Question

You are constructing a railing to place on a ramp for your handicapped grandmother. You have been given a 12-ft rail and 5 posts to support the railing. The posts are equally spaced under the rail and the first and last posts are positioned at each end of the rail. The ramp rises at an angle of 12° in relation to the ground. After construction is complete, the rail is parallel to the ramp and the posts are perpendicular to the ground.

a. What part(s) will represent the transversal(s)?
b. In order to make the posts parallel, what angles must they form with the transversal?
c. What angle would they form with the ramp?
d. Which angles would be alternate interior, alternate exterior, corresponding, and consecutive interior? Explain your reasoning.
e. What do you know about each of these angles?
f. How far apart will the posts be?

Answer 1

a. The rail and the ramp will represent the transversals.

b. In order for the posts to be parallel, they must form equal angles with the transversals. Since the rail is parallel to the ramp, the posts must form angles of 12° with the rail.

c. The posts will also form angles of 12° with the ramp, since the rail is parallel to the ramp.

d. The angles that are formed between parallel lines and a transversal have special relationships. In this case, the rail and the ramp are parallel lines, and the posts are the transversal. The pairs of angles that have special relationships are:

Alternate interior angles: These are pairs of angles that are on opposite sides of the transversal, and are between the parallel lines. In this case, the alternate interior angles would be the angles between the rail and the ramp, on opposite sides of each post.

Alternate exterior angles: These are pairs of angles that are on opposite sides of the transversal, and are outside the parallel lines. In this case, the alternate exterior angles would be the angles outside the rail and the ramp, on opposite sides of each post.

Corresponding angles: These are pairs of angles that are in the same position relative to the transversal, but on opposite sides of the parallel lines. In this case, the corresponding angles would be the angles on the same side of each post, but on opposite sides of the rail and the ramp.

Consecutive interior angles: These are pairs of angles that are on the same side of the transversal, and are between the parallel lines. In this case, the consecutive interior angles would be the angles between the rail and the ramp, on the same side of each post.

e. Each pair of corresponding angles is congruent, each pair of alternate interior angles is congruent, each pair of alternate exterior angles is congruent, and the consecutive interior angles are supplementary (their sum is 180 degrees).

f. Since there are 5 posts supporting the rail, there are 6 sections of the rail between the posts. These 6 sections must be equal in length. To find the length of each section, we can use trigonometry. The rise of the ramp is 12°, and the length of the rail is 12 feet. Therefore, the height of the ramp is:

h = 12 tan(12°) ≈ 2.4 feet

The total length of the rail is 12 feet, so the length of the 6 sections is:

12 ft - 2(0.5 ft) = 11 ft

Dividing this length by 6, we get:

11 ft ÷ 6 = 1.8333 ft

Therefore, the posts will be spaced 1.8333 feet apart (or approximately 22 inches apart).