Question

given ad = bc and ad || bc. prove: abcd is a parallelogram. statements: 1. ad = bc, ad || bc; 2. ∠cad and ∠acb are alternate - interior ∠s; 3. ∠cad = ∠acb; 4. ac = ac; 5. △cad ≅ △acb. reasons: 1. given; 2. definition of alternate - interior angles; 3. alternate - interior angles are congruent; 4. reflexive property; 5. sas congruency theorem.

Explanation:

Step1: Recall parallelogram criteria

One way to prove a quadrilateral is a parallelogram is to show that one - pair of opposite sides is both parallel and congruent.

Step2: Analyze given information

We are given that \(AD = BC\) (congruent sides) and \(AD\parallel BC\) (parallel sides).

Step3: Use angle - side relationships

Since \(AD\parallel BC\), \(\angle CAD\) and \(\angle ACB\) are alternate interior angles. By the property of alternate interior angles (\(\angle CAD\cong\angle ACB\)). Also, \(AC\) is common to \(\triangle CAD\) and \(\triangle ACB\) (reflexive property). So, \(\triangle CAD\cong\triangle ACB\) by SAS (Side - Angle - Side) congruency theorem. But we don't actually need the triangle - congruence to prove \(ABCD\) is a parallelogram. Since we have a pair of opposite sides (\(AD\) and \(BC\)) that are both parallel and congruent, \(ABCD\) is a parallelogram.

Answer:

Since one pair of opposite sides (\(AD\) and \(BC\)) of quadrilateral \(ABCD\) is both parallel and congruent, \(ABCD\) is a parallelogram.