Proving a Quadrilateral is a Parallelogram: Challenge

austint

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Oct 20, 2017
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Hello all!

I have a challenge problem here regarding how to prove a quadrilateral is a parallelogram. Here it is:

Given that a quadrilateral has one pair of congruent sides and one pair of opposite angles congruent, prove that it is a parallelogram or provide a counterexample.

Thank you for any responses!
 
Hello all!

I have a challenge problem here regarding how to prove a quadrilateral is a parallelogram. Here it is:

Given that a quadrilateral has one pair of congruent sides and one pair of opposite angles congruent, prove that it is a parallelogram or provide a counterexample.

Thank you for any responses!
Did you draw some figures to investigate?

Note that the problem does not mention whether the given congruent sides are adjacent or not!

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33
 
My diagram consisted of a quadrilateral that consisted of the givens. I then drew a diagonal to split the shape into two triangles. Then, using the SAS postulate I proved the triangles congruent. From that, I found that there were two sets of alternate interior angles that allow both sides of the quadrilateral be parallel. So, because it is a quadrilateral with two sets of parallel sides it is a parallelogram.
 
Draw AB = 10 cm and AC = 10 cm and angle BAC = 60°.

Draw BD = 6 cm and CD = 6 cm.

Did you get a parallelogram?

Do you have two sets of congruent sides (AB = AC and BD = CD?

Do you have a set of congruent opposite angles (angles ABD = angle ACD)?
 
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Draw AB = 10 cm and AC = 10 cm and angle BAC = 60°.

Draw BD = 6 cm and CD = 6 cm.

Did you get a parallelogram?

Do you have two sets of congruent sides (AB = AC and BD = CD?

Do you have a set of congruent opposite angles (angles ABD = angle ACD)?

I have two sets of congruent sides (AB = AC and BD = CD). I drew diagonal CB and proved that triangle CAB is equiangular. Therefore, since angles BCD and CBD are congruent, then angle ABD and ACD are both equal to (x+60), if I let x = the unknown angles. Because the two triangles are not congruent, I can't say there are alternate interior angles or same-side interior angles (since 60 + 60 is 120 and not 180). So, I'm not sure if I have a parallelogram because I can't prove the sides parallel. Also, the sides congruent are not opposite so I don't think I can use them to prove the quadrilateral is a parallelogram.
 
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