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Lets use a geometric argument.
If we draw a line such it it intersects two sides.
Since the lengths on the opposite sides are the same, according to the parallel lines theorem, angle a must equal angle d.
Similarly, angle must equal angle c.
If we make two triangles, Triangle abc and triangle def
180-e-d must equal angle f.
Are you following me so far?
yes I am, I am writing this down
Similarly, 180 - a -c must equal b.
ok I dont understand what 180-a-c means
Since a = d and c = e , therefore we can conclude f equal b.
I mean I am subtracting going: 180 degrees - angle a - angle c.
Because in a triangle there is 180 degrees. Do you understand or shall I explain more?
oh.. so does that mean 180 degrees has to be subtracted by angle a and c to find B? or am I completely wrong?
Yes you are correct but do you understand why?
because.... nope I got nothing... I just remember thats a rule somewhere right?
Does that make sense?
so that means both ABC and DEF are equal because they were cut equally in half. So each triangle equal 180 and both together is 360?
yeah okay, you could reason it that way too :P . But you had to prove they were equal first ;) .
You can only prove they are equal if you cut the quadrilateral in half like I said. THen you had to prove that I did cut it in half by proving all the angles were the same.
Ohh yeah, that's valid too :) .
Good! :) .
so to answer the question how do I conclude its a parallelogram?
Now since you have proved all the angles are the same on opposite sides, you can conclude that this quadrilateral in indeed a parralelogram because the opposite sides have the same length and the angles on opposite sides of each other are the same as well.
Thank you so much for that. You were very patient and kind. I will remember this!! Thanks. :D
You are welcome :) .