Question

select the correct answer from each drop - down menu. complete the following proof. given: wxyz is a parallelogram with diagonals \\( \overline{xz} \\) and \\( \overline{wy} \\) intersecting at point v. prove: \\( \overline{xz} \\) bisects \\( \overline{wy} \\) 1. wxyz is a parallelogram with diagonals \\( \overline{xz} \\) and \\( \overline{wy} \\) intersecting at point v. given 2. \\( \overline{wx}\parallel\overline{zy} \\) and \\( \overline{xy}\parallel\overline{wz} \\) definition of a parallelogram 3. \\( \angle wxy\cong\angle yzw \\) \\( \angle wxz\cong\angle yzx \\) 4. \\( \overline{xy}\cong\overline{zw} \\) opposite sides of a parallelogram are congruent 5. \\( \triangle wzv\cong\triangle yxv \\) 6. \\( \overline{wv}\cong\overline{yv} \\) corresponding parts of congruent triangles are congruent 7 \\( \overline{xz} \\) bisects \\( \overline{wy} \\) definition of segment bisector

Explanation:

Step1: Justify \(\angle WXY\cong\angle YZW\) and \(\angle WXZ\cong\angle YZX\)

Since \(WX\parallel ZY\) (from statement 2, definition of a parallelogram), by the Alternate - Interior Angles Theorem, when a transversal cuts two parallel lines. For transversal \(WY\) cutting \(WX\) and \(ZY\), \(\angle WXY\cong\angle YZW\) (alternate - interior angles). For transversal \(XZ\) cutting \(WX\) and \(ZY\), \(\angle WXZ\cong\angle YZX\) (alternate - interior angles).

Step2: Justify \(\triangle WZV\cong\triangle YXV\)

We know that \(XY\cong ZW\) (statement 4), \(\angle WXZ\cong\angle YZX\) (from step 1), and \(\angle WXY\cong\angle YZW\) (from step 1). By the Angle - Side - Angle (\(ASA\)) congruence criterion, in \(\triangle WZV\) and \(\triangle YXV\), \(\angle WZV=\angle YXV\) (from \(\angle WXZ\cong\angle YZX\)), \(ZW = XY\) (statement 4), \(\angle ZWV=\angle X YV\) (from \(\angle WXY\cong\angle YZW\)). So, \(\triangle WZV\cong\triangle YXV\) by \(ASA\).

Answer:

For statement 3: Alternate - Interior Angles Theorem. For statement 5: \(ASA\) (Angle - Side - Angle)