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Hannibal lecter

Member

From: Iraq

Registered: 2016-02-11

Posts: 402

David Halber III.4 axiom

Definition By an angle is meant a point (called the vertex of the angle) and two rays (called the sides of the angle) emanating from the point. If the vertex of the angle is point A and if B and C are any two points other than A on the two sides of the angle, we speak of the angle /BAC or LCAB or simply of angle ZA.

III-4. If <BAC is an angle whose sides do not lie on the same line and if A'B is a ray emanating from A', then there is one and only one ray A'C' on a given side of line A'B', such that <B'A'C' <BAC. In short, a given angle can be laid off on a given side of a given ray in one and only one way. Every angle is congruent to itself.

The Axiom's Goal: It guarantees that if you have an angle in one place, you can perfectly recreate it in another place. It proves that measurement is portable.

I understand only the goal when I read it.
but the reast the axiom itself I can't understand it

that's another modern explanation:

Hilbert’s Axiom III, 4 (The Angle Construction Axiom)
    "Let angle(h, k) be an angle in a plane alpha and let a be a straight line in a plane alpha'. Let a definite side of a in alpha' be assigned. Let h' be a ray of the line a emanating from a point O'. Then there exists a unique ray k' such that the angle angle(h', k') is congruent to the angle angle(h, k)

I can't understand is there a step by step explanation please and a diagrams examples

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

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Registered: 2010-06-20

Posts: 10,823

Re: David Halber III.4 axiom

Axioms are the starting statements for any mathematical model.  They are assumed to be true. From them, other statements can be proved. These are called theorems.

Angle BAC is defined as made by the rays AB and AC.

If A' and B' are two new points then the next axiom says that there is only one point C' so that angle ABC = angle A'B'C'.

It may not seem much at an early stage, but later in the development of theorems it may be important that C' is unique. This axiom also introduces that idea that two angles may be equal.

Bob

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Children are not defined by school ...........The Fonz
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