Help with proving a quadrilateral is a parallelogram.

In summary, the conversation discussed a proof for showing that a quadrilateral is a parallelogram when given a pair of congruent and parallel opposite sides. The suggested solution involved constructing both diagonals and using SAS congruency theorem to prove the congruence of the two triangles formed. This ultimately leads to the conclusion that the quadrilateral is a parallelogram.
  • #1
Scorpino
7
0
[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?
 
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  • #2
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?
 
  • #3
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.
 
  • #4
Nice job! By using SAS you avoided the need to construct both diagonals.
 
  • #5
Sweet! Thanks for the help and for checking. :biggrin:
 
  • #6
You're welcome. Thread marked as [ SOLVED ].
 

1. How do I prove that a quadrilateral is a parallelogram?

To prove that a quadrilateral is a parallelogram, you can use the properties of parallelograms, such as opposite sides being parallel and congruent, opposite angles being congruent, and consecutive angles being supplementary.

2. Can I use the properties of a parallelogram to prove that a quadrilateral is a parallelogram?

Yes, you can use the properties of a parallelogram, such as opposite sides being parallel and congruent, opposite angles being congruent, and consecutive angles being supplementary, to prove that a quadrilateral is a parallelogram.

3. What other methods can I use to prove that a quadrilateral is a parallelogram?

Other methods to prove that a quadrilateral is a parallelogram include using the definition of a parallelogram, which states that it is a quadrilateral with two pairs of parallel sides. You can also use theorems, such as the opposite sides of a quadrilateral bisect each other, or the diagonals of a parallelogram bisect each other.

4. Can I use only one property to prove that a quadrilateral is a parallelogram?

No, you typically need to use multiple properties or methods to prove that a quadrilateral is a parallelogram. This is because there are different ways to prove that a shape is a parallelogram, and using only one property may not provide enough evidence to support the claim.

5. Is it possible for a quadrilateral to have some properties of a parallelogram but not be a parallelogram?

Yes, a quadrilateral may have some properties of a parallelogram, such as opposite sides being congruent, but not be a parallelogram. This is because there are other shapes, such as rhombuses and rectangles, that also have these properties but are not parallelograms.

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