Question 614081: Two adjacent angles of a parallelogram are bisected. How many degrees are in the angle made by the two bisectors?
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! The key to this is to know that consecutive angles (adjacent is not really the proper term for any of the angle in a parallelogram) are supplementary. IOW, they always add up to 180 degrees.
Let x = the degrees in one of the consecutive angles. Then, since tghe angle consecutive to this must add, with x, to 180, we can use
(180-x)
for the degrees of the other angle. (Add x and (180-x) to see that they do add up to 180.)
The bisected angle of x will be  degrees. And the other bisected angle will be  degrees.
The two bisectors will intersect somewhere. (It may help to make a drawing to see all this.) The two bisectors and the side between the consecutive angles form a triangle.
One of the angles in this triangle is and another is and the third angle is the angle we want to find.
As we all know the three angles of every triangle add up to 180. And as we have probably done many times before, if we have two angles of a triangle and we want to find the third, you add the two angles you have and then subtract that from 180. We will do exactly this now.
Add the two angles we "know":
These fractions have the same denominators so we can go ahead and add them:
which simplifies as follows:
90
Now we subtract this from 180 to find the third angle:
180 - 90 = 90
So the angle we want is 90 degrees (i.e. a right angle).
Note, since the x's disappeared, this means that the value of x does not matter. IOW, the angle formed by the angle bisectors is always 90 degrees no matter what the original parallelogram looks like.
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