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Math III · Mathematics

Geometric solids · Geometric solids

Parallel and Perpendicular Lines and Planes in Space

Before you start, make sure you know: Space (Math 7)

Before you start, make sure you know: Lines (Math 6)

Look around your classroom. The floor and the ceiling never touch. Two walls meet in a corner. The top edge of the whiteboard and a vertical corner on the opposite wall don't meet and aren't parallel. In space there are more possibilities than on a flat sheet of paper, and each one has a name. By the end of this page you'll be able to name them all and use them to analyze solids.

Two lines in space

On a flat sheet of paper, two lines either cross or are parallel. In space there is a third possibility: the lines can be skew. Skew lines never meet and are not parallel, because there is no single plane that contains both.

Position	Do they meet?	Are they in one plane (coplanar)?
Parallel	No	Yes
Intersecting	Yes, at one point	Yes
Skew	No	No

If two intersecting lines form a $90^\circ$ angle, they are perpendicular.

Rotate the cube and pick edges. For each pair, ask yourself: do they meet? Are they in a common face or plane?

Practice★☆☆

Imagine you have a shoebox (cuboid shape) on your table. If we look at the edges of the lid: a) State two edges that are parallel to each other. b) State two edges that…

Imagine you have a shoebox (cuboid shape) on your table. If we look at the edges of the lid:

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

Practice★☆☆

Two airplanes are flying in the sky. Airplane A flies North at a height of $10.000$ feet. Airplane B flies East at a height of $8.000$ feet. Knowing that their paths do…

Two airplanes are flying in the sky. Airplane A flies North at a height of $10.000$ feet. Airplane B flies East at a height of $8.000$ feet. Knowing that their paths do not intersect and are not parallel:

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

Practice★☆☆

Imagine a soccer goal. Let's analyze its posts as lines: a) What is the relative position of the two vertical posts? b) What is the relative position of the crossbar…

Imagine a soccer goal. Let's analyze its posts as lines:

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

Skew lines and the angle between them

Skew lines don't meet, but they still point in directions that have an angle between them. To measure it, keep one line where it is and slide the other, without turning it, until it crosses the first. The angle between the two lines that now meet is the angle between the skew lines.

Angle between two skew edges of a cube

Use the cube $ABCDEFGH$ from before ($ABCD$ on the bottom, $E$ above $A$, and so on). Find the angle between the face diagonal $AC$ and the top edge $EF$.

Are they skew? $AC$ lies in the bottom plane and $EF$ in the top plane. These planes are parallel, so the lines don't meet. They aren't parallel either, so they are skew.

Slide $EF$ down without turning it. Edge $EF$ is parallel to $AB$, so $AB$ is a parallel copy that passes through $A$, a point of $AC$.

The angle between $AC$ and $AB$ is $\angle CAB$. In the square $ABCD$ the diagonal bisects the right angle, so the angle is $45^\circ$.

Your turn: find the angle between $AC$ and the vertical edge $DH$.

$DH$ is parallel to $AE$, which is perpendicular to the bottom face. So $DH$ is perpendicular to every line of the bottom face, including $AC$. The angle is $90^\circ$: the lines are skew and perpendicular.

Practice★★☆

In the design of a modern cube-shaped shelving unit (as shown in the figure), some of its edges have been numbered. Identify the relative position in space of the…

In the design of a modern cube-shaped shelving unit (as shown in the figure), some of its edges have been numbered. Identify the relative position in space of the following pairs of lines containing these edges:

a) The lines passing through edges $r$ and $s$.
b) The lines passing through edges $s$ and $t$.
c) The lines passing through edges $r$ and $p$.
d) State an edge that is perpendicular to the top face and does not intersect edge $s$.

Practice★★☆

Two lines in space, $a$ and $b$, cross without intersecting. If line $a$ is parallel to a horizontal plane $\pi$ and line $b$ is perpendicular to that same plane $\pi$:…

Two lines in space, $a$ and $b$, cross without intersecting. If line $a$ is parallel to a horizontal plane $\pi$ and line $b$ is perpendicular to that same plane $\pi$:

a) What angle do the directions of lines $a$ and $b$ form?
b) Is it possible to find a plane that contains both lines at the same time? Why?
c) Can we state that the two lines are perpendicular even though they have no points in common?

Practice★★★

In an exhibition hall in the shape of a cuboid with dimensions $10 \times 6 \times 4$ feet, a taut cable is installed joining a vertex on the floor with the midpoint of…

In an exhibition hall in the shape of a cuboid with dimensions $10 \times 6 \times 4$ feet, a taut cable is installed joining a vertex on the floor with the midpoint of the opposite top edge. A second cable joins the center of the ceiling with the midpoint of one of the vertical edges.

a) Determine if there is any point where the cables touch (intersecting lines) or if, on the contrary, they cross in space without touching.
b) Is there any plane that is parallel to both cables simultaneously? Justify your answer based on the directions of the cables.

Lines and planes

A line and a plane can be in three positions:

Contained: every point of the line is in the plane.
Parallel: they never meet.
Intersecting: they meet at exactly one point.

An intersecting line is perpendicular to the plane if it is perpendicular to every line of the plane that passes through the point of intersection. Think of a column standing on the gym floor.

Why two lines? A leaning pole can be perpendicular to one line on the floor and still lean. A second line, crossing the first at the pole's base, forces the pole to stand straight up.

Practice★☆☆

In the school gym, one of the vertical columns (line $r$) meets the floor (plane $\pi$). a) What is the relative position of the column and the floor? b) If we draw any…

In the school gym, one of the vertical columns (line $r$) meets the floor (plane $\pi$).

a) What is the relative position of the column and the floor?
b) If we draw any line on the floor passing through the base of the column, what angle will it form with it?

Practice★☆☆

Look at the following cube and identify: a) An edge that is parallel to the plane of the bottom base. b) An edge that is perpendicular to the plane of the bottom base.

Look at the following cube and identify:

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

Practice★★☆

Imagine that the floor of your room is plane $\pi$ and one of the side walls is plane $\beta$. If you install a fiber optic cable that runs along the corner where the…

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

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

Using perpendicularity to a plane

Perpendicularity to a plane is the main tool for solid-geometry problems.

When a problem has a cone, a pyramid or a sphere, find a plane that contains the pieces you care about. Cutting through the axis usually gives a right triangle.

Distance from the center of a cone's base to a slant side

A cone has radius $3$ in. and height $4$ in. A ray leaves the center $O$ of the base perpendicular to the generator $VA$ (a slant side from the apex $V$ to the rim point $A$). How far is the point $P$ where it meets the generator?

Cut the cone through its axis. Triangle $VOA$ is right-angled at $O$ with legs $VO = 4$ and $OA = 3$.

The hypotenuse is $VA = \sqrt{4^2 + 3^2} = 5$ in.

$OP$ is the altitude to the hypotenuse. Compute the area of the triangle in two ways: $\tfrac{1}{2}(3)(4) = \tfrac{1}{2}(5)(OP)$.

So $OP = \dfrac{12}{5} = 2.4$ in.

Does the ray really reach the lateral surface? Triangles $VPO$ and $VOA$ are similar, so $\dfrac{VP}{VO} = \dfrac{VO}{VA}$ and $VP = \dfrac{4^2}{5} = 3.2$ in. Since $3.2 < 5$, the foot $P$ lies on the segment $VA$, so the ray does hit the lateral surface.

Practice★★★

A point light source is placed at the center of the base of a right cone with radius $R = 5$ in and height $H = 12$ in. The source emits a ray of light that is…

A point light source is placed at the center of the base of a right cone with radius $R = 5$ in and height $H = 12$ in. The source emits a ray of light that is perpendicular to one of the generators of the cone.

a) Is it possible for this ray to pass through the lateral surface of the cone?
b) Calculate the distance from the light source to the point on the generator where the ray would be perpendicular. To do this, visualize the right-angled triangle formed by the axial section of the cone.

Checking with coordinates

Sometimes it is easier to put a solid in a coordinate system and compute. A line has a direction vector $\langle a, b, c \rangle$, and a plane $Ax + By + Cz = D$ has a perpendicular direction $\langle A, B, C \rangle$ (its normal vector).

Coordinate tests

Lines with directions $\mathbf{u}$ and $\mathbf{v}$ are parallel if one is a multiple of the other.
If they are not parallel, they meet only if their parametric equations have a common solution. If there is none, the lines are skew.
They are perpendicular if $\mathbf{u} \cdot \mathbf{v} = 0$, where $\langle a, b, c \rangle \cdot \langle d, e, f \rangle = ad + be + cf$.
The angle $\theta$ between them satisfies $\cos\theta = \dfrac{|\mathbf{u} \cdot \mathbf{v}|}{\|\mathbf{u}\|\,\|\mathbf{v}\|}$.
A line is perpendicular to the plane $Ax + By + Cz = D$ when its direction is a multiple of $\langle A, B, C \rangle$.
Two planes are parallel when their normal vectors are multiples of each other.
A plane that meets the axes at $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$ has equation $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$.

Skew lines in a box

A box has dimensions $4 \times 3 \times 2$, with one corner at the origin: $A(0,0,0)$, $C(4,3,0)$, $H(0,3,2)$, $G(4,3,2)$. Line $p$ is $AC$ and line $q$ is $HG$. Show they are skew and find the angle between them.

Directions: $p$ has $\langle 4, 3, 0 \rangle$ and $q$ has $\langle 4, 0, 0 \rangle$, which is a multiple of $\langle 1, 0, 0 \rangle$. They are not multiples of each other, so the lines are not parallel.

Every point of $p$ has $z = 0$ and every point of $q$ has $z = 2$. They never meet, so they are skew.

$\cos\theta = \dfrac{|4 \cdot 1 + 3 \cdot 0 + 0 \cdot 0|}{5 \cdot 1} = 0.8$, so $\theta = \cos^{-1}(0.8) \approx 36.9^\circ$.

Your turn: is line $p$ perpendicular to a line $r$ with direction $\langle 3, -4, 5 \rangle$?

$\langle 4, 3, 0 \rangle \cdot \langle 3, -4, 5 \rangle = 12 - 12 + 0 = 0$. Yes, they are perpendicular.

A plane and a diagonal

Now take a $6 \times 3 \times 2$ box with a corner at the origin. A plane passes through $(6,0,0)$, $(0,3,0)$ and $(0,0,2)$. Is the main diagonal from $(0,0,0)$ to $(6,3,2)$ perpendicular to it?

The plane is $\dfrac{x}{6} + \dfrac{y}{3} + \dfrac{z}{2} = 1$. Multiply by 6: $x + 2y + 3z = 6$.

The normal vector is $\langle 1, 2, 3 \rangle$. The diagonal has direction $\langle 6, 3, 2 \rangle$, which is not a multiple of $\langle 1, 2, 3 \rangle$.

So the diagonal is not perpendicular to the plane. Now try a cube with edge $4$ and the plane through $(4,0,0)$, $(0,4,0)$ and $(0,0,4)$. Is the diagonal from the origin to $(4,4,4)$ perpendicular to it?

The plane is $x + y + z = 4$ with normal $\langle 1, 1, 1 \rangle$. The diagonal direction is $\langle 4, 4, 4 \rangle = 4\langle 1, 1, 1 \rangle$, so yes, it is perpendicular.

Cutting the corners off a cube

Take a cube of edge $a$ occupying $0 \le x, y, z \le a$. A plane $\pi$ passes through the midpoints of the three edges that meet at the vertex $V(0,0,0)$: $\left(\tfrac{a}{2},0,0\right)$, $\left(0,\tfrac{a}{2},0\right)$ and $\left(0,0,\tfrac{a}{2}\right)$. Is $\pi$ perpendicular to the cube's diagonal from $V$?

The plane meets the axes at $\tfrac{a}{2}$, so $\dfrac{x}{a/2} + \dfrac{y}{a/2} + \dfrac{z}{a/2} = 1$, which is $x + y + z = \tfrac{a}{2}$. Its normal vector is $\langle 1, 1, 1 \rangle$.

The diagonal from $V$ goes to $(a, a, a)$, so its direction is $\langle a, a, a \rangle = a\langle 1, 1, 1 \rangle$. That is a multiple of the normal, so the diagonal is perpendicular to $\pi$.

Your turn: repeat the construction at the opposite vertex $(a,a,a)$. The midpoints of the three edges there are $\left(\tfrac{a}{2},a,a\right)$, $\left(a,\tfrac{a}{2},a\right)$ and $\left(a,a,\tfrac{a}{2}\right)$. Find the equation of this plane and compare it with $\pi$.

Each of the three points satisfies $x + y + z = \tfrac{5a}{2}$. The normal is again $\langle 1, 1, 1 \rangle$ but the constant is different, so the two planes are parallel and distinct. Both are perpendicular to the same diagonal. The same happens for every pair of opposite vertices.

Planes at adjacent vertices are not parallel. At $(a,0,0)$ the plane is $-x + y + z = -\tfrac{a}{2}$, with normal $\langle -1, 1, 1 \rangle$, not a multiple of $\langle 1,1,1 \rangle$.

What solid is left after cutting all eight corners? Each cut removes a small corner pyramid whose three edges at the vertex are $\tfrac{a}{2}$ long. Each original square face becomes a smaller square whose vertices are the midpoints of its four edges (6 squares). Each cut adds an equilateral triangle of side $\tfrac{a\sqrt{2}}{2}$ (8 triangles). The result has 14 faces, 12 vertices (the midpoints of the cube's edges) and all edges of equal length. It is called a cuboctahedron (the problem calls it a truncated cube). It is not an octahedron, which has only 8 faces. Its 8 triangular faces form 4 pairs of parallel faces, one pair for each diagonal of the cube.

Practice★★★

Consider a cube of edge $a$. A plane $\pi$ is drawn passing through the midpoints of three edges that meet at the same vertex. a) Justify whether the plane $\pi$ is…

Consider a cube of edge $a$. A plane $\pi$ is drawn passing through the midpoints of three edges that meet at the same vertex.

a) Justify whether the plane $\pi$ is perpendicular to the diagonal of the cube starting from that vertex.
b) If we repeat the process at the eight vertices of the cube, the eight planes cut the cube to form a new solid (a truncated cube). Analyze the relative position of the planes that have cut opposite vertices of the original cube.

Two planes

Two planes are either parallel (floor and ceiling) or intersecting. Intersecting planes always meet along a whole line, like two adjacent walls that meet at a corner.

Why is fact 1 true? The lines $r$ and $s$ lie in the same plane $\pi_3$, and they can't meet because they are in two parallel planes. Coplanar and never meeting means parallel.

Practice★☆☆

Look at the walls of your room. a) What is the relative position of two opposite walls? b) What is the line formed when two adjacent walls intersect called?

Look at the walls of your room.

a) What is the relative position of two opposite walls?
b) What is the line formed when two adjacent walls intersect called?

Practice★☆☆

Draw three planes ($\pi_1, \pi_2, \pi_3$) in your notebook that meet the following conditions: 1. $\pi_1$ and $\pi_2$ are parallel to each other. 2. $\pi_3$ is…

Draw three planes ($\pi_1, \pi_2, \pi_3$) in your notebook that meet the following conditions:

$\pi_1$ and $\pi_2$ are parallel to each other.
$\pi_3$ is perpendicular to $\pi_1$.

What is the relative position between $\pi_3$ and $\pi_2$?

Practice★★☆

Given three planes in space $\pi_1, \pi_2$ and $\pi_3$, we know that $\pi_1$ is parallel to $\pi_2$ and that $\pi_3$ intersects $\pi_1$ forming a line $r$. a) What will…

Given three planes in space $\pi_1, \pi_2$ and $\pi_3$, we know that $\pi_1$ is parallel to $\pi_2$ and that $\pi_3$ intersects $\pi_1$ forming a line $r$.

a) What will be the relative position of $\pi_3$ with respect to $\pi_2$?
b) If we call $s$ the line of intersection between $\pi_3$ and $\pi_2$, what relationship of parallelism or perpendicularity do lines $r$ and $s$ have?
c) If a line $t$ is perpendicular to $\pi_1$, will it also be perpendicular to $\pi_2$? Why?

Dihedral and trihedral angles

An open book shows the idea of a dihedral angle: the region between two half-planes that share a common line, the edge. Each half-plane is a face of the dihedral angle.

To measure a dihedral angle, cut it with a plane perpendicular to the edge. In each face you get a ray perpendicular to the edge, and the angle between those rays is the measure of the dihedral angle.

A closed book measures $0^\circ$. A book lying flat measures $180^\circ$.
When the dihedral angle is $90^\circ$, the planes are perpendicular.

Three planes that meet at one point, like the corner of a room, form a trihedral angle. It has three face angles (the angle between two edges, inside each face) and three dihedral angles (between each pair of faces).

The corner of a room

The floor and two walls meet at a corner $O$. Three edges leave $O$: the baseboard of wall 1, the baseboard of wall 2, and the vertical corner line where the walls meet. What do the three face angles and the three dihedral angles measure, and what do the dihedral angles add up to?

Each face (the floor, wall 1, wall 2) is bounded at $O$ by two of the edges, and those edges are perpendicular. So each of the three face angles measures $90^\circ$.

Look at the dihedral angle along the vertical edge, between the two walls. Cut with the floor, which is perpendicular to that edge. In the floor, the two baseboards are the rays perpendicular to the edge, and they form $90^\circ$. So this dihedral angle is $90^\circ$.

The same argument works for the other two edges: each edge is perpendicular to the plane of the other two. (Equivalently, each wall is perpendicular to the floor, and the walls are perpendicular to each other.) All three dihedral angles are $90^\circ$.

Your turn: what is the sum of the three dihedral angles?

$90^\circ + 90^\circ + 90^\circ = 270^\circ$.

Notice what an edge being perpendicular to the plane of the other two gives you: a line parallel to one edge is perpendicular to the plane of the other two.

A revolving door is another good picture. From above, its vertical axis looks like a single point and the three leaves look like three segments from that point.

The regions between consecutive leaves are dihedral angles, and the axis is the common edge of all of them. With three leaves, each leaf is a face of two of the dihedral angles.

Practice★☆☆

When you open a math book and leave it resting on the table, the two pages form a dihedral angle. a) If we open the book completely until it lies flat on the table, what…

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

a) If we open the book completely until it lies flat on the table, what does this dihedral angle measure?
b) If the book is half-open and the pages form an angle of $90^\circ$, what are the planes of the two pages like with respect to each other?

Practice★★☆

A revolving door in a shopping center has three identical glass leaves that intersect at a central vertical axis. a) What name is given to each of the three regions of…

A revolving door in a shopping center has three identical glass leaves that intersect at a central vertical axis.

a) What name is given to each of the three regions of space bounded by two consecutive leaves?
b) If the dihedral angle between the first and second leaf is $120^\circ$, and the three leaves are distributed symmetrically, what do the other two dihedral angles measure?
c) If a fly lands exactly on the central axis, how many faces of the dihedral angles does it belong to?

Practice★★☆

Look at the corner of a room where two walls and the floor meet. This set forms a trihedral angle at the vertex of the bottom corner. a) What do the three rectilinear…

Look at the corner of a room where two walls and the floor meet. This set forms a trihedral angle at the vertex of the bottom corner.

a) What do the three rectilinear angles that define the faces of this trihedron measure?
b) What is the sum of the measures of the three dihedral angles formed between the walls and the floor?
c) If a fly flies following a path parallel to the baseboard line of one wall, is that path also parallel to the plane of the other wall? Explain why.

Prisms and pyramids

Now we put everything to work on solids.

In a right prism, the lateral edges are perpendicular to the bases. That means:

The two bases lie in parallel planes.
Every lateral face is a rectangle perpendicular to the bases.
A vertical plane (one that contains a vertical line) is perpendicular to the base plane. If it cuts through the prism, the section is a rectangle: its width is the chord the plane cuts in the base, and its height is the height of the prism.
A plane perpendicular to all the lateral faces must be horizontal (parallel to the bases). It cuts the prism in a polygon congruent to the base. So the only plane through a line of the bottom base that is perpendicular to the lateral faces is the base plane itself, and its section is the base polygon.
In a regular polygon with an even number of sides, opposite sides are parallel. A regular hexagon has 3 pairs, so a regular hexagonal prism has 3 pairs of parallel lateral faces.

In a regular pyramid, the apex is directly above the center of the base. The line from the apex to the center, the axis, is perpendicular to the base plane, so it is perpendicular to every line in the base.

In a square pyramid (some problems say quadrangular pyramid) with apex $V$ and base $ABCD$:

A lateral edge such as $VA$ touches the base plane only at $A$, so it can't be parallel to any line of the base. If it doesn't share a vertex with a base edge, the two lines are skew.
The planes of two opposite lateral faces intersect along a line through the apex that is parallel to the base edges.

Planes of symmetry can also be found with perpendicularity. A plane of symmetry perpendicular to the axis of a prism must swap the two bases, so there is exactly one such plane: the one halfway between the bases.

Practice★★☆

Consider a right prism with a regular hexagonal base. Analyze the relationship between its elements: a) How many lateral faces are perpendicular to the bottom base? b)…

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

a) How many lateral faces are perpendicular to the bottom base?
b) How many pairs of lateral faces are parallel to each other?
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?

Practice★★☆

In a regular quadrangular pyramid (square base), the top vertex is called the apex. a) Justify why the axis of the pyramid (the line joining the apex to the center of…

In a regular quadrangular pyramid (square base), the top vertex is called the apex.

a) Justify why the axis of the pyramid (the line joining the apex to the center of the base) is perpendicular to any diagonal of the square base.
b) Is there any lateral edge that is parallel to a base edge? Reason your answer.
c) What are the planes containing two opposite lateral faces like? Are they intersecting or parallel?

Practice★★★

Analyze a regular hexagonal prism. Consider a line $r$ passing through two opposite vertices of the bottom base and a line $s$ passing through two opposite vertices of…

Analyze a regular hexagonal prism. Consider a line $r$ passing through two opposite vertices of the bottom base and a line $s$ passing through two opposite vertices of the top base, but in such a way that they are not parallel to each other.

a) Are lines $r$ and $s$ coplanar?
b) Find the number of planes of symmetry of the prism that are perpendicular to the line joining the centers of both bases.
c) If we draw a plane containing $r$ and perpendicular to the lateral faces it crosses, what geometric figure is obtained in the cross-section?

Spheres and regular solids

A plane and a sphere of radius $R$ with center $O$ can be in three positions, depending on the distance $d$ from $O$ to the plane:

$d > R$: they don't meet.
$d = R$: the plane is tangent and touches the sphere at one point, perpendicular to the radius there.
$d < R$: the intersection is a circle.

The center of that circle is the foot of the perpendicular from $O$ to the plane. By the Pythagorean Theorem, its radius is $r = \sqrt{R^2 - d^2}$. When $d = 0$, the circle is a great circle with radius $R$.

A plane that passes through $O$ and is perpendicular to another plane $\pi$ contains the perpendicular from $O$ to $\pi$. In particular, it passes through the center of the circle that $\pi$ cuts on the sphere. So the line where the two planes meet is a diameter of that circle, and it is a secant line of the sphere.

Two regular solids come up often.

Regular tetrahedron: 4 equilateral triangles. Two edges that share a vertex form $60^\circ$. Edges that share no vertex are called opposite edges. The four altitudes meet at one point, the centroid.
Regular octahedron: 8 equilateral triangles. Opposite faces are parallel.

Opposite edges of a regular tetrahedron

Regular tetrahedron $ABCD$. Show that the opposite edges $AB$ and $CD$ are perpendicular.

Let $M$ be the midpoint of $AB$. Triangles $ABC$ and $ABD$ are equilateral, so their medians $CM$ and $DM$ are also altitudes: $CM \perp AB$ and $DM \perp AB$.

$CM$ and $DM$ intersect at $M$, so they span the plane $CDM$. Then $AB$ is perpendicular to two intersecting lines of that plane, so $AB \perp$ plane $CDM$.

The edge $CD$ lies in plane $CDM$, so $AB \perp CD$. The two edges are skew and perpendicular.

Your turn: how many pairs of perpendicular edges does the tetrahedron have?

It has 6 edges, and each one has exactly one opposite edge, so there are $6 \div 2 = 3$ pairs. The same argument works for each pair. Edges that share a vertex form $60^\circ$, not $90^\circ$.

Now set the tetrahedron on a table with base $BCD$. The four altitudes pass through the centroid $G$. A plane through $G$ parallel to the base is perpendicular to the altitude from the top vertex $A$, because that altitude is perpendicular to the base. The other three altitudes also pass through $G$ but are not perpendicular to the plane: if one were, it would be parallel to the altitude from $A$ (two lines perpendicular to the same plane are parallel), yet the two lines are different and meet at $G$.

A plane that only touches an octahedron

Use the octahedron with vertices $(\pm 1, 0, 0)$, $(0, \pm 1, 0)$, $(0, 0, \pm 1)$. The face with vertices $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ has center $\left(\tfrac13, \tfrac13, \tfrac13\right)$, and the opposite face has center $\left(-\tfrac13, -\tfrac13, -\tfrac13\right)$. Line $r$ joins them, so it has direction $\langle 1, 1, 1 \rangle$. Let $s$ be the line of the edge from $(1,0,0)$ to $(0,-1,0)$, which is in neither of those two faces. How are $r$ and $s$ related? What happens with a plane through $s$ parallel to $r$?

The direction of $s$ is $\langle 1, 1, 0 \rangle$, not a multiple of $\langle 1, 1, 1 \rangle$, so the lines are not parallel.

Points of $s$ are $(1 + t, t, 0)$ and points of $r$ are $(u, u, u)$. The $z$-coordinates force $u = 0$, then $1 + t = 0$ and $t = 0$ at the same time, which is impossible. The lines never meet: they are skew.

The vector $\langle 1, -1, 0 \rangle$ is perpendicular to both directions ($1 - 1 = 0$ and $1 - 1 + 0 = 0$). The plane $x - y = 1$ contains $s$ (since $(1+t) - t = 1$) and is parallel to $r$.

Every point of the octahedron satisfies $x - y \le |x| + |y| \le |x| + |y| + |z| \le 1$. Equality $x - y = 1$ only happens for $z = 0$, $x \ge 0$, $y \le 0$ and $|x| + |y| = 1$, which are exactly the points of the edge. The plane touches the octahedron only along the edge $s$.

Practice★★★

There is a sphere of radius $13$ in. A plane $\pi_1$ cuts the sphere at a distance of $5$ in from its center, generating a circle $C_1$. A second plane $\pi_2$,…

There is a sphere of radius $13$ in. A plane $\pi_1$ cuts the sphere at a distance of $5$ in from its center, generating a circle $C_1$. A second plane $\pi_2$, perpendicular to the first, also cuts the sphere passing through its center.

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

Practice★★★

Imagine a regular tetrahedron resting on a table (horizontal plane). a) How many of its edges are perpendicular to each other? Justify your answer by analyzing the…

Imagine a regular tetrahedron resting on a table (horizontal plane).

a) How many of its edges are perpendicular to each other? Justify your answer by analyzing the angles formed by the edges that do not share a vertex.
b) If we draw a plane parallel to the base passing through the centroid of the tetrahedron, what is the relative position of this plane with respect to the four heights of the solid?

Practice★★★

In an avant-garde architectural design, a building has been designed in the shape of a regular octahedron. If we draw a straight line $r$ that joins the centers of two…

In an avant-garde architectural design, a building has been designed in the shape of a regular octahedron. If we draw a straight line $r$ that joins the centers of two opposite faces and another line $s$ that contains one of the edges not belonging to either of those two faces, determine with reasoning:

a) What is the relative position between line $r$ and line $s$?
b) If we introduce a plane $\pi$ that contains the edge $s$ and is parallel to line $r$, what type of intersection would this plane have with the octahedron?

Back to the start: In your classroom, the floor and ceiling are parallel planes, two walls are intersecting planes that meet in a corner line, and the top edge of the whiteboard and a vertical corner on the far wall are skew lines.

Summary

Summary

Two lines in space: parallel, intersecting or skew (no common point, not coplanar). Their angle is found by sliding one line parallel to itself until the lines meet.
A line and a plane: contained, parallel or intersecting. A line is perpendicular to a plane if it is perpendicular to two intersecting lines of the plane, and then to every line in it.
Two planes: parallel or intersecting along a line. A plane cutting two parallel planes gives parallel lines. Planes are perpendicular when one contains a line perpendicular to the other.
A dihedral angle is measured with lines perpendicular to the edge. Perpendicular planes form $90^\circ$. At the corner of a room, the three face angles and the three dihedral angles are all $90^\circ$, so the dihedral angles add up to $270^\circ$.
In solids, use axes and heights (perpendicular to the base) and cut by a plane to get a right triangle. In coordinates, perpendicular directions have dot product $0$ and parallel planes have normal vectors that are multiples of each other.

Let's practice! 8 medium exercises in this section

★★☆ In the design of a modern cube-shaped shelving unit (as shown in the figure), some of its…★★☆ Imagine that the floor of your room is plane $\pi$ and one of the side walls is plane…★★☆ Two lines in space, $a$ and $b$, cross without intersecting. If line $a$ is parallel to a…

Let's practice! 7 hard exercises in this section

★★★ In an avant-garde architectural design, a building has been designed in the shape of a…★★★ Consider a cube of edge $a$. A plane $\pi$ is drawn passing through the midpoints of…

See all the exercises in this section