Question

1. wxyz is a parallelogram with diagonals \\( \overline{xz} \\) and \\( \overline{wy} \\) intersecting at point v. 2. \\( \overline{wx} \parallel \overline{zy} \\) and \\( \overline{xy} \parallel \overline{wz} \\) 3. \\( \angle wxy \cong \angle yzw \angle wxz \cong \angle yzx \\) 4. \\( \overline{xy} \cong \overline{zw} \\) 5. \\( \triangle wzv \cong \triangle yxv \\) 6. \\( \overline{wv} \cong \overline{yv} \\) 7. \\( \overline{xz} \\) bisects \\( \overline{wy} \\)

Explanation:

Step1: Find the reason for $\angle WXY\cong\angle YZW,\angle WXZ\cong\angle YZX$

Since \(WX\parallel ZY\) (from step 2), by the property of parallel lines (alternate - interior angles), when a transversal cuts two parallel lines. For \(\angle WXY\) and \(\angle YZW\), \(XY\parallel WZ\) (from step 2) and \(WX\) and \(ZY\) are transversals. Also, for \(\angle WXZ\) and \(\angle YZX\), \(WX\parallel ZY\) and \(XZ\) is a transversal. The reason is "Alternate - interior angles theorem".

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

We know that \(\angle WXZ\cong\angle YZX\) (from step 3), \(\angle WZV\) and \(\angle YXV\) are vertical angles (\(\angle WZV\cong\angle YXV\) because vertical angles are congruent), and \(\overline{XY}\cong\overline{ZW}\) (from step 4). By the Angle - Side - Angle (ASA) congruence criterion, \(\triangle WZV\cong\triangle YXV\).

Answer:

1. Alternate - interior angles theorem; 5. ASA (Angle - Side - Angle)