Question

1. by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are pa

use this definition in your proof.
given: ( a b c d ) is a parallelogram
prove ( overline{a d} cong overline{b c} )

1. for each part below, use the figure to fill in the blank.

if necessary, you may learn what the markers on a figure indicate

Explanation:

Step1: Use the property of parallelogram

Since \(ABCD\) is a parallelogram, by the definition of a parallelogram, \(\overline{AB}\parallel\overline{DC}\) and \(\overline{AD}\parallel\overline{BC}\).

Step2: Use the alternate - interior angles theorem

When \(\overline{AB}\parallel\overline{DC}\) and \(AC\) is a transversal, \(\angle1\cong\angle3\) (alternate - interior angles). When \(\overline{AD}\parallel\overline{BC}\) and \(AC\) is a transversal, \(\angle2\cong\angle4\).

Step3: Use the reflexive property

\(AC = AC\) (reflexive property of equality).

Step4: Prove triangle congruence

In \(\triangle ABC\) and \(\triangle CDA\), we have \(\angle1\cong\angle3\), \(AC = AC\), \(\angle2\cong\angle4\). So, \(\triangle ABC\cong\triangle CDA\) (by \(ASA\) - Angle - Side - Angle congruence criterion).

Step5: Use the property of congruent triangles

Since \(\triangle ABC\cong\triangle CDA\), then \(\overline{AD}\cong\overline{BC}\) (corresponding parts of congruent triangles are congruent).

Answer:

The proof is completed as above to show \(\overline{AD}\cong\overline{BC}\).