Question

in the diagram, a || b and c || f, forming a parallelogram. write a proof that proves that opposite sides of the parallelogram are congruent. you can write a two - column proof or a paragraph proof, but be sure to include a reason for each of your statements. (4 points)

Explanation:

Step1: State the parallel lines and transversal

Given \(a\parallel b\) and \(e\parallel f\). When two parallel lines are cut by a transversal, alternate - interior angles are congruent.
For lines \(a\) and \(b\) with transversal \(e\): \(\angle1\cong\angle4\) (alternate - interior angles).
For lines \(a\) and \(b\) with transversal \(f\): \(\angle2\cong\angle3\) (alternate - interior angles).
For lines \(e\) and \(f\) with transversal \(a\): \(\angle6\cong\angle4\) (alternate - interior angles).
For lines \(e\) and \(f\) with transversal \(b\): \(\angle1\cong\angle2\) (alternate - interior angles).

Step2: Use the ASA (Angle - Side - Angle) congruence criterion

Consider \(\triangle YWZ\) and \(\triangle XZW\).
We know that \(\angle1\cong\angle4\) (from Step 1, alternate - interior angles for \(a\parallel b\) and transversal \(e\)), \(WZ = ZW\) (common side), \(\angle2\cong\angle3\) (from Step 1, alternate - interior angles for \(a\parallel b\) and transversal \(f\)).
By ASA congruence criterion, \(\triangle YWZ\cong\triangle XZW\).

Step3: Use the property of congruent triangles

Since \(\triangle YWZ\cong\triangle XZW\), then \(YW\cong XZ\) and \(YZ\cong XW\) (corresponding parts of congruent triangles are congruent). In a parallelogram, opposite sides are \(YW\) and \(XZ\), \(YZ\) and \(XW\).

Answer:

We have proved that opposite sides of the parallelogram (\(YW\) and \(XZ\), \(YZ\) and \(XW\)) are congruent by showing \(\triangle YWZ\cong\triangle XZW\) using the ASA congruence criterion (based on the alternate - interior angles formed by parallel lines \(a\parallel b\) and \(e\parallel f\)) and then using the property that corresponding parts of congruent triangles are congruent.