diagram 1 shows the ramp and the rail and the posts in real life situation.
the rail is ABCDE
the ramp is PQRST
the ground is KLMNT
diagram 2 shows the angle relationships that are formed.
the parallel posts are line segments FAK, GBL, HCM, IDN, JET.
the transversaal is the rail which is the line segment ABCDE.
the ground is the line segment KLMNT.
the ramp is the line segment PQRST.
the ramp makes a 12 degree angle with the ground.
this is shown as triangle SNT in the diagram.
since angle STN is 12 degrees, then angle TSN must be 78 degrees because the acute angles in a right triangle are complementary.
this is what makes the angles that the rail makes with the parallel posts equal to 78 degrees and 102 degrees.
we'll just look at two of the parallel posts in order to show the angle relationships.
the other parallel posts will have the same properties with each other.
the parallel posts are representedby the line segments GBQL and HCRM.
the points on the top side of the rail are just there to identify angles.
they aren't part of the real posts, which terminate at the rail on one end and at the ground on the other end.
since the posts are parallel with each other and the rail is the transversal, the following relationships must hold.
1.
alternate interior angles must be equal.
this can be seen with angles QBC and HCB which are both 78 degrees.
2.
alternate exterior angle must be equal.
this can be seen with angles GBA and RCD which are both 78 degrees.
3.
corresponding angles must be equal.
this can be seen with angles ABG and BCH which are both 78 degrees.
4.
same side interior angles must be supplementary.
this can be seen with angles GBC and HCB which are 78 degrees and 102 degrees which adds up to 180 degrees, making them supplementary.
5.
same side exterior angles must be supplementary.
this can be seen with angles ABQ and RCD which are 102 degrees and 78 degrees which adds up to 180 degrees, making them supplementary.
the posts are 3 feet apart.
this is calculated as follows.
there are 5 posts.
this is equal to the first post on plus 4 additional.
divide 12 by 4 and you get 3 feet between each post.
the positions are:
post 1 at 0 feet.
post 2 at 3 feet.
post 3 at 6 feet.
post 4 at 9 feet.
post 5 at 12 feet.
the end posts are posts 0 and 5.
the intermediate posts are posts 1, 2, and 3.
i believe this answers your questions.
if you have any further questions regarding this, drop me a line.
a good reference on parallel lines and transversals can be found here.
https://www.mathsisfun.com/geometry/parallel-lines.html
there are lots more references on the web.
just do a search for parallel lines cut by a transversal.
this problem may have presented some visual problems since the parallel lines are vertical and most references show the parallel lines at horizontal.
the relationships are still the same, regardless.