Question

wx ≅ yz and yz || wx. complote the proof that ∠ywz ≅ ∠wyx.
1 wx ≅ yz given
2 yz || wx given
3 ∠wyz ≅ ∠xwy
4 wy ≅ wy
5 △wxy ≅ △yzw
6 ∠ywz ≅ ∠wyx

Explanation:

Step1: Alternate interior angles

Since \(\overline{YZ}\parallel\overline{WX}\), by the alternate - interior angles theorem, \(\angle WYZ\cong\angle XWY\) (when two parallel lines are cut by a transversal, alternate interior angles are congruent).

Step2: Reflexive property

For any segment \(a\), \(a = a\). So, \(\overline{WY}\cong\overline{WY}\) by the reflexive property of congruence.

Step3: SAS congruence

We know that \(\overline{WX}\cong\overline{YZ}\) (given), \(\angle WYZ\cong\angle XWY\) (from step 1), and \(\overline{WY}\cong\overline{WY}\) (from step 2). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle WXY\cong\triangle YZW\).

Step4: CPCTC

If two triangles \(\triangle WXY\) and \(\triangle YZW\) are congruent, then their corresponding parts are congruent. So, \(\angle YWZ\cong\angle WYX\) by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Answer:

1. Given
2. Given
3. Alternate - interior angles theorem
4. Reflexive property of congruence
5. SAS (Side - Angle - Side) congruence criterion
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)