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In the given quadrilateral ABCD, we have two congruent sides AB = CD and AD = CB. We need to prove that two sides DC and AB are parallel. We can write the following two-column proof.

Statement #1: AB = CD and AD = CB.

Reason #1: Given.

Statement #2: AC = AC.

Reason #2: Reflexive property (x = x).

Statement #3: ΔABC ≅ ΔCDA.

Reason #3: SSS (Side-Side-Side).

Statement #4: ∠1 = ∠4.

Reason #4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Statement #5: DC║AB.

Reason #5: Alternate Interior Angles Postulate.

Answer:

Statement                                                       Reason

1. AB=CD and AD=CB                                     1. Given

2. AC=AC                                                        2. Reflexive property

3. Triangle ACD is congruent to                   3. SSS (Side-Side-Side)

   triangle ABC

4. <1=<4                                                           4. CPCTE

5. DC // AB                                                      5. Equal alternate interior angles


Solution:

1. AB=CD and AD=CB is given by the problem.

2. AC=AC: Each value is equal to itself by the reflexive property.

3. The triangles ACD and ABC are congruent because they have their three sides congruent (SSS: Side-Side-Side): AC in triangle ACD with AC in triangle ABC (by point 2); AB in triangle ABC with CD in triangle ACD, and CB in triangle ABC with AD in triangle ACD (this is given).

4. <1=<4, because of Corresponding Parts of Congruent Triangles are Equal (CPCTE), the angles 1 and 4 are correspondings in triangles ACD and ABC because they are opposite to congruent sides: <1 is opposite to side CB, that is congruent to side AD (given), opposite to <4.

5. If the alternate interior angles in two lines cut by a transversal are equal, the lines must be parallels: Then alternate interior angles <1 and <4 are equal by point 4, in the two lines DC and AB that are cut by the transversal AC, then the lines DC and AB must be parallel.


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