[Geometry] Need help with straightedge only constructions.

ieatStairs

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Hello!
I'm taking a geometry course, and was assigned a project where we have to make some constructions, using ONLY two parallel edges of a straightedge. (No compass or measurements.) There constructions are:
Construction 1: Construct the angle bisector of a given angle. Show your method works on both acute and obtuse angles.
Construction 2: Construct a perpendicular bisector of a given segment (where the length of the given segment is greater than the distance between your parallel lines.)
Construction 3: Construct the perpendicular through a given point on a given line.
Construction 4: Construct a perpendicular bisector of a given segment (where the length of the given segment is less than the distance between your parallel lines.
Construction 5: Construct a line parallel to a given line through a given point not on the given line.
Construction 6: Construct a perpendicular to a given line through a given point not on the given line.
I've tried creating 1 & 2, and I think I've gotten somewhere by creating a rhombus and following its properties, but I'm not really sure how to prove all the side lengths are equal.
On the other ones, I tried creating a few, but I always end up with something like, "how can I be sure that this angle is 90 degrees," or "how can I be sure that this line is half the length of another?"
I'm struggling on these, and if anyone can give me some hints or point me in the right direction on any of the constructions, that would be greatly appreciated. :)
Thanks!
P.S. Here is a link to the assignment, the uploader wouldn't work for me so here:

< link to objectionable site removed >
 
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...we have to make some constructions, using ONLY two parallel edges of a straightedge. (No compass or measurements.) There constructions are:
Construction 1: Construct the angle bisector of a given angle. Show your method works on both acute and obtuse angles.
Would the following work?

Line up the straightedge with one of the parallel lines, with the straightedge on the in-side of that side of the angle. Draw a parallel line along the other side of the straightedge.

Do the same with the other side of the angle.

Draw a line from the vertex of the angle through the intersection of the two parallels you just drew.

Construction 2: Construct a perpendicular bisector of a given segment (where the length of the given segment is greater than the distance between your parallel lines.)
Place the straightedge so that one side touches one endpoint, and the other side touches the other. Draw the parallel lines along either side of the straightedge.

Place the straightedge so the sides are swapped from however you oriented it previously, with respect to the endpoints.

Draw the line through the off-segment intersections of these pairs of parallel lines.

Construction 3: Construct the perpendicular through a given point on a given line.
Place the straightedge across the line at an angle to the line, with one edge touching the given point. Draw the two parallel lines along the two sides of the straightedge.

Using the current (new) line through the given point, move the straightedge so its other side is along this line. Draw the parallel line along the other side of the straightedge.

The new parallels you've drawn pass through the given point, plus now other points, one on each side of the given point. Follow the sort of procedure as in the previous construction.

Construction 4: Construct a perpendicular bisector of a given segment (where the length of the given segment is less than the distance between your parallel lines.
Can you extend the segment, and then apply what you did in (2) above?

Construction 5: Construct a line parallel to a given line through a given point not on the given line.
Try drawing three parallels, one passing through that point not on the line, and all three crossing the given line at some angle. Can you see anything useful coming from this?

Construction 6: Construct a perpendicular to a given line through a given point not on the given line.
Try doing something along the lines of (4) above, but with the middle straightedge parallel crossing the point not on the given line. ;)
 
Would the following work?

Line up the straightedge with one of the parallel lines, with the straightedge on the in-side of that side of the angle. Draw a parallel line along the other side of the straightedge.

Do the same with the other side of the angle.

Draw a line from the vertex of the angle through the intersection of the two parallels you just drew.


Place the straightedge so that one side touches one endpoint, and the other side touches the other. Draw the parallel lines along either side of the straightedge.

Place the straightedge so the sides are swapped from however you oriented it previously, with respect to the endpoints.

Draw the line through the off-segment intersections of these pairs of parallel lines.


Place the straightedge across the line at an angle to the line, with one edge touching the given point. Draw the two parallel lines along the two sides of the straightedge.

Using the current (new) line through the given point, move the straightedge so its other side is along this line. Draw the parallel line along the other side of the straightedge.

The new parallels you've drawn pass through the given point, plus now other points, one on each side of the given point. Follow the sort of procedure as in the previous construction.


Can you extend the segment, and then apply what you did in (2) above?


Try drawing three parallels, one passing through that point not on the line, and all three crossing the given line at some angle. Can you see anything useful coming from this?


Try doing something along the lines of (4) above, but with the middle straightedge parallel crossing the point not on the given line. ;)

Alright, I'll try this out.
Thanks so so much for the help!
 
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