Question

prove: wx ≅ yz
place reasons in the table to complete the proof.
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1. wxyz is a parallelogram.
2. wx = yz

wz = xy

1. ∠zwy ≅ ∠xyw

∠zyw ≅ ∠xwy

1. wy = wy
2. △wyz ≅ △ywx
3. wx ≅ yz
4. given
5. definition of a parallelogram

3.
4.
5.
6.
sss
angle addition postulate
reflexive property
corresponding angles are congruent.
transitive property
asa
alternate interior angles are congruent.
alternate exterior angles are congruent.
corresponding parts of congruent triangles are congruent.
corresponding parts of congruent triangles are similar.
aa
sas

Explanation:

Step1: Analyze statement 3

Since \(WXYZ\) is a parallelogram (\(WX\parallel YZ\)), by the property of alternate interior angles (when two parallel lines are cut by a transversal), \(\angle ZWY\) and \(\angle XYW\) are congruent. So the reason for statement 3 is "Alternate interior angles are congruent".

Step2: Analyze statement 4

\(WY = WY\) is due to the reflexive property (a segment is congruent to itself). So the reason for statement 4 is "Reflexive property".

Step3: Analyze statement 5

We have two triangles \(\triangle WYZ\) and \(\triangle YWX\). We know \(\angle ZWY\cong\angle XYW\) (from step 1), \(WY = WY\) (from step 2), and \(WZ = XY\) (property of parallelogram \(WXYZ\), opposite sides of a parallelogram are equal). By the \(ASA\) (Angle - Side - Angle) congruence criterion, \(\triangle WYZ\cong\triangle YWX\). So the reason for statement 5 is "\(ASA\)".

Step4: Analyze statement 6

Since \(\triangle WYZ\cong\triangle YWX\), by the property of corresponding parts of congruent triangles, \(WX\cong YZ\). So the reason for statement 6 is "Corresponding parts of congruent triangles are congruent".

Answer:

1. Given
2. Definition of a parallelogram
3. Alternate interior angles are congruent
4. Reflexive property
5. \(ASA\)
6. Corresponding parts of congruent triangles are congruent