Question 526318
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Draw a diagonal in the quadrilateral forming two triangles.


Label the angles in one triangle *[tex \Large a], *[tex \Large b], and *[tex \Large c].  Label the angles in the other triangle *[tex \Large d], *[tex \Large e], and *[tex \Large f].  Arrange the labels so that *[tex \Large b] and *[tex \Large e] are adjacent and *[tex \Large c] and *[tex \Large f] are adjacent.  Then label the angle that is the combination of *[tex \Large b] and *[tex \Large e] with a *[tex \Large g] and the angle that is the combination of *[tex \Large c] and *[tex \Large f] with a *[tex \Large h].  The four interior angles of the quadrilateral should then be *[tex \Large a], *[tex \Large g], *[tex \Large d], and *[tex \Large h] and we want to prove that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ g\ +\ d\ +\ h\ =\ 360]


The interior angles of a triangle sum to 180 degrees (you find the theorem that says so; it is in your Geometry book somewhere.)


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ =\ 180]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ +\ e\ +\ f\ =\ 180]


so


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ +\ e\ +\ f\ =\ 360]


But


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ g\ =\ b\ +\ e]


by the Angle Sum Postulate and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ h\ =\ c\ +\ f]


by the Angle Sum Postulate


Then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ (b\ +\ e)\ +\ d\ +\ (c\ +\ f)\ =\ 360]


By Addition Associativity


And then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ g\ +\ d\ +\ h\ =\ 360]


By Substitution of Equality.  Q.E.D.


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