Help with proving a quadrilateral is a parallelogram.

[Scorpino]

[SOLVED] Help with proving a quadrilateral is a parallelogram.

Hi this is my first post here and I'm glad to see that this is a well visited board. I'm having trouble with this one proof though that I have to do for geometry due tomorrow. Only a few other people I know have been assigned this specific one but they haven't done it yet.

Given: A quadrilateral with a pair of congruent and parallel opposite sides.

Prove: The quadrilateral is a parallelogram

I know that I need to get the other two sides of the quad to be parallel, but I have no idea how to do that. I'm assuming first you have to make another construction, like connecting the opposite sides forming two triangles. I'd need to prove those triangles are congruent, but I can't because of the limited information given. So can someone try and help?

PS: Why can't I attach Geometer's Sketchpad files to the post?

 

[D H]

Form both diagonals (the lines that onnect both opposite vertices of the quadrilateral). Those two lines intersect somewhere in the middle of the quadrilateral, forming four angles. What can you say about these four angles? Now look at just one of the diagonals. It intersects both of the known parallel lines. What can you say about the angles formed by these intersecting lines?

 

[Scorpino]

Hmm...thanks for the reply. I managed to get a solution before you posted though, so it didn't include what you were saying. Do you or anyone else mind checking it for me?

1) BC congruent and Parallel to AD (Given)

2) Construct AC (Construction)

3) AC congruent AC (Reflexive prop. of equality)

4) <CAD congruent to <ACB (Parallel lines form equal alternate interior angles)

5) Triangle BCA congruent to Triangle DAC (SAS congruency theorem)

6) <CAB congruent to <ACD (CPCTC)

7) BA Parallel to CD (Equal alternate interior angles form parallel lines)

8) Quadrilateral ABCD is a parallelogram (Definition of a parallelogram)

Thanks.

 

[D H]

Nice job! By using SAS you avoided the need to construct both diagonals.

 

[Scorpino]

Sweet! Thanks for the help and for checking.

 

[D H]

You're welcome. Thread marked as [ SOLVED ].