I have already shown how:

1) to bisect an angle,

2)bisect a line segment,

3)construct a perpendicular line at a given point on the line,

4)and how to construct a perpendicular line from a point not on the line,

all based on Euclid's propositions 9-12 in book I. I am then asked to prove:

5)the bisector in 1 is unique

6) the bisector in 2 is unique

7) the perpendicular line in 3 is unique

8) the perpendicular line in 4 is unique

it also hints that these proofs are usually done by contradiction. now i have done number five, but i am stuck at the other three. how are they unique and how can i prove them? because i can easily create another line that bisects the original line segment, but just at a different line segment. and for 7 and 8, i can create other lines as well that bisect the are perpendicular. any help would be great. my prof barely touched on uniqueness at all and now i have to construct all of this.