Question 357957
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Given quadrilateral ABCD:


{{{drawing(
500, 500, -5, 5, -5, 5,
line(-2,3,3,3),
line(-2,3,-3,-3),
line(-3,-3,2,-3),
line(2,-3,3,3),
locate(-2.3,3.4,"A"),
locate(3.3,3.4,"B"),
locate(-3.3,-3.1,"D"),
locate(2.3,-3.1,"C")
)}}}


If 


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \overline{AB}\ \parallel\ \overline{CD}] and *[tex \LARGE \overline{DA}\ \parallel\ \overline{CB}] 


then ABCD is a parallelogram.


OR if


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\angle DAB\ =\ m\angle BCD] and *[tex \LARGE \ m\angle ABC\ =\ m\angle CDA]


then ABCD is a parallelogram.


OR if


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\angle DAB\ +\ m\angle ABC\ =\ 180^\circ] and *[tex \LARGE \ m\angle ABC\ +\ m\angle BCD\ =\ 180^\circ]


then ABCD is a parallelogram.


OR if


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \overline{AC}] bisects *[tex \LARGE \overline{BD}] and *[tex \LARGE \overline{BD}] bisects *[tex \LARGE \overline{AC}]


then ABCD is a parallelogram.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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