How can we conclude that a proof applies to all situations

[CoolShank8]

When we do a parallelogram proof we do something like this:

https://gyazo.com/f8987046ad28142673daff9bd844c6fe

But we've only proven it to this specific pallelegram, how can we conclude that this applies for everything (this proof is showing how the two opposite side sets are congruent)

And when can we conclude that it only works for that specific shape?

 

[R.M.]

Actually, you have proven it for EVERY parallelogram. You and I could each draw a parallelogram ABCD that look different (mine might have long AB, yours might have short AB), but our proofs would be identical, meaning that the theorem is proven for ALL cases.

 

[HallsofIvy]

What do you mean by "this specific parallelogram"? A "specific" parallelogram will have specific side lengths and angles. This proof does not assume any specific lengths or angles.

 

[Dr.Peterson]

When we do a parallelogram proof we do something like this:

https://gyazo.com/f8987046ad28142673daff9bd844c6fe

But we've only proven it to this specific pallelegram, how can we conclude that this applies for everything (this proof is showing how the two opposite side sets are congruent)

And when can we conclude that it only works for that specific shape?

You omitted the statement of the theorem being proved. That will state the conditions assumed for the proof (the "givens"); the proof then shows that the conclusion is true whenever those conditions are satisfied (in this case, whenever ABCD is a parallelogram). Of course, the particular letters used in naming the figure are not essential to what it is; they are just "pseudonyms"; all that matters is what it is, not what it is called.

A proof will work for any case that meets the conditions, and it will not work for any other case. It tells you when it works!

In particular, this proof only works for parallelograms, not for trapezoids, nor for pallelegrams, whatever they are