Stand a line straight up, open a book — and measure space with right angles.
Parallelism was about lines and planes that never meet, no matter how far you run them. Perpendicularity is the opposite instinct — it is about lines and planes that meet as squarely as possible. A flagpole rises straight out of the ground; a wall stands straight up from the floor; a book, opened on a table, makes an angle you can actually measure. In this lesson we turn each of those everyday pictures into a precise test. The whole subject rests on one surprisingly strict idea: a line is perpendicular to a plane not when it is square to one line of the plane, but when it is square to two intersecting ones. From that single test flow the dihedral angle, perpendicular planes, the distance from a point to a plane — and a quiet doorway to the vector method waiting in Stage 29.
Start with two lines. In the plane we called two lines perpendicular when they crossed at 90°. In space the definition is gentler about where they cross: two lines are perpendicular when the angle between them is 90° — and remember from 28.5 that the angle between two lines is taken by sliding one parallel to itself until it meets the other. So two lines can be perpendicular even if they are skew and never actually touch: in a room, the floor edge running east–west and a vertical edge on the far wall make a 90° angle, though they are nowhere near each other.
Now the heart of the lesson. When is a whole line perpendicular to a whole plane? Picture a flagpole the instant it is planted. Lay a string on the ground in one direction and check: the pole is square to it. Good — but a pole leaning sideways can still be square to one ground line (the one running directly under its lean). The honest test demands more:
A line l is perpendicular to a plane α exactly when it is perpendicular to two intersecting lines that lie in α. Once that holds, l is automatically perpendicular to every line of α. We write l ⊥ α.
Why must the two lines intersect? Two intersecting lines pin down the whole plane — they are like two different directions you can walk from one spot, and every other direction in the plane is a blend of those two. If l is square to both basic directions, it is square to every blend of them, hence to every line in the plane. Two parallel floor lines, by contrast, only show one direction; a leaning pole can be ⊥ to that single direction and still tip along the other.
Let us watch the test pass and fail on a cube. Build cube ABCD-A₁B₁C₁D₁ with the base on the table. The edge AA₁ rises straight up from corner A. Two base lines through A are AB and AD — and they intersect at A. Compute the dot products: AA₁ · AB = 0 and AA₁ · AD = 0. Both are right angles, the two base lines cross, so AA₁ ⊥ plane ABCD. The slider below also offers a slanted line AD₁: it is ⊥ to AB (its dot is 0) but not ⊥ to AD — so it pierces the floor at a tilt and is not ⊥ the plane.
Perpendicular to one line of the plane is not enough — a leaning line can manage that. You need two intersecting lines of the plane. And the two lines must cross: two parallel lines of the plane only show one direction and never close the loophole.
The test tells us how to recognise a line ⊥ a plane. The property tells us what such lines do. Here is the cleanest one, and it has a homely picture: every flagpole on a level field stands the same way — straight up. They cannot lean toward or away from one another, because each is locked square to the ground.
If two lines are each perpendicular to the same plane, then the two lines are parallel to each other. And through any point in space there is exactly one line perpendicular to a given plane.
On our cube this is visible at a glance: AA₁ and DD₁ are both ⊥ the base ABCD (each is a vertical edge meeting two base lines at right angles), and indeed AA₁ ∥ DD₁ — they are opposite vertical edges of the same face. The same holds for all four vertical edges: four lines, each ⊥ the floor, all parallel. The "exactly one" half explains why a dropped plumb line settles in a single definite direction, and why the height of a pyramid is a single well-defined segment, not a family of them.
Two half-planes sharing a common edge make a dihedral angle, like the two covers of an open book hinged along its spine, or two walls meeting at a room corner, or the open lid of a laptop and its base. But here is the subtlety: how "open" the book is cannot be read from just any two lines in the two faces. Tilt your eyes and a wide-open book can look nearly shut. We need a measurement that does not depend on viewpoint.
Pick a point on the edge. In each face draw a ray from that point perpendicular to the edge. The angle between those two rays is the plane angle of the dihedral angle — and it is the same whatever point on the edge you chose. That number, in [0°, 180°], is the size of the dihedral angle.
The "perpendicular to the edge" clause is the whole game. Both rays lie in the plane that cuts the edge squarely, so the angle you read is the genuine opening of the book — independent of how you look at it. Drag the hinge below from shut toward flat: when the readout hits 90° the two faces are perpendicular planes, which is exactly the bridge to the next section.
The dihedral angle is measured by rays perpendicular to the edge — not by any two lines you happen to draw in the faces. Two slanted lines in the faces can make almost any angle; only the ⊥-to-the-edge rays give the true opening.
Two planes are perpendicular when the dihedral angle they form is 90° — when one stands bolt upright out of the other, the way a wall meets a floor. Checking a dihedral angle ray-by-ray is fussy, so geometry gives a shortcut that needs only a single perpendicular line.
If a plane β contains a line that is perpendicular to a plane α, then β ⊥ α. (One upright line in a plane drags the whole plane upright.)
Run it on the cube: the side face ABB₁A₁ contains the edge AA₁, and we already showed AA₁ ⊥ base ABCD. So the side face is perpendicular to the base — a wall meets the floor at 90°, exactly as a room should. Every side face of an upright box is ⊥ the base for the same reason.
The converse runs backward and is just as useful. It lets you produce a line ⊥ a plane once you already know two planes are perpendicular:
If β ⊥ α and they meet along the line m, then any line of β that is perpendicular to m is perpendicular to the whole plane α.
A pyramid P-ABCD has its apex P directly above the centre O of a square base of side 6, with PO = 4. Since PO ⊥ base (it is ⊥ two base diagonals through O), the segment PO is the pyramid's height. The plane POM through the apex and the midpoint M of a base edge is ⊥ the base, because it contains the upright line PO. The slant height to that edge is PM = √(PO² + OM²) = √(4² + 3²) = √25 = 5 — a clean right-triangle calculation, made legal by perpendicularity.
How far is a point from a plane? Not "along some convenient slope" — distance always means the shortest way, and in space the shortest way to a plane is the perpendicular.
From the point P drop the perpendicular to the plane α; it lands at the foot F. The length of the segment PF is the distance from P to α. Any other segment from P to the plane is the hypotenuse of a right triangle with leg PF, so it is longer.
That last sentence is the whole proof that PF is shortest: if Q is any other point of the plane, triangle PFQ has its right angle at F (because PF ⊥ α makes PF ⊥ FQ), so PQ is the hypotenuse and beats the leg PF every time. The figure shows it on the cube: A₁A is the perpendicular from A₁ to the base, its foot is A, and the slanted A₁B is visibly longer.
The same idea measures other distances built from perpendiculars: the distance between two parallel planes is the length of a perpendicular segment from one to the other (the same everywhere, as we saw in 28.6); the distance from a point to a line is the perpendicular dropped to that line; a pyramid's height is the perpendicular from the apex to the base plane. Perpendicularity is the universal yardstick.
Look back at what just powered every result. To prove a line ⊥ a plane we hunted for two intersecting lines and checked two right angles. To find the dihedral angle we constructed two auxiliary rays ⊥ the edge. Pure geometry works, but it asks you to find the right auxiliary line every time — and sometimes that is genuinely hard.
There is a calmer way coming. In Stage 29 · Spatial Vectors we will plant coordinate axes in the room, so every point becomes a triple (x, y, z) and every line and plane gets a direction written as a vector. Then "perpendicular" stops being a search and becomes a one-line calculation: two directions are perpendicular exactly when their dot product is 0. The very fact we proved by hand here — AA₁ · AB = 0 and AA₁ · AD = 0 ⇒ AA₁ ⊥ plane ABCD — is already a dot-product statement in disguise. The pure-geometry chapter closes; the algebra of space opens. Same truths, settled by arithmetic.
Five facts, each turning a right angle into a tool. Notice how often the word two appears — that is the whole secret of perpendicularity in space.
| Idea | The rule | Why it works / the trap |
|---|---|---|
| Line ⊥ plane (test) | l ⊥ α ⇔ l is ⊥ two intersecting lines of α | Two intersecting lines pin the plane; ⊥ to one line is not enough. |
| Line ⊥ plane (property) | Two lines ⊥ the same plane are parallel | Both stand equally straight; exactly one ⊥ line through a point. |
| Dihedral angle | Rays ⊥ the edge, one in each face → the plane angle | ⊥ the edge, or the angle is wrong; independent of the point chosen. |
| Plane ⊥ plane | β ⊥ α if β holds a line ⊥ α; reverse: ⊥ the common edge ⇒ ⊥ the other plane | One upright line lifts a whole plane upright (dihedral = 90°). |
| Point-to-plane distance | Length of the perpendicular segment to its foot | The shortest one; any slanted segment is a hypotenuse, so longer. |
Next stop: Stage 29 · Spatial Vectors, where coordinates turn "perpendicular" into "dot product = 0," and these hand-built proofs become one-line calculations.
A line l is perpendicular to two lines lying in a plane α, yet l turns out not to be perpendicular to α. What must be true of those two lines?
They must be parallel (or the same line). The test requires two intersecting lines of the plane. Two parallel lines display only one direction, and a slanted line can be ⊥ that single direction while still tilting along the plane's other direction.
In cube ABCD-A₁B₁C₁D₁, name a plane to which the edge DD₁ is perpendicular, and justify it with two intersecting lines.
DD₁ ⊥ base ABCD. Through D the base holds the two intersecting edges DA and DC; the vertical edge DD₁ is ⊥ to each of them (both dot products are 0), and they cross at D, so DD₁ is ⊥ the whole base. (By symmetry it is also ⊥ the top face A₁B₁C₁D₁.)
Two walls of a room and the floor meet at one corner. How many of the three dihedral angles there are right angles, and what does each "wall ⊥ floor" fact rest on?
All three dihedral angles are 90°: wall-to-floor, wall-to-floor, and wall-to-wall. Each wall contains a vertical edge that is ⊥ the floor (it meets two floor lines at the corner squarely), so by the test each wall ⊥ floor; the two walls share that vertical edge, and each is ⊥ the other for the same reason.
A regular pyramid has a square base of side 8 and apex P a height of 3 above the base centre O. Find the slant height — the distance from P to the midpoint M of a base edge.
Because PO ⊥ base, it is ⊥ the segment OM, so triangle POM is right-angled at O. Here OM = ½·8 = 4 (centre to edge midpoint) and PO = 3, so the slant height is PM = √(3² + 4²) = √25 = 5.
Point P sits 12 above a plane α (perpendicular distance). A point Q in α is 5 from the foot F of that perpendicular. How long is the slanted segment PQ, and why is it longer than the distance?
Since PF ⊥ α, it is ⊥ FQ, so triangle PFQ is right-angled at F. Thus PQ = √(12² + 5²) = √169 = 13. It exceeds the distance PF = 12 because PQ is the hypotenuse while PF is a leg — the perpendicular is always the shortest path to the plane.
"To measure the dihedral angle along edge m, I drew any line in each face from a point on m and measured the angle between them." Find the flaw, and state the correct recipe.
The flaw: arbitrary lines in the two faces can make almost any angle, so the number is meaningless. Correct recipe: from one point on the edge m, draw in each face the ray that is perpendicular to m; the angle between those two rays is the plane angle of the dihedral angle, and it does not depend on which point of m you picked.
Six questions to lock it in. Tap the answer you think is right.
This lesson closes the pure-geometry treatment of space in CCSS G-GMD (visualizing relationships between two- and three-dimensional objects) and G-CO (precise definitions of perpendicularity and angle), and it feeds G-MG (modeling with geometry — heights, slant heights, and distances built on perpendicular segments). The single load-bearing idea is the two-intersecting-lines test for a line perpendicular to a plane; the dihedral angle's plane angle and the point-to-plane distance both follow from it. Every figure here is rendered by a real 3-D engine — the right-angle squares and the dihedral arc are honest projections, and the dot products in the readouts are computed, not asserted. It sets up Stage 29 · Spatial Vectors, where the same facts are re-derived with coordinates and the condition "dot product equals zero."