Perpendicular Bisectors

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skinski43

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Prove that bisectors of two supplementary angles are perpendicular to each other.
please help
 
skinski43 said:
Prove that bisectors of two supplementary angles are perpendicular to each other.
please help
Click to expand...

This is fairly straight-forward. Exactly where are you stuck. Draw a pair of supplementary angles - with a common side. Then bisect those angles and proceed.

Please show your work, indicating exactly where you are stuck - so that we know where to begin to help you.
 
i have a horizontal line with a vertical line coming up from it. the angle on the right of the vertical line is 60 degrees and bisected 30 degrees. that leaves the other angle on the left of the vertical line as 120 degrees and bisected = 60 degrees. sure enough 60 plus 30 is 90 degrees so the bisectors are perpendicular.
but how do i prove it will always happen this way?
 
skinski43 said:
i have a horizontal line with a vertical line coming up from it. the angle on the right of the vertical line is 60 degrees and bisected 30 degrees. that leaves the other angle on the left of the vertical line as 120 degrees and bisected = 60 degrees. sure enough 60 plus 30 is 90 degrees so the bisectors are perpendicular.
but how do i prove it will always happen this way?
Click to expand...

Unless your two supplementary angles are ADJACENT angles (if you're not sure what adjacent angles are, please look up the definition in your book), then the bisectors of those two supplementary angles are NOT necessarily perpendicular.

Consider the following possibility: there is an angle on YOUR ceiling that has a measure of 144 degrees. There is an angle on MY floor which has a measure of 36 degrees. Would you agree that those angles are supplementary? (I hope you do!)

Now, suppose you draw the bisector of your 144-degree angle, and I draw the bisector of my 36-degree angle. There is NO GUARANTEE that your angle bisector and my angle bisector will even intersect! And if they don't intersect, then they can't possibly be perpendicular....

Please double-check the wording of your problem....I'm not going to go any further until I know exactly what your problem says.
 
the problem reads "prove bisectors of two supplementary angles are perpendicular to each other"
 
my drawing didn't come out properly but my angles are adjacent to each other
 
skinski43 said:
the problem reads "prove bisectors of two supplementary angles are perpendicular to each other"
Click to expand...

That's just plain not true....unless the angles are adjacent, which is not the condition stated in the problem.

If the angles ARE adjacent, you can proceed as follows:

Code: 
                       /D
                      /
                     /
--------------------/---------------------------------
A                  B                        C

B is between A and C, so <ABC is a straight angle, with measure 180 (definition of a straight angle)

m<ABD + m<DBC = m<ABC
m<ABD + m<DBC = 180

So, <ABD and <DBC are supplementary.

Now, draw BE as the bisector of <ABD, and BF as the bisector of <DBC

Multiply both sides of this equation by 1/2:
m<ABD + m<DBC = 180
(1/2)*m<ABD + (1/2)*m<DBC = (1/2)*180
(1/2)*m<ABD + (1/2)*m<DBC = 90

Your goal is to prove that BE is perpendicular to BF.

m<EBD = (1/2)m<ABD and m<FBD = (1/2) m<DBC by the definition of bisect.

So, substituting for (1/2)*m<ABD and (1/2)*m<DBC, we have

m<EBD + m<FBD = 90

And m<EBD + m<FBD = m<EBF (by angle addition)

So, m<EBF = 90 and <EBF is a right angle.

Since BE and BF form a right angle, EB is perpendicular to BF.
 
horizontal line with a vertical line coming up from the horizontal line the angles on either side of the vertical line are supplementary, are they not?
 
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