Given a parallelogram, prove that the opposite sides are congruent.

Question

Using a two-column proof, prove that the opposite sides of a parallelogram are congruent, given that the shape is a parallelogram.

Answer 1

Given parallelogram ABCD with AB || CD and AD || BC.

 

Construct diagonal AC.

 

Angle DAC ≅ Angle BCA b/c Alternate Interior Angles of a Transversal are ≅.

 

Angle BAC ≅ Angle DCA b/c Alternate Interior Angles of a Transversal are ≅.

 

AC ≅ AC b/c it's the same line segment.

 

Triangle ABC ≅ Triangle CDA by ASA.

 

AB ≅ CD and BC ≅ DA b/c corresponding parts of ≅ Triangles are ≅ .

 

Q.E.D.