mathematics

Point–line–plane postulate

Point–line–plane postulate is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Point–line–plane postulate rather than just read about it. In short: In geometry, the point–line–plane postulate is a collection of assumptions (axioms) that can be used in a set of postulates for Euclidean geometry in two (plane geometry), three (solid geometry) or more dimensions. Assumptions The following are the assumptions of the point-line-plane postulate: Unique line assumption.

Key takeaways

Point–line–plane postulate belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Point–line–plane postulate to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Point–line–plane postulate from memory before moving on to harder problems.

Reference excerpt

In geometry, the point–line–plane postulate is a collection of assumptions (axioms) that can be used in a set of postulates for Euclidean geometry in two (plane geometry), three (solid geometry) or more dimensions.

Assumptions The following are the assumptions of the point-line-plane postulate:

Unique line assumption. There is exactly one line passing through two distinct points. Number line assumption. Every line is a set of points which can be put into a one-to-one correspondence with the real numbers. Any point can correspond with 0 (zero) and any other point can correspond with 1 (one). Dimension assumption. Given a line in a plane, there exists at least one point in the plane that is not on the line. Given a plane in space, there exists at least one point in space that is not in the plane. Flat plane assumption. If two points lie in a plane, the line containing them lies in the plane. Unique plane assumption. Through three non-collinear points, there is exactly one plane. Intersecting planes assumption. If two different planes have a point in common, then their intersection is a line. The first three assumptions of the postulate, as given above, are used in the axiomatic formulation of the Euclidean plane in the secondary school geometry curriculum of the University of Chicago School Mathematics Project (UCSMP).

History

The axiomatic foundation of Euclidean geometry can be dated back to the books known as Euclid's Elements (circa 300 B.C.). These five initial axioms (called postulates by the ancient Greeks) are not sufficient to establish Euclidean geometry. Many mathematicians have produced complete sets of axioms which do establish Euclidean geometry. One of the most notable of these is due to Hilbert who created a system in the same style as Euclid. Unfortunately, Hilbert's system requires 21 axioms. Other systems have used fewer (but different) axioms. The most appealing of these, from the viewpoint of having the fewest axioms, is due to G.D. Birkhoff (1932) which has only four axioms. These four are: the Unique line assumption (which was called the Point-Line Postulate by Birkhoff), the Number line assumption, the Protractor postulate (to permit the measurement of angles) and an axiom that is equivalent to Playfair's axiom (or the parallel postulate). For pedagogical reasons, a short list of axioms is not desirable and starting with the New math curricula of the 1960s, the number of axioms found in high school level textbooks has increased to levels that even exceed Hilbert's system.

References

External links The Point-Line-Plane Postulate as described in the Oracle Education Foundation's "ThinkQuest" online description of basic geometry postulates and theorems. The Point-Line-Plane Postulate as described in Professor Calkins' (Andrews University) online listing of basic geometry concepts.

Worked examples

Example 1 — a first encounter with Point–line–plane postulate

Start with the simplest possible case. Write down what Point–line–plane postulate claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Point–line–plane postulate before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Point–line–plane postulate ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Point–line–plane postulate

In research: Point–line–plane postulate appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Point–line–plane postulate in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Point–line–plane postulate is common in secondary-school and first-year university syllabi. It links to neighbouring topics Foundations of geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Point–line–plane postulate outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Point–line–plane postulate in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Point–line–plane postulate means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Point–line–plane postulate out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Point–line–plane postulate in simple terms?

Why does Point–line–plane postulate matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Point–line–plane postulate?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Point–line–plane postulate.