What are the postulates that can be used to derive geometry?

What are the various sets of postulates that can used to derive Euclidean geometry?

It might be nice to have several different approaches together for comparison purposes and for ready reference.

It might also be interesting to include an axiomatization (or two) of elliptical geometry.

Solution 1:

Axioms used in David Hilbert's The Foundations of Geometry (1899), as translated by E. J. Townsend in 1902

The undefined terms in Hilbert's axiomatization are point, line, plane, lies upon, between and congruent.

Group I: Axioms of Connection.

The axioms of this group establish a connection between the concepts indicated above; namely, points, straight lines, and planes. These axioms are as follows:

1. Two distinct points $A$ and $B$ always completely determine a straight line $a$. We write $AB = a$ or $BA = a$.

2. Any two distinct points of a straight line completely determine that line; that is, if $AB = a$ and $AC=a$, where $B \neq C$, then is also $BC=a$.

3. Three points $A$, $B$, $C$ not situated in the same straight line always completely determine a plane $\alpha$. We write $ABC=a$.

4. Any three points $A$, $B$, $C$ of a plane $\alpha$, which do not lie in the same straight line, completely determine that plane.

5. If two points $A$, $B$ of a straight line $a$ lie in a plane $\alpha$, then every point of $a$ lies in $a$.

6. If two planes $\alpha$, $\beta$ have a point $A$ in common, then they have at least a second point $B$ in common.

7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.

Group II: Axioms of Order.

The axioms of this group define the idea expressed by the word between, and make possible, upon the basis of this idea, an order of sequence of the points upon a straight line, in a plane, and in space. The points of a straight line have a certain relation to one another which the word between serves to describe. The axioms of this group are as follows:

1. If $A$, $B$, $C$ are points of a straight line and $B$ lies between $A$ and $C$, then $B$ lies also between $C$ and $A$.

2. If $A$ and $C$ are two points of a straight line, then there exists at least one point $B$ lying between $A$ and $C$ and at least one point $D$ so situated that $C$ lies between $A$ and $D$.

3. Of any three points situated on a straight line, there is always one and only one which lies between the other two.

4. Any four points $A$, $B$, $C$, $D$ of a straight line can always be so arranged that $B$ shall lie between $A$ and $C$ and also between $A$ and $D$, and, furthermore, that $C$ shall lie between $A$ and $D$ and also between $B$ and $D$

5. Let $A$, $B$, $C$ be three points not lying in the same straight line and let $a$ be a straight line lying in the plane $ABC$ and not passing through any of the points $A$, $B$, $C$. Then, if the straight line $a$ passes through a point of the segment $AB$, it will also pass through either a point of the segment $BC$ or a point of the segment $AC$.

Group III: Axiom of Parallels (The Axiom of Euclid).

The introduction of this axiom simplifies greatly the fundamental principles of geometry and facilitates in no small degree its development. This axiom may be expressed as follows:

1. In a plane $\alpha$ there can be drawn through any point $A$, lying outside of a straight line $a$, one and only one straight line which does not intersect the line $a$. This straight line is called the parallel to $a$ through the given point $A$.

Group IV. Axioms of Congruence.

The axioms of this group define the idea of congruence or displacement.

Segments stand in a certain relation to one another which is described by the word congruent.

1. If $A$, $B$ are two points on a straight line $a$, and if $A'$ is a point upon the same or another straight line $a'$, then, upon a given side of $A'$ on the straight line $a'$, we can always find one and only one point $B'$ so that the segment $AB$ (or $BA$) is congruent to the segment $A'B'$. We indicate this relation by writing $AB\equiv A'B'$. Every segment is congruent to itself; that is, we always have $AB\equiv AB$.

2. If a segment $AB$ is congruent to the segment $A'B'$ and also to the segment $A''B''$, then the segment $A'B'$ is congruent to the segment $A''B''$; that is, if $AB \equiv A'B$ and $AB \equiv A''B''$, then $A'B' \equiv A''B''$.

3. Let $AB$ and $BC$ be two segments of a straight line $a$ which have no points in common aside from the point $B$, and, furthermore, let $A'B'$ and $B'C'$ be two segments of the same or of another straight line $a'$ having, likewise, no point other than $B'$ in common. Then, if $AB \equiv A'B'$ and $BC \equiv B'C'$, we have $AC \equiv A'C'$.

4. Let an angle $(h,k)$ be given in the plane $\alpha$ and let a straight line $a'$ be given in a plane $\alpha'$. Suppose also that, in the plane $\alpha$, a definite side of the straight line $a'$ be assigned. Denote by $h'$ a half-ray of the straight line $a'$ emanating from a point $O'$ of this line. Then in the plane $\alpha'$ there is one and only one half-ray $k'$ such that the angle $(h,k)$, or $(k,h)$, is congruent to the angle $(h',k')$ and at the same time all interior points of the angle $(h',k')$ lie upon the given side of $a'$. We express this relation by means of the notation $\angle (h,k) \equiv \angle (h',k')$ Every angle is congruent to itself; that is, $\angle (h,k) \equiv \angle (h,k)$ or $\angle (h,k) \equiv \angle (k,h)$.

5. f the angle $(h,k)$ is congruent to the angle $(h',k')$ and to the angle $(h'',k'')$, then the angle $(h',k')$ is congruent to the angle $(h'',k'')$; that is to say, if $\angle (h, k) \equiv \angle (h', k')$ and $\angle (h, k) \equiv \angle (h'',k'')$, then $\angle (h',k') \equiv \angle (h'',k'')$.

6. If, in the two triangles $ABC$ and $A'B'C'$ the congruences $AB \equiv A'B', \: AC \equiv A'C', \: \angle BAC \equiv \angle B'A'C'$ hold, then the congruences $\angle ABC \equiv \angle A'B'C' \:\mbox{and}\; \angle ACB \equiv \angle A'C'B'$ also hold.

Some definitions relevant to the axioms of congruence:

Let $\alpha$ be any arbitrary plane and $h$, $k$ any two distinct half-rays lying in $\alpha$ and emanating from the point $O$ so as to form a part of two different straight lines. We call the system formed by these two half-rays $h$, $k$ an angle and represent it by the symbol $\angle(h, k)$ or $\angle(k, h)$. From axioms II, 1--5, it follows readily that the half-rays $h$ and $k$, taken together with the point $O$, divide the remaining points of the plane a into two regions having the following property: If $A$ is a point of one region and $B$ a point of the other, then every broken line joining $A$ and $B$ either passes through $O$ or has a point in common with one of the half-rays $h$, $k$. If, however, $A$, $A'$ both lie within the same region, then it is always possible to join these two points by a broken line which neither passes through $O$ nor has a point in common with either of the half-rays $h$, $k$. One of these two regions is distinguished from the other in that the segment joining any two points of this region lies entirely within the region. The region so characterised is called the interior of the angle $(h,k)$. To distinguish the other region from this, we call it the exterior of the angle $(h,k)$. The half rays $h$ and $k$ are called the sides of the angle, and the point $O$ is called the vertex of the angle.

Group V. Axiom of Continuity (The Axiom of Archimedes).

This axiom makes possible the introduction into geometry of the idea of continuity. In order to state this axiom, we must first establish a convention concerning the equality of two segments. For this purpose, we can either base our idea of equality upon the axioms relating to the congruence of segments and define as equal the correspondingly congruent segments, or upon the basis of groups I and II, we may determine how, by suitable constructions, a segment is to be laid off from a point of a given straight line so that a new, definite segment is obtained equal to it. In conformity with such a convention, the axiom of Archimedes may be stated as follows:

1. Let $A_1$ be any point upon a straight line between the arbitrarily chosen points $A$ and $B$. Take the points $A_2$, $A_3$, $A_4,\ldots$ so that $A_1$ lies between $A$ and $A_2$, $A_2$ between $A_1$ and $A_3$, $A_3$ between $A_2$ and $A_4$ etc. Moreover, let the segments $ A A_1, \; A_1 A_2, \; A_2 A_3, \; A_3 A_4, \;\ldots$ be equal to one another. Then, among this series of points, there always exists a certain point $A_n$ such that $B$ lies between $A$ and $A_n$.

Solution 2:

Axioms used in Afred Tarski's and Steven Givant's Tarski's System of Geometry (1991).
Link: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012

(This is basically a summary of the first part of Tarski's and Givant's paper.)

Alfred Tarski proved around 1930 that his system of geometry is complete, decidable and that there is a constructive concistency proof for the theory.

History
Through out the years, the set of Tarski's axioms have changed. At first, there were 20 (1-4, 5$_1$, 6, 7$_2$, 8$^{(2)}$, 9$_1^{(2)}$, 10, 12-21), as well as all instances of axiom schema 11. This made it an axiom system for elementary 2-dimensional geometry.

It turned out that axiom 13 and 19 was derivable from the remaining axioms, and they where omitted. Axiom 20 was replaced by a more precise variant, 20$_1$.

In 1956-57 a substantial simplification of the axiom set was obtained by the efforts of Eva Kallin, Scott Taylor and Tarski. The axioms 5$_1$, 7$_2$, 9$_1^{(2)}$ and 10 was respectively replaced by 5, 7$_1$, 9$^{(2)}$ and 10$_1$. In the modified axiom set, the axioms 12, 14, 16, 17, 20$_1$ and 21 are shown to be derivable from the remaining ones. Then we have only the twelve axioms 1-6, 7$_1$, 8$^{(2)}$, 9$^{(2)}$, 10$_1$, 15 and 18, and all instances of the axiom schema 11.

In Tarski's course on the foundations of geometry at the University of California (1956-57), he pointed out that by replacing the axiom schema (11) with the second-order sentence 11 for the full (non-elementary) 2-dimensional Euclidean geometry. It was also pointed out that the axioms 8$^{(2)}$, 9$_1^{(2)}$ could be changed in order to obtain an axiom set for $n$-dimensional geometry.

Gupta then showed that axioms 6 and 18 can be derived from the remaining axioms. So what people now usually mean when they talk about Tarski's axioms of geometry, is the axioms 1-5, 7$_1$, 8$^{(n)}$, 9$^{(n)}$ ($n=2,3,...$), 10$_1$, 15 and 11 (either the first-order axiom schema or the second-order single sentence). These are the ones marked with *.

Definitions

• The ternary relation $B(abc)$ expresses that betweenness holds between $a,b$ and $c$, which intuitively means that $b$ lies on the line segment joining $a$ and $d$.
• The quarternary relation $ab \equiv cd$ expresses that the relation of equidistance (or congruence of segments) holds among the points $a,b,c,d$, which intuitively means that the distance from $a$ to $b$ is the same as the distance from $c$ to $d$.

The only primitive geometrical object in Tarski's axiom system, are points. All variables $a,b,c,...$ are assumed to range over points.

*Axiom 1 Reflexivity Axiom for Equidistance. $$ab\equiv ba$$ The distance between the points $a$ and $b$ is the same as the distance between $b$ and $a$.

*Axiom 2 Transitivity Axiom for Equidistance. $$ab \equiv pq \land ab \equiv rs \to pq \equiv rs$$ If the distance between the points $a$ and $b$ is the same as the distance between $p$ and $q$ and also the same as the distance between $r$ and $s$, then the distance between $p$ and $q$ is the same as the distance between $r$ and $s$.

*Axiom 3 Identity Axiom for Equidistance. $$ab\equiv cc\to a = b$$ If the distance between $a$ and $b$ is the same as the distance between $c$ and $c$, then $a$ and $b$ are the same point.

*Axiom 4 Axiom of Segment Construction. $$\exists x (B(qax)\land ax \equiv bc)$$ There is a point $x$ such that $a$ lies between $q$ and $x$ and the distance between $a$ and $x$ is equal to the distance between $b$ and $c$.

Intuitively this means that, given any line segment $bc$, it is possible to construct a line segment congruent to it (of equal length), starting on any point $a$ and going in the direction of ray that is determined by $a$ and the endpoint $q$ on the ray.

*Axiom 5 Five-Segment Axiom. $$[a \neq b \land B(abc)\land B(a'b'c')\land ab \equiv a'b'\land bc \equiv b'c' \land ad \equiv a'd' \land bd \equiv b'd']\to cd \equiv c'd'$$

If

• $a$ and $b$ is not the same point and
• $b$ lies between $a$ and $c$ and
• $b'$ lies between $a'$ and $c'$ and
• the distance between $a$ and $b$ is the same as the distance between $a'$ and $b'$ and
• the distance between $b$ and $c$ is the same as the distance between $b'$ and $c'$ and
• the distance between $a$ and $d$ is the same as the distance between $a'$ and $d'$ and
• the distance between $b$ and $d$ is the same as the distance between $b'$ and $d'$,

then the distance between $c$ and $d$ is the same as the distance between $c'$ and $d'$.

"The Five-Segment Axiom asserts (in the non-degenerate case) that, given two triangles $\triangle acd$ and $\triangle a'c'd'$, and given interior points $b$ and $b'$ of the sides $ac$ and $a'c'$, from the congruences of certain corresponding pairs of line segments, one can conclude the congruence of another pair of corresponding line segments. Thus, this axiom is similar in character to the well-known theorems of Euclidean geometry that allow one to conclude, from hypotheses about the congruence of certain corresponding sides and angles in two triangles, the congruence of other corresponding sides and angles."

(There is another variant of this axiom at page 179 in the paper i linked to.)

Axiom 6 Identity Axiom for Betweenness. $$B(aba) \to a = b$$

Axiom 7 First (or Inner) form of the Pasch Axiom. $$B(apc) \land B(bqc) \to \exists x [B(pxb)\land B(qxa)]$$

*Axiom 7$_1$ Second (or Outer) form of Pasch Axiom. $$B(apc) \land B(qcb) \to \exists x [B(axq)\land B(bpx)]$$

If $p$ lies between $a$ and $c$, and $c$ lies between $q$ and $b$, then there is an $x$ such that $x$ lies between $a$ and $q$ and $p$ lies between $b$ and $x$.

"In the outer form of the Pasch Axiom [...] the point $b$ lies on the extension of the side $cq$ in the direction from $q$ to $c$, and the line is assumed to intersect the “inner” side of the triangle (from the perspective of $bp$). The conclusion is that it must intersect the side $aq$ in some point $x$ on the extension of the side $bp$; this is expressed by the assertion $B(bpx)$. In other words, it intersects the "outer" side of the triangle."

(Page 180 contains another variant of this axiom.)

Axiom 7$_3$ Weak Pasch Axiom. $$B(atd) \land B(bdc) \to \exists x \, \exists y [B(axb) \land B(ayc)\land B(ytx)]$$

Axiom 8$^{(1)}$ Lower 1-Dimensional Axiom. $$\exists a \,\exists b \, (a \neq b)$$

Axiom 8$^{(2)}$ Lower 2-Dimensional Axiom. $$\exists a\, \exists b\, \exists c\, [\neg B(abc) \land \neg B(bca) \land \neg B(cab)]$$

"The Lower 2-Dimensional Axiom asserts that there exist three non-collin- ear points."

*Axiom 8$^{(n)}$ Lower $n$-Dimensional Axiom for $n = 3, 4, ...$.

$\exists a \, \exists b \, \exists c \, \exists p_1 \, \exists p_2 \,\cdots \, \exists p_{n-1} \,$ $$\left[ \bigwedge_{1\leq i < j < n} p_i \neq p_j \land \bigwedge_{i=2}^{n-1} a p_1 \equiv a p_i \land \bigwedge_{i=2}^{n-1} b p_1 \equiv b p_i \land \bigwedge_{i=2}^{n-1} c p_1 \equiv c p_i \land [\neg B(abc)\land \neg B(bca) \land \neg B(cab)]\right]$$

"The Lower $n$-Dimensional Axiom for $n = 3, 4, . . .$ asserts that there exist $n − 1$ distinct points $p_1 , p_2 , . . . , p_{n−1}$ and three points $a, b, c$ such that each of the three points is equidistant from each of the $n-1$ points, but the three points are not collinear."

Axiom 9$^{(0)}$ Upper 0-Dimensional Axiom. $$a=b$$

Axiom 9$^{(1)}$ Upper 1-Dimensional Axiom. $$B(abc) \lor B(bca) \lor B(cab)$$

Axiom 9$_1^{(2)}$ Upper 2-Dimensional Axiom.

$\exists y \,$ {$([B(xya) \lor B(yax) \lor B(axy)] \land B(byc))$
$([B(xyb) \lor B(ybx) \lor B(bxy)] \land B(cya))$
$([B(xyc) \lor B(ycx) \lor B(cxy)] \land B(ayb))$}

(Page 183 contains another variant of this axiom.)

*Axiom 9$^{(n)}$ Upper $n$-Dimensional Axiom (for $n=2,3,...$).

$$\left[ \bigwedge_{1\leq i < j \leq n} p_i \neq p_j \land \bigwedge_{i=2}^{n} a p_1 \equiv a p_i \land \bigwedge_{i=2}^{n} b p_1 \equiv b p_i \land \bigwedge_{i=2}^{n} c p_1 \equiv c p_i\right]\to [B(abc)\lor B(bca) \lor B(cab)]$$

"The Upper $n$-Dimensional Axiom for $n = 2, 3, . . .$ asserts that any three points $a$, $b$, $c$ which are equidistant from each of $n$ distinct points $p_1, p_2, ..., p_n$ must be collinear."

*Axiom 10$_1$ First Form of Euclid’s Axiom. $B(adt) \land B(bdc) \land a \neq d \to \exists x \, \exists y \, [B(abx) \land B(acy) \land B(ytx)]$$

"The First Form of Euclid’s Axiom says that through any point $t$ in the interior of an angle $\triangle bac$ there is a line—here, the line $xy$—that intersects both sides of the angle—here, in the points $x$ and $y$."

(Page 183 contains another variant of this axiom.)

Axiom 10$_2$ Second Form of Euclid’s Axiom. $$B(abc) \lor B(bca) \lor B(cab) \lor \exists x \, [ax \equiv bx\land ax \equiv cx]$$

Axiom 10$_3$ Third Form of Euclid’s Axiom. $$[B(abf) \land ab \equiv bf \land B(ade) \land ad\equiv de \land B(bdc) \land bd \equiv dc] \to bc \equiv fe$$

*Axiom 11 Axiom of Continuity. $$\exists a\, \forall x \, \forall y [x\in X \land y \in Y \to B(axy)] \to \exists b\, \forall x \, \forall y [x\in X \land y \in Y \to B(xby)]$$

"The Axiom of Continuity asserts: any two sets $X$ and $Y$ such that the elements of $X$ precede the elements of $Y$ with respect to some point $a$ (that is, $B(axy)$ whenever $x$ is in $X$ and $y$ is in $Y$ ) are separated by a point $b$."

*Axiom Schema 11(alternative to axiom 11.) Axiom Schema of Continuity. $$\exists a\, \forall x \, \forall y [\alpha \land \beta \to B(axy)] \to \exists b\, \forall x \, \forall y [\alpha \land \beta \to B(xby)]$$ where $\alpha, \beta$ are first-order formulas, the first of which does not contain any free occurrences of $a, b, y$ and the second any free occurrences of $a, b, x$.

We can use the Axiom Schema in stead of axiom 11 to keep the axioms within the framework of first-order logic.

Axiom 12 Reflexivity Axiom for Betweenness. $$B(abb)$$

Axiom 13 $$a=b \to B(aba)$$

Axiom 14 Symmetry Axiom for Betweenness. $$B(abc) \to B(cba)$$

*Axiom 15 Inner Transitivity Axiom for Betweenness. $$B(abd) \land B(bcd) \to B(abc)$$

Axiom 16 Outer Transitivity Axiom for Betweenness. $$B(abc) \land B(bcd) \land b \neq c \to B(abd)$$

Axiom 17 Inner Connectivity Axiom for Betweenness. $$B(abd) \land B(acd) \to [B(abc) \lor B(acb)]$$

Axiom 18 Outer Connectivity Axiom for Betweenness. $$B(abc) \land B(abd) \land a \neq b \to [B(acd) \lor B(adc)]$$

Axiom 19 $$a=b \to ac \equiv bc$$

Axiom 20 Uniqueness Axiom for Triangle Construction. $$[ac\equiv ac'\land bc \equiv bc'\land B(adb) \land B(ad'b)\land B(cdx)\land B(c'd'x) \land d \neq x\land d' \neq x] \to c=c'$$

(Page 187 contains another variant of this axiom.)

Axiom 21 Existence Axiom for Triangle Construction. $$ab \equiv a'b' \to \exists c \, \exists x \, (ac\equiv a'c' \land bc \equiv b'c' \land B(cxp) \land [B(abx) \lor B(bxa) \lor B(xab)])$$

Axiom 22 Density Axiom for Betweenness. $$x \neq z \to \exists y[x\neq y \land z \neq y \land B(xyz)]$$

Axiom 23 $$[B(xyz) \land B(x'y'z')\land xy \equiv x'y'\land yz \equiv y'z'] \to xz \equiv x'z'$$

Axiom 24 $$[B(xyz) \land B(x'y'z')\land xz \equiv x'z' \land yz \equiv y'z'] \to xy \equiv x'y'$$

Solution 3:

Postulates used in John M. Lee's Axiomatic Geometry (Draft, 2011)

Used with author's permission

Postulates of Neutral Geometry

The undefined terms in Jack Lee's axiomatization are point, line, distance (between points) and measure (of an angle).

Postulate 1 (The Set Postulate). Every line is a set of points, and there is a set of point called the plane.

Postulate 2 (The Existence Postulate). There exist at least two distinct points.

Postulate 3 (The Unique Line Postulate). Given any two points, there is a unique line that contains both of them.

Postulate 4 (The Distance Postulate). For every pair of points $A$ and $B$, the distance between $A$ and $B$ is a non-negative real number determined by $A$ and $B$.

Postulate 5 (The Ruler Postulate). For every line $\ell$, there is a bijective function $\mathcal{f}:\ell \rightarrow \mathbb{R}$ with the property that for any two points $A, B \in \ell$, we have

$$ AB = | \mathcal{f}(B) - \mathcal{f}(A)|.$$

Postulate 6 (The Plane Separation Postulate). For any line $\ell$, the set of all points not on $\ell$ is the union of two disjoint, non-empty subsets called sides of $\ell$. If $A$ anf $B$ are distinct points not on $\ell$, then both of the following conditions are satisfied:

1. $A$ and $B$ are on the same side of line $\ell$ if and only if $\overline{AB} \cap \ell= \oslash$

2. $A$ and $B$ are on the opposite side of line $\ell$ if and only if $\overline{AB} \cap \ell \neq \oslash$

Postulate 7 (The Angle Measure Postulate). For every angle $\angle ab$, the measure of $\angle ab$, written as $m\angle ab$, is a real number in the closed interval $[0,180]$ determined by $\angle ab$.

• Definition: If $\overrightarrow r$ is a ray starting at point $O$, and $P$ is a point not on the line $\overleftrightarrow r$, the half-rotation of rays determined by $\overrightarrow r$ and $P$, denoted by $HR(\overrightarrow r, P )$, is the set whose elements are all the rays that start at $O$ and whose points are either in line $\overleftrightarrow r$ or on the same side of $\overleftrightarrow r$ as $P$.

Postulate 8 (The Protractor Postulate). For every ray $\overrightarrow r$ and every point $P$ not on line $\overleftrightarrow{r}$, there is a bijective function $g:HR(\overrightarrow{r}, P) \rightarrow [0,180]$ such that the following two conditions are satisfied:

1. the function $g$ assigns the number $0$ to $\overrightarrow{r}$ and the number $180$ to the ray opposite $\overrightarrow{r}$.

2. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are any two rays in $HR(\overrightarrow{r}, P)$ then $$ m\angle ab = |g(\overrightarrow b)-g(\overrightarrow b)|.$$

Postulate 9 (The SAS Postulate). If there is a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding sides and angle of the other triangle, then the triangles are congruent under that correspondence.

Postulates of Euclidean Geometry

Postulates for Euclidean geometry are the postulates are Postulates 1–9 of neutral geometry, plus the following:

Postulate 10E (The Euclidean Parallel Postulate). For each line $\ell$ and each point $A$ that does not lie on $\ell$, there is a unique line that contains $A$ and is parallel to $\ell$.

• Definition: Lines $\ell$ and $m$ are called parallel if they do not intersect.

There is one additional undefined term for Euclidean geometry: area.

Postulate 11E (The Euclidean Area Postulate). For every polygonal region $\mathcal{R}$, there is a positive real number $\alpha(\mathcal{R})$ called the area of $\mathcal{R}$, which satisfies the following three conditions:

1. (Area Congruence Property) If $\mathcal{R}_1$ and $\mathcal{R}_2$ are congruent simple regions, then $$ \alpha(\mathcal{R}_1)=\alpha(\mathcal{R}_2).$$

2. (Area Addition Property) If $\mathcal{R}_1, \dots , \mathcal{R}_n$ are non-overlapping simple regions, then $$\alpha(\mathcal{R}_1 \cup \cdots \cup \mathcal{R}_n)= \alpha(\mathcal{R}_1) + \cdots + \alpha( \mathcal{R}_n).$$

3. (Unit Area Property) If $\mathcal{R}$ is a square region with sides of length $1$, then $\alpha(\mathcal{R})=1$.

Some useful definitions:

• A polygon is a broken line segment that is simple closed and proper.

• A broken line segment is the union of finitely many segments $\overline{A_1 A_2},\, \overline{A_2 A_3}, \dots ,\, \overline{A_{n} A_{n+1}}$ determined by points $A_1, \dots , A_{n+1}$, not necessarily distinct. We denote this $\overline{A_1, \dots , A_{n+1}}$.

• Given a broken line segment $\overline{A_1, \dots , A_{n+1}}$, we call it

  1. simple if the first $n$ vertices are distinct and no two of its constituent segments, edges, intersect except at a common end point.

  2. closed if the first and last points points, $A_1$ and $A_{n+1}$, are the same.

  3. proper if for each $i=1, \dots , n-1$, the three end points $A_i, A_{i+1}, A_{i+2}$ are non-collinear and if the broken line segment is closed, we require that $A_n, A_1, A_2$ are non-collinear as well.

• We call two polygons congruent if there is a correspondence between their vertices such that consecutive vertices correspond to consecutive vertices, corresponding edges are congruent, and corresponding interior angle measures are equal.

• Given a polygon $\mathcal{P}$, a point $Q$ not on $\mathcal{P}$ is said to be an interior point of $\mathcal{P}$ if a ray starting with $Q$ and not containing any vertices of the polygon $\mathcal{P}$ has an odd number of intersections with the polygon.

• We call a set of points $\mathcal{R}$ a simple region if, for some polygon $\mathcal{P}$, $\mathcal{R}$ is the union of $\mathcal{P}$ and its interior.

• Two regions are congruent if their associated polygons are congruent.

• Two simple regions are said to be non-overlapping if their interiors are disjoint.

• A polygon is called a square if it has four sides, all four sides are congruent and all four angles have measure 90.

• We describe a region as a square region if its associated polygon is a square.

Postulates of Hyperbolic Geometry

Postulates for hyperbolic geometry are the postulates are Postulates 1–9 of neutral geometry, plus the following:

Postulate 10H (The Hyperbolic Parallel Postulate). For each line $\ell$ and each point $A$ that does not lie on $\ell$, there are at least two distinct lines that contain $A$ and are parallel to $\ell$.

Solution 4:

Postulates used in George D. Birkhoff's A set of postulates for plane geometry, based on scale and protractor (1932)

The undefined terms in Birkhoff's axiomatization are point, line, distance and angle.

Postulate I (The Postulate of Line Measure). The points $A, B, \dots$ of any line can be put into $1:1$ correspondence with the real numbers $x$ so that $|x_B-x_A| = d(A,B)$ for all points $A$ and $B$.

Postulate II (The Point-line Postulate). One and only one line, $\ell$, contains any two distinct points $P$ and $Q$.

Postulate III (The Postulate of Angle Measure). The half-lines, or rays, $\ell, m, \dots$ through any point $O$ can be put into $1:1$ correspondence with the real numbers $a \; ( \textrm{ mod }2 \pi)$ so that if $A \neq 0$ and $B \neq 0$ are points of $\ell$ and $m$, respectively, the difference $a_m - a_l \; ( \textrm{ mod }2 \pi)$ of the numbers associated with lines $\ell$ and $m$ is $\angle AOB$. Furthermore, if the point $B$ on $m$ varies continuously in a line $r$ not containing the vertex $O$, the number $a_m$ varies continuously also.

Postulate IV (Postulate of Similarity). If in two triangles $\triangle ABC$ and $\triangle A'B'C'$ and for some constant $k > 0,\; d(A', B') = kd(A, B),\; d(A', C') = kd(A, C)$, and $\angle B'A'C' = \pm \angle BAC$, then also $d(B', C') = kd(B, C),\; \angle C'B'A' = \pm \angle CBA$, and $\angle A'C'B' = \pm \angle ACB$.

Solution 5:

Axioms used in Euclid's elements as translated by J. L. Heiberg
Link: http://farside.ph.utexas.edu/euclid/Elements.pdf

Definitions

1. A point is that of which there is no part.
2. And a line is a length without breadth.
3. And the extremities of a line are points.
4. A straight-line is (any) one which lies evenly with points on itself.
5. And a surface is that which has length and breadth only.
6. And the extremities of a surface are lines.
7. A plane surface is (any) one which lies evenly with the straight-lines on itself.
8. And a plane angle is the inclination of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line.
9. And when the lines containing the angle are straight then the angle is called rectilinear.
10. And when a straight-line stood upon (another) straight-line makes adjacent angles (which are) equal to one another, each of the equal angles is a right-angle, and the former straight-line is called a perpendicular to that upon which it stands.
11. An obtuse angle is one greater than a right-angle.
12. And an acute angle (is) one less than a right-angle.
13. A boundary is that which is the extremity of something.
14. A figure is that which is contained by some boundary or boundaries.
15. A circle is a plane figure contained by a single line [which is called a circumference], (such that) all of the straight-lines radiating towards [the circumference] from one point amongst those lying inside the figure are equal to one another.
16. And the point is called the center of the circle.
17. And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each direction by the circumference of the circle. (And) any such (straight-line) also cuts the circle in half.
18. And a semi-circle is the figure contained by the diameter and the circumference cuts off by it. And the center of the semi-circle is the same (point) as (the center of) the circle.
19. Rectilinear figures are those (figures) contained by straight-lines: trilateral figures being those contained by three straight-lines, quadrilateral by four, and multi- lateral by more than four.
20. And of the trilateral figures: an equilateral trian- gle is that having three equal sides, an isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having three unequal sides.
21. And further of the trilateral figures: a right-angled triangle is that having a right-angle, an obtuse-angled (triangle) that having an obtuse angle, and an acute- angled (triangle) that having three acute angles.
22. And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one an- other which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia.
23. Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direc- tion, meet with one another in neither (of these direc- tions).

Postulates

1. Let it have been postulated† to draw a straight-line from any point to any point.
2. And to produce a finite straight-line continuously in a straight-line.
3. And to draw a circle with any center and radius.
4. And that all right-angles are equal to one another.
5. And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).

Common Notions

1. Things equal to the same thing are also equal to one another.
2. And if equal things are added to equal things then the wholes are equal.
3. And if equal things are subtracted from equal things then the remainders are equal.
4. And things coinciding with one another are equal to one another.
5. And the whole [is] greater than the part.