Question

in the diagram, ( a parallel b ) and ( e parallel 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: Identify congruent angles

Since \(a\parallel b\) and \(e\) is a transversal, \(\angle1=\angle3\) (alternate - interior angles). Since \(e\parallel f\) and \(a\) is a transversal, \(\angle6 = \angle4\) (alternate - interior angles). Also, since \(a\parallel b\) and \(f\) is a transversal, \(\angle5=\angle3\) (alternate - interior angles), and since \(e\parallel f\) and \(b\) is a transversal, \(\angle2=\angle4\) (alternate - interior angles).

Step2: Consider triangles \(\triangle WXY\) and \(\triangle ZYX\)

In \(\triangle WXY\) and \(\triangle ZYX\):

• \(\angle1=\angle3\) (from Step 1, alternate - interior angles as \(a\parallel b\) and \(e\) is a transversal)
• \(XY = YX\) (common side)
• \(\angle2=\angle4\) (from Step 1, alternate - interior angles as \(e\parallel f\) and \(b\) is a transversal)

By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle WXY\cong\triangle ZYX\)

Step3: Use congruent triangles to find congruent sides

Since \(\triangle WXY\cong\triangle ZYX\), \(WY = ZX\) (corresponding parts of congruent triangles). Also, consider \(\triangle YWZ\) and \(\triangle XZW\):

• \(\angle6=\angle4\) (alternate - interior angles as \(e\parallel f\) and \(a\) is a transversal)
• \(YZ=WX\) (common side)
• \(\angle5=\angle3\) (alternate - interior angles as \(a\parallel b\) and \(f\) is a transversal)

By ASA, \(\triangle YWZ\cong\triangle XZW\), so \(YW = ZX\) and \(YZ = WX\)

Answer:

We have proved that the opposite sides of the parallelogram (formed by \(a\parallel b\) and \(e\parallel f\)) are congruent. For example, using the ASA congruence criterion on \(\triangle WXY\) and \(\triangle ZYX\) (and also on \(\triangle YWZ\) and \(\triangle XZW\)) we get that \(WY = ZX\) and \(YZ = WX\) which are the pairs of opposite sides of the parallelogram.