: \( \angle 1 \cong \angle 3 \).

Step2: Reflexive Property for Side

The side \( \overline{XZ} \) is common to both triangles \( \triangle WXZ \) and \( \triangle YZX \), so by reflexive property, \( \overline{XZ} \cong \overline{XZ} \). So statement 4: \( \overline{XZ} = \overline{XZ} \) (or \( \overline{XZ} \cong \overline{XZ} \)).

Step3: SAS Congruence

We have \( \overline{WX} \cong \overline{YZ} \) (given), \( \angle 1 \cong \angle 3 \) (alternate interior angles), and \( \overline{XZ} \cong \overline{XZ} \) (reflexive). So by SAS (Side - Angle - Side) congruence criterion, \( \triangle WXZ \cong \triangle YZX \). So statement 5: \( \triangle WXZ \cong \triangle YZX \), Reason: SAS Congruence Postulate.

Step4: Corresponding Parts of Congruent Triangles

Since \( \triangle WXZ \cong \triangle YZX \), their corresponding parts (angles) are congruent. So \( \angle 2 \cong \angle 4 \), Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

Step5: Identify Parallel Sides from Alternate Interior Angles

Since \( \angle 2 \cong \angle 4 \) and these are alternate interior angles formed by transversal \( \overline{XZ} \) with sides \( \overline{XY} \) and \( \overline{WZ} \), we can say \( \overline{XY} \parallel \overline{WZ} \), Reason: If alternate interior angles are congruent, then lines are parallel.

Step6: Definition of Parallelogram

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

Filling the Table:

• Statement 3: \( \boldsymbol{\angle 1} \cong \boldsymbol{\angle 3} \)
• Statement 4: \( \boldsymbol{\overline{XZ}} \cong \boldsymbol{\overline{XZ}} \)
• Statement 5: \( \boldsymbol{\triangle YZX} \), Reason: SAS Congruence Postulate
• Statement 6: Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
• Statement 7: \( \boldsymbol{\overline{XY}} \parallel \boldsymbol{\overline{WZ}} \), Reason: If alternate interior angles are congruent, then lines are parallel
• Statement 8: Reason: Definition of a parallelogram (both pairs of opposite sides parallel)

Answer:

• Statement 3: \( \angle 1 \cong \angle 3 \)
• Statement 4: \( \overline{XZ} \cong \overline{XZ} \)
• Statement 5: \( \triangle WXZ \cong \triangle YZX \), Reason: SAS Congruence Postulate
• Statement 6: Reason: CPCTC
• Statement 7: \( \overline{XY} \parallel \overline{WZ} \), Reason: Alternate Interior Angles Converse
• Statement 8: Reason: Definition of Parallelogram (both pairs of opposite sides parallel)