Perpendicularity and parallelism

Imagine you have a shoebox (shaped like a cuboid) on your desk. Looking at the edges of the lid:

1. a) State two edges that are parallel to each other.
2. b) State two edges that are perpendicular and intersect at a vertex.

Two aeroplanes are flying in the sky. Aeroplane A is flying North at an altitude of $10,000$ metres. Aeroplane B is flying East at an altitude of $8,000$ metres. Given that their flight paths do not intersect and are not parallel:

1. a) What is the relative position of these two paths in space called?
2. b) Are these lines coplanar? Explain why.

Imagine a football goal. Let us analyse its posts as lines:

1. a) What is the relative position of the two vertical posts?
2. b) What is the relative position of the crossbar (horizontal post) with respect to each of the vertical posts?

In the school gym, a straight column (line $r$) meets the floor (plane $\pi$) at a point, which is its base. On the floor there are two distinct lines, $s$ and $t$, that pass through the base of the column, and it is known that the column forms a right angle with each of them.

1. a) What is the relative position of the column and the floor? Justify your answer.

1. b) If we draw any other line on the floor that passes through the base of the column, what angle will it form with the column?

Look at the following cube and identify:

1. a) An edge that is parallel to the plane of the bottom base.
2. b) An edge that is perpendicular to the plane of the bottom base.

Consider a right prism with a regular hexagonal base. Analyse the relationship between its elements:

1. a) How many lateral faces are perpendicular to the bottom base?
2. b) How many pairs of lateral faces are parallel to each other?
3. c) If we draw the diagonal of a lateral face and a base edge that shares no vertex with that diagonal, what is the relative position of the lines containing them?

Look at the walls of your room.

1. a) What is the relative position of two opposite walls?
2. b) What is the line called where two adjacent walls intersect?

When you open a maths textbook and leave it resting on the table, the two pages form a dihedral angle.

1. a) If we open the book completely until it lies flat on the table, what is the size of that dihedral angle?
2. b) If the book is half-open and the pages form an angle of $90^\circ$, how are the planes of the two pages related to each other?

Imagine that the floor of your room is plane $\pi$ and one of the side walls is plane $\beta$. If you install a fibre optic cable along the corner where the ceiling meets the front wall (line $r$):

1. a) What is the relation between line $r$ and the floor plane $\pi$?
2. b) What is the relative position of line $r$ with respect to the side wall plane $\beta$?
3. c) If we draw a vertical line on the side wall (line $s$), will lines $r$ and $s$ intersect? Justify whether they are perpendicular.

A revolving door in a shopping centre has three identical glass panels that intersect along a central vertical axis.

1. a) What is the name of each of the three regions of space bounded by two consecutive panels?
2. b) If the dihedral angle between the first and second panels is $120^\circ$, and the three panels are symmetrically distributed, what are the sizes of the other two dihedral angles?
3. c) If a fly lands exactly on the central axis, how many faces of the dihedral angles does it lie on?

To provide shade to a school playground in Cork, the city council is going to install a rectangular tensioned awning $ABCD$. The upper side $AB$ is attached horizontally to a flat vertical wall $\alpha$ at a height of $3$ m above the ground. The lower side $CD$ is anchored directly to the horizontal ground $\beta$, at a horizontal distance of $4$ m from the wall. The canvas forms an inclined plane $\gamma$.\na) Before calculating anything, estimate whether the dihedral angle formed by the canvas $\gamma$ with the ground $\beta$ will be smaller than, equal to, or greater than $45^\circ$. Justify your estimate by comparing the height of the anchor on the wall with the horizontal anchor distance.\nb) Visualise the perpendicular cross-section (a right-angled triangle with legs of $3$ m and $4$ m) and calculate the length of the canvas slope (segment $BC$).\nc) Check whether your initial estimate was correct by calculating the tangent trigonometric ratio of the acute dihedral angle formed with the ground ($\tan\theta = 3/4 = 0.75$) and assessing whether an angle with that tangent is greater than or smaller than $45^\circ$ (taking into account that $\tan 45^\circ = 1$).