Exercises: 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.

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?

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.

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?

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.

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 math class, Aoife is analyzing the edges of a cube-shaped box whose vertices are $A, B, C, D$ on the bottom base and $A', B', C', D'$ on the top base. Looking at the vertical edge containing segment $AA'$ and the horizontal edge containing $BC$, Aoife states: "Since these two lines never intersect in space no matter how far we extend them, they must necessarily be parallel, exactly as happens with lines in the plane."\na) Explain with reasoning what conceptual error Aoife makes by applying a property in three-dimensional space that is only valid in plane geometry.\nb) Determine the correct relative position between the line containing $AA'$ and the line containing $BC$, and justify why they cannot be parallel.\nc) Give an example using the edges of that same cube of two lines that do not intersect and that are indeed parallel to each other.

Consider a right regular triangular prism whose bases are the horizontal triangles $ABC$ (bottom) and $A'B'C'$ (top), with vertical lateral edges $AA'$, $BB'$, and $CC'$.\na) Write the name of a line determined by two vertices of the prism that is secant to the line containing edge $AB$, but is not contained in the plane of the bottom base.\nb) Find and name a line determined by two vertices of the prism that is skew to line $AA'$ (that is, neither secant nor parallel to it).\nc) If we want to draw an imaginary plane $\pi$ that cuts the three lateral edges at a fixed height of $2$ cm with respect to the bottom base, what relative position must this plane $\pi$ necessarily have with respect to the plane of the base $ABC$? Justify your deduction.

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?

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?

A famous tower in Dublin has an inclination of $15^\circ$ with respect to the vertical. Model this tower as an oblique prism where the horizontal ground of the square is the plane $\pi$ and the four lateral edges of the building represent the inclined corners of the facade.\na) What is the relative position of the lateral edges of the building with respect to the ground plane $\pi$? Justify whether they are perpendicular intersecting lines or oblique intersecting lines, stating the measure of the acute dihedral angle that the inclined facade forms with the horizontal plane of the ground.\nb) During maintenance work on the tower, an external freight hoist goes up along a guide rail parallel to one of the inclined edges of the facade, while an internal elevator ascends following a completely vertical rectilinear path (perpendicular to the ground plane $\pi$). If both paths do not intersect at any point and are not on the same vertical wall, what is their relative position in space? Reason whether they are parallel, intersecting, or skew.

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 $\tan 45^\circ = 1$).

A point light source is located at the centre of the base of a right cone of radius $R = 5$ cm and height $H = 12$ cm. The light source emits a light ray that is perpendicular to one of the slant heights (generators) of the cone.

1. a) Is it possible for this ray to pass through the curved surface of the cone?
2. b) Calculate the distance from the light source to the point on the slant height where the ray is perpendicular. To do this, picture the right-angled triangle formed by the axial cross-section of the cone.

A sphere has a radius of $13$ cm. A plane $\pi_1$ intersects the sphere at a distance of $5$ cm from its centre, forming a circle $C_1$. A second plane $\pi_2$, perpendicular to the first, also intersects the sphere, passing through its centre.

1. a) Describe the relative position of the line of intersection of both planes with respect to the sphere.
2. b) What is the minimum distance between the boundary of circle $C_1$ and plane $\pi_2$? Use a diagram of the circular cross-section to justify your answer.

In a regular square-based pyramid $ABCD$ with apex $V$, let $r$ be the line that contains the height of the pyramid (which joins $V$ to the centre $O$ of the base) and let $s$ be the line that contains the base edge $AB$. The aim is to rigorously prove the relative position between $r$ and $s$.\nTwo classmates propose different approaches:\n- Cian's method: He tries to find out whether there is any plane that simultaneously contains lines $r$ and $s$, analysing whether vertices $V, O, A$, and $B$ can lie in the same plane.\n- Aoife's method: She relies on the line-plane relationship: she uses the fact that line $r$ is perpendicular to the base plane and intersects it at point $O$, to then check whether line $s$ (contained in the base) passes through $O$ or not.\na) Apply Cian's procedure and conclude the relative position between $r$ and $s$.\nb) Apply Aoife's procedure and obtain the same conclusion.\nc) Compare both methods and justify which of the two is more direct and generalisable to prove that a line perpendicular to a plane is skew to any line in that plane that does not pass through the foot of the perpendicular.

An architecture team is designing the structure of an industrial building with a gable roof. The two slopes of the roof form two intersecting planes $\pi_1$ and $\pi_2$ whose intersection is the horizontal ridge edge $r$. The floor of the building is a horizontal plane $\pi_0$ parallel to line $r$.\na) A metal reinforcing beam contained in a line $s$ needs to be added. Deduce what geometric conditions $s$ must satisfy with respect to planes $\pi_1$ and $\pi_2$ to be parallel to both slopes at the same time without being contained in either of them. From there, determine what relative position $s$ will necessarily have with respect to the ridge $r$ and with respect to the floor $\pi_0$.\nb) At the lower ends of each slope, drainage gutters are placed along lines $t_1$ (in $\pi_1$) and $t_2$ (in $\pi_2$), both parallel to the ridge $r$. Prove whether $t_1$ and $t_2$ are parallel to each other and justify whether they determine a plane that turns out to be parallel to the floor $\pi_0$.\nc) Design and describe the geometric conditions of a line $p$ that is perpendicular to the ridge $r$, parallel to the floor plane $\pi_0$, and skew to $r$ at a fixed vertical distance of $1.5$ m without ever intersecting it.