1 The Five Groups of Axioms

Chapter 1
The Five Groups of Axioms

1. The Elements of Geometry and the Five Groups of Axioms

Let us consider three distinct systems of things. The things composing the ﬁrst system, we will call points and designate them by the letters $A$ , $B$ , $C$ ,…; those of the second, we will call straight lines and designate them by the letters $a$ , $b$ , $c$ ,…; and those of the third system, we will call planes and designate them by the Greek letters $α$ , $β$ , $γ$ ,…The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space.

We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry. These axioms may be arranged in ﬁve groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows:

I,	1–7.	Axioms of connection.
II,	1–5.	Axioms of order.
III.		Axiom of parallels (Euclid’s axiom).
IV,	1–6.	Axioms of congruence.
V.		Axiom of continuity (Archimedes’s axiom).

2. 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, 1.: Two distinct points $A$ and $B$ always completely determine a straight line $a$ . We write $A B = a$ or $B A = a$ .
Instead of “determine,” we may also employ other forms of expression; for example, we may say $A$ “lies upon” $a$ , $A$ “is a point of” $a$ , $a$ “goes through” $A$ “and through” $B$ , $a$ “joins” $A$ “and” or “with” $B$ , etc. If $A$ lies upon $a$ and at the same time upon another straight line $b$ , we make use also of the expression: “The straight lines” $a$ “and” $b$ “have the point $A$ in common,” etc.
I, 2.: Any two distinct points of a straight line completely determine that line; that is, if $A B = a$ and $A C = a$ , where $B \neq C$ , then is also $B C = a$ .
I, 3.: Three points $A$ , $B$ , $C$ not situated in the same straight line always completely determine a plane $α$ . We write $A B C = a$ .
We employ also the expressions: $A$ , $B$ , $C$ , “lie in” $a$ ; $A$ , $B$ , $C$ “are points of” $a$ , etc.
I, 4.: Any three points $A$ , $B$ , $C$ of a plane $α$ , which do not lie in the same straight line, completely determine that plane.
I, 5.: If two points $A$ , $B$ of a straight line $a$ lie in a plane $α$ , then every point of $a$ lies in $a$ . In this case we say: “The straight line a lies in the plane $α$ ,” etc.
I, 6.: If two planes $α$ , $β$ have a point $A$ in common, then they have at least a second point $B$ in common.
I, 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.

Axioms I, 1–2 contain statements concerning points and straight lines only; that is, concerning the elements of plane geometry. We will call them, therefore, the plane axioms of group I, in order to distinguish them from the axioms I, 3–7, which we will designate brieﬂy as the space axioms of this group.

Of the theorems which follow from the axioms I, 3–7, we shall mention only the following:

Theorem 1. Two straight lines of a plane have either one point or no point in common; two planes have no point in common or a straight line in common; a plane and a straight line not lying in it have no point or one point in common.

Theorem 2. Through a straight line and a point not lying in it, or through two distinct straight lines having a common point, one and only one plane may be made to pass.

3. Group II: Axioms of order⁹

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:

II,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 $A$ .

Figure 1
II,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 $D$ .

Figure 2
II,3.: Of any three points situated on a straight line, there is always one and only one which lies between the other two.
II,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 $D$ .

Deﬁnition 1. We will call the system of two points $A$ and $B$ , lying upon a straight line, a segment and denote it by $A B$ or $B A$ . The points lying between $A$ and $B$ are called the points of the segment $A B$ or the points lying within the segment $A B$ . All other points of the straight line are referred to as the points lying outside the segment $A B$ . The points $A$ and $B$ are called the extremities of the segment $A B$ .

Figure 3
II,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 $A B C$ and not passing through any of the points $A$ , $B$ , $C$ . Then, if the straight line $a$ passes through a point of the segment $A B$ , it will also pass through either a point of the segment $B C$ or a point of the segment $A C$ . Axioms II, 1–4 contain statements concerning the points of a straight line only, and, hence, we will call them the linear axioms of group II. Axiom II, 5 relates to the elements of plane geometry and, consequently, shall be called the plane axiom of group II.

4. Consequences of The Axioms of Connection and Order

By the aid of the four linear axioms II, 1–4, we can easily deduce the following theorems:

Theorem 3. Between any two points of a straight line, there always exists an unlimited number of points.

Theorem 4. If we have given any ﬁnite number of points situated upon a straight line, we can always arrange them in a sequence $A$ , $B$ , $C$ , $D$ , $E$ , $\dots$ , $K$ so that $B$ shall lie between $A$ and $C$ , $D$ , $E$ , $\dots$ , $K$ ; $C$ between $A$ , $B$ and $D$ , $E$ ,…, $K$ ; $D$ between $A$ , $B$ , $C$ and $E$ ,… $K$ , etc. Aside from this order of sequence, there exists but one other possessing this property namely, the reverse order $K$ ,…, $E$ , $D$ , $C$ , $B$ , $A$ .

pict
Figure 4

Theorem 5. Every straight line $a$ , which lies in a plane $α$ , divides the remaining points of this plane into two regions having the following properties: Every point $A$ of the one region determines with each point $B$ of the other region a segment $A B$ containing a point of the straight line $a$ . On the other hand, any two points $A$ , $A'$ of the same region determine a segment $A A'$ containing no point of $a$ .

pict
Figure 5

If $A$ , $A'$ , $O$ , $B$ are four points of a straight line $a$ , where $O$ lies between $A$ and $B$ but not between $A$ and $A'$ , then we may say: The points $A$ , $A'$ are situated on the line a upon one and the same side of the point $O$ , and the points $A$ , $B$ are situated on the straight line $a$ upon diﬀerent sides of the point $O$ .

pict
Figure 6

All of the points of $a$ which lie upon the same side of $O$ , when taken together, are called the half-ray emanating from $O$ . Hence, each point of a straight line divides it into two half-rays.

Making use of the notation of Theorem 5, we say: The points $A$ , $A'$ lie in the plane $α$ upon one and the same side of the straight line $a$ , and the points $A$ , $B$ lie in the plane $α$ upon diﬀerent sides of the straight line $a$ .

Deﬁnition 2.

A system of segments $A B$ , $B C$ , $C D$ , …, $K L$ is called a broken line joining $A$ with $L$ and is designated, brieﬂy, as the broken line $A B C D E$ … $K L$ . The points lying within the segments $A B$ , $B C$ , $C D$ , …, $K L$ , as also the points $A$ , $B$ , $C$ , $D$ , …, $K$ , $L$ , are called the points of the broken line. In particular, if the point $A$ coincides with $L$ , the broken line is called a polygon and is designated as the polygon $A B C D \dots K$ . The segments $A B$ , $B C$ , $C D$ , …, $K A$ are called the sides of the polygon and the points $A$ , $B$ , $C$ , $D$ , …, $K$ the vertices. Polygons having $3$ , $4$ , $5$ , …, $n$ vertices are called, respectively, triangles, quadrangles, pentagons, …, n-gons. If the vertices of a polygon are all distinct and none of them lie within the segments composing the sides of the polygon, and, furthermore, if no two sides have a point in common, then the polygon is called a simple polygon.

With the aid of Theorem 5, we may now obtain, without serious diﬃculty, the following theorems:

Theorem 6. Every simple polygon, whose vertices all lie in a plane $α$ , divides the points of this plane, not belonging to the broken line constituting the sides of the polygon, into two regions, an interior and an exterior, having the following properties: If $A$ is a point of the interior region (interior point) and $B$ a point of the exterior region (exterior point), then any broken line joining $A$ and $B$ must have at least one point in common with the polygon. If, on the other hand, $A$ , $A'$ are two points of the interior and $B$ , $B'$ two points of the exterior region, then there are always broken lines to be found joining $A$ with $A'$ and $B$ with $B'$ without having a point in common with the polygon. There exist straight lines in the plane $α$ which lie entirely outside of the given polygon, but there are none which lie entirely within it.

pict
Figure 7

Theorem 7. Every plane $α$ divides the remaining points of space into two regions having the following properties: Every point $A$ of the one region determines with each point $B$ of the other region a segment $A H$ , within which lies a point of $α$ . On the other hand, any two points $A$ , $A'$ lying within the same region determine a segment $A A'$ containing no point of $α$ .

Making use of the notation of Theorem 7, we may now say: The points $A$ , $A'$ are situated in space upon one and the same side of the plane $α$ , and the points $A$ , $B$ are situated in space upon diﬀerent sides of the plane $α$ .

Theorem 7 gives us the most important facts relating to the order of sequence of the elements of space. These facts are the results, exclusively, of the axioms already considered, and, hence, no new space axioms are required in group II.

5. Group III: Axiom of Parallels (Euclid’s Axiom)

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:

III.: In a plane $α$ 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 $A$ .

This statement of the axiom of parallels contains two assertions. The ﬁrst of these is that, in the plane $α$ , there is always a straight line passing through $A$ which does not intersect the given line $a$ . The second states that only one such line is possible. The latter of these statements is the essential one, and it may also be expressed as follows:

Theorem 8. If two straight lines $a$ , $b$ of a plane do not meet a third straight line $c$ of the same plane, then they do not meet each other.

For, if $a$ , $b$ had a point $A$ in common, there would then exist in the same plane with $c$ two straight lines $a$ and $b$ each passing through the point $A$ and not meeting the straight line $c$ . This condition of aﬀairs is, however, contradictory to the second assertion contained in the axiom of parallels as originally stated. Conversely, the second part of the axiom of parallels, in its original form, follows as a consequence of Theorem 8.

The axiom of parallels is a plane axiom.

6. 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.”

IV,I.: 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 $A B$ (or $B A$ ) is congruent to the segment $A' B'$ . We indicate this relation by writing $A B \equiv A' B' .$ Every segment is congruent to itself; that is, we always have $A B \equiv A B .$

We can state the above axiom brieﬂy by saying that every segment can be laid oﬀ upon a given side of a given point of a given straight line in one and and only one way.

IV,2.: If a segment $A B$ 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 $A B \equiv A' B$ and $A B \equiv A'' B''$ , then $A' B' \equiv A'' B''$ .
IV,3.: Let $A B$ and $B C$ 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 $A B \equiv A' B'$ and $B C \equiv B' C'$ , we have $A C \equiv A' C'$ .

pict
Figure 8

Deﬁnition 3. Let $α$ be any arbitrary plane and $h$ , $k$ any two distinct half-rays lying in $α$ 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 $∠ (h, k)$ or $∠ (k, h)$ . 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.

IV,4.: Let an angle $(h, k)$ be given in the plane $α$ and let a straight line $a'$ be given in a plane $α'$ . Suppose also that, in the plane $α$ , 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 $α'$ 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 $∠ (h, k) \equiv ∠ (h', k')$ Every angle is congruent to itself; that is, $∠ (h, k) \equiv ∠ (h, k)$ or $∠ (h, k) \equiv ∠ (k, h)$

We say, brieﬂy, that every angle in a given plane can be laid oﬀ upon a given side of a given half-ray in one and only one way.

IV,5.: If 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 $∠ (h, k) \equiv ∠ (h', k')$ and $∠ (h, k) \equiv ∠ (h'', k'')$ , then $∠ (h', k') \equiv ∠ (h'', k'')$ .

Suppose we have given a triangle $A B C$ . Denote by $h$ , $k$ the two half-rays emanating from $A$ and passing respectively through $B$ and $C$ . The angle $(h, k)$ is then said to be the angle included by the sides $A B$ and $A C$ , or the one opposite to the side $B C$ in the triangle $A B C$ . It contains all of the interior points of the triangle $A B C$ and is represented by the symbol $∠ B A C$ , or by $∠ A$ .

IV,6.: If, in the two triangles $A B C$ and $A' B' C'$ the congruences $A B \equiv A' B', A C \equiv A' C', ∠ B A C \equiv ∠ B' A' C'$ hold, then the congruences $∠ A B C \equiv ∠ A' B' C' and ∠ A C B \equiv ∠ A' C' B'$ also hold.

Axioms IV, 1–3 contain statements concerning the congruence of segments of a straight line only. They may, therefore, be called the linear axioms of group IV. Axioms IV, 4, 5 contain statements relating to the congruence of angles. Axiom IV, 6 gives the connection between the congruence of segments and the congruence of angles. Axioms IV, 4–6 contain statements regarding the elements of plane geometry and may be called the plane axioms of group IV.

7. Consequences of the Axioms of Congruence

Suppose the segment $A B$ is congruent to the segment $A' B'$ . Since, according to axiom IV, 1, the segment $A B$ is congruent to itself, it follows from axiom IV, 2 that $A' B'$ is congruent to $A B$ ; that is to say, if $A B \equiv A' B'$ , then $A' B' \equiv A B$ . We say, then, that the two segments are congruent to one another.

Let $A$ , $B$ , $C$ , $D, \dots,$ $K$ , $L$ and $A'$ , $B'$ , $C'$ , $D', \dots,$ $K'$ , $L'$ be two series of points on the straight lines $a$ and $a'$ , respectively, so that all the corresponding segments $A B$ and $A' B'$ , $A C$ and $A' C'$ , $B C$ and $B' C', \dots,$ $K L$ and $K' L'$ are respectively congruent, then the two series of points are said to be congruent to one another. $A$ and $A'$ , $B$ and $B', \dots,$ $L$ and $L'$ are called corresponding points of the two congruent series of points.

From the linear axioms IV, 1–3, we can easily deduce the following theorems:

Theorem 9. If the ﬁrst of two congruent series of points $A$ , $B$ , $C$ , $D, \dots,$ $K$ , $L$ and $A'$ , $B'$ , $C'$ , $D', \dots,$ $K'$ , $L'$ is so arranged that $B$ lies between $A$ and $C$ , $D, \dots,$ $K$ , $L$ , and $C$ between $A$ , $B$ and $D, \dots,$ $K$ , $L$ , etc., then the points $A'$ , $B'$ , $C'$ , $D', \dots,$ $K'$ , $L'$ of the second series are arranged in a similar way; that is to say, $B'$ lies between $A'$ and $C'$ , $D', \dots,$ $K'$ , $L'$ , and $C'$ lies between $A'$ , $B'$ and $D', \dots,$ $K'$ , $L'$ , etc.

Let the angle $(h, k)$ be congruent to the angle $(h', k')$ . Since, according to axiom IV, 4, the angle $(h, k)$ is congruent to itself, it follows from axiom IV, 5 that the angle $(h', k')$ is congruent to the angle $(h, k)$ . We say, then, that the angles $(h, k)$ and $(h', k')$ are congruent to one another.

Deﬁnition 4. Two angles having the same vertex and one side in common, while the sides not common form a straight line, are called supplementary angles. Two angles having a common vertex and whose sides form straight lines are called vertical angles. An angle which is congruent to its supplementary angle is called a right angle.

Two triangles $A B C$ and $A' B' C'$ are said to be congruent to one another when all of the following congruences are fulﬁlled:

\begin{matrix} A B \equiv A' B', & A C \equiv A' C', & B C \equiv B' C', \\ ∠ A \equiv ∠ A', & ∠ B \equiv ∠ B', & ∠ C \equiv ∠ C' . \end{matrix}

Theorem 10 (First theorem of congruence for triangles.). If, for the two triangles $A B C$ and $A' B' C'$ , the congruences

A B \equiv A' B', A C \equiv A' C', ∠ A \equiv ∠ A'

hold, then the two triangles are congruent to each other.

Proof. From axiom IV, 6, it follows that the two congruences

∠ B \equiv ∠ B' and ∠ C \equiv ∠ C'

are fulﬁlled, and it is, therefore, suﬃcient to show that the two sides

B C

and

B' C'

are congruent. We will assume the contrary to be true, namely, that

B C

and

B' C'

are not congruent, and show that this leads to a contradiction. We take upon

B' C'

a point

D'

such that

B C \equiv B' D'

. The two triangles

A B C

and

A' B' D'

have, then, two sides and the included angle of the one agreeing, respectively, to two sides and the included angle of the other. It follows from axiom IV,

6

that the two angles

B A C

and

B' A' D'

are also congruent to each other. Consequently, by aid of axiom IV,

5

, the two angles

B' A' C'

and

B' A' D'

must be congruent.

pict
Figure 9

This, however, is impossible, since, by axiom IV, 4, an angle can be laid oﬀ in one and only one way on a given side of a given half-ray of a plane. From this contradiction the theorem follows. □

We can also easily demonstrate the following theorem:

Theorem 11 (Second theorem of congruence for triangles). If in any two triangles one side and the two adjacent angles are respectively congruent, the triangles are congruent.

We are now in a position to demonstrate the following important proposition.

Theorem 12. If two angles $A B C$ and $A' B' C'$ are congruent to each other, their supplementary angles $C B D$ and $C' B' D'$ are also congruent.

pict
Figure 10

Proof. Take the points $A'$ , $C'$ , $D'$ upon the sides passing through $B'$ in such a way that

A' B' \equiv A B, C' B' \equiv C B, D' B' \equiv D B .

Then, in the two triangles

A B C

and

A' B' C'

, the sides

A B

and

B C

are respectively congruent to

A' B'

and

C' B'

. Moreover, since the angles included by these sides are congruent to each other by hypothesis, it follows from Theorem 10 that these triangles are congruent; that is to say, we have the congruences

A C \equiv A' C, ∠ B A C \equiv ∠ B' A' C' .

On the other hand, since by axiom IV, 3 the segments

A D

and

A' D'

are congruent to each other, it follows again from Theorem 10 that the triangles

C A D

and

C' A' D'

are congruent, and, consequently, we have the congruences:

C D \equiv C' D', ∠ A D C \equiv ∠ A' D' C' .

From these congruences and the consideration of the triangles

B C D

and

B' C' D'

, it follows by virtue of axiom IV, 6 that the angles

C B D

and

C' B' D'

are congruent. □

As an immediate consequence of Theorem 12, we have a similar theorem concerning the congruence of vertical angles.

Theorem 13. Let the angle ( $h$ , $k$ ) of the plane $α$ be congruent to the angle ( $h'$ , $k'$ ) of the plane $α'$ , and, furthermore, let $l$ be a half-ray in the plane $α$ emanating from the vertex of the angle ( $h$ , $k$ ) and lying within this angle. Then, there always exists in the plane $α'$ a half-ray $l'$ emanating from the vertex of the angle ( $h'$ , $k'$ ) and lying within this angle so that we have

∠ (h, l) \equiv ∠ (h', l'), ∠ (k, l) \equiv ∠ (k', l') .

pict
Figure 11

Proof. will represent the vertices of the angles ( $h$ , $k$ ) and ( $h'$ , $k'$ ) by $O$ and $O'$ , respectively, and so select upon the sides $h$ , $k$ , $h'$ , $k'$ the points $A$ , $B$ , $A'$ , $B'$ that the congruences

O A \equiv O' A', O B \equiv O' B'

are fulﬁlled. Because of the congruence of the triangles

O A B

and

O' A' B'

, we have at once

A B \equiv A' B', ∠ O A B \equiv O' A' B', ∠ O B A \equiv ∠ O' B' A' .

Let the straight line

A B

intersect

l

C

. Take the point

C'

upon the segment

A' B'

so that

A' C' \equiv A C

. Then,

O' C'

is the required half-ray. In fact, it follows directly from these congruences, by aid of axiom IV, 3, that

B C \equiv B' C'

. Furthermore, the triangles

O A C

and

O' A' C'

are congruent to each other, and the same is true also of the triangles

O C B

and

O' B' C'

. With this our proposition is demonstrated. □

In a similar manner, we obtain the following proposition.

Theorem 14. Let $h$ , $k$ , $l$ and $h'$ , $k'$ , $l'$ be two sets of three half-rays, where those of each set emanate from the same point and lie in the same plane. Then, if the congruences

∠ (h, l) \equiv ∠ (h', l'), ∠ (k, l) \equiv ∠ (k', l')

are fulﬁlled, the following congruence is also valid; viz.:

∠ (h, k) \equiv ∠ (h', k') .

By aid of Theorems 12 and 13, it is possible to deduce the following simple theorem, which Euclid held–although it seems to me wrongly–to be an axiom.

Theorem 15. All right angles are congruent to one another.

Proof. the angle $B A D$ be congruent to its supplementary angle $C A D$ , and, likewise, let the angle $B' A' D'$ be congruent to its supplementary angle $C' A' D'$ . Hence the angles $B A D$ , $C A D$ , $B' A' D'$ , and $C' A' D'$ are all right angles. We will assume that the contrary of our proposition is true, namely, that the right angle $B' A' D'$ is not congruent to the right angle $B A D$ , and will show that this assumption leads to a contradiction. We lay oﬀ the angle $B' A' D'$ upon the half-ray $A B$ in such a manner that the side $A D''$ arising from this operation falls either within the angle $B A D$ or within the angle $C A D$ . Suppose, for example, the ﬁrst of these possibilities to be true. Because of the congruence of the angles $B' A' D'$ and $B A D''$ , it follows from Theorem 12 that angle $C' A' D'$ is congruent to angle $C A D''$ , and, as the angles $B' A' D'$ and $C' A' D'$ are congruent to each other, then, by IV, 5, the angle $B A D''$ must be congruent to $C A D''$ . □

pict
Figure 12

Furthermore, since the angle $B A D$ is congruent to the angle $C A D$ , it is possible, by Theorem 13, to ﬁnd within the angle $C A D$ a half-ray $A D'''$ emanating from $A$ , so that the angle $B A D''$ will be congruent to the angle $C A D'''$ , and also the angle $D A D''$ will be congruent to the angle $D A D'''$ . The angle $B A D''$ was shown to be congruent to the angle $C A D''$ and, hence, by axiom IV, 5, the angle $C A D''$ , is congruent to the angle $C A D'''$ . This, however, is not possible; for, according to axiom IV, 4, an angle can be laid oﬀ in a plane upon a given side of a given half-ray in only one way. With this our proposition is demonstrated. We can now introduce, in accordance with common usage, the terms “acute angle” and “obtuse angle.”

The theorem relating to the congruence of the base angles $A$ and $B$ of an equilateral triangle $A B C$ follows immediately by the application of axiom IV, 6 to the triangles $A B C$ and $B A C$ . By aid of this theorem, in addition to Theorem 14, we can easily demonstrate the following proposition.

Theorem 16 (Third theorem of congruence for triangles). If two triangles have the three sides of one congruent respectively to the corresponding three sides of the other, the triangles are congruent.

Any ﬁnite number of points is called a ﬁgure. If all of the points lie in a plane, the ﬁgure is called a plane ﬁgure.

Two ﬁgures are said to be congruent if their points can be arranged in a one-to-one correspondence so that the corresponding segments and the corresponding angles of the two ﬁgures are in every case congruent to each other.

Congruent ﬁgures have, as may be seen from Theorems 9 and 12, the following properties: Three points of a ﬁgure lying in a straight line are likewise in a straight line in every ﬁgure congruent to it. In congruent ﬁgures, the arrangement of the points in corresponding planes with respect to corresponding lines is always the same. The same is true of the sequence of corresponding points situated on corresponding lines.

The most general theorems relating to congruences in a plane and in space may be expressed as follows:

Theorem 17. If $(A, B, C, \dots)$ and $(A', B', C', \dots)$ are congruent plane ﬁgures and P is a point in the plane of the ﬁrst, then it is always possible to ﬁnd a point $P$ in the plane of the second ﬁgure so that $(A, B, C, \dots, P)$ and $(A', B', C', \dots, P')$ shall likewise be congruent ﬁgures. If the two ﬁgures have at least three points not lying in a straight line, then the selection of $P'$ can be made in only one way.

Theorem 18. If $(A, B, C, \dots)$ and $(A', B', C', \dots =$ are congruent ﬁgures and $P$ represents any arbitrary point, then there can always be found a point $P'$ so that the two ﬁgures $(A, B, C, \dots, P)$ and $(A', B', C', \dots, P')$ shall likewise be congruent. If the ﬁgure $(A, B, C, \dots, P)$ contains at least four points not lying in the same plane, then the determination of $P'$ can be made in but one way.

This theorem contains an important result; namely, that all the facts concerning space which have reference to congruence, that is to say, to displacements in space, are (by the addition of the axioms of groups I and II) exclusively the consequences of the six linear and plane axioms mentioned above. Hence, it is not necessary to assume the axiom of parallels in order to establish these facts.

If we take, in, addition to the axioms of congruence, the axiom of parallels, we can then easily establish the following propositions:

Theorem 19. If two parallel lines are cut by a third straight line, the alternate-interior angles and also the exterior-interior angles are congruent Conversely, if the alternate-interior or the exterior-interior angles are congruent, the given lines are parallel.

Theorem 20. The sum of the angles of a triangle is two right angles.

Deﬁnition 5. If $M$ is an arbitrary point in the plane $a$ , the totality of all points $A$ , for which the segments $M A$ are congruent to one another, is called a circle. $M$ is called the centre of the circle.

From this deﬁnition can be easily deduced, with the help of the axioms of groups III and IV, the known properties of the circle; in particular, the possibility of constructing a circle through any three points not lying in a straight line, as also the congruence of all angles inscribed in the same segment of a circle, and the theorem relating to the angles of an inscribed quadrilateral.

8. Group V. Axiom of Continuity (Archimedean Axiom)

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 (see Chap. V, Section 21), 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:

V. Let $A_{1}$ be any point upon a straight line between the arbitrarily chosen points $A$ and $B$ . Take the points $A_{2}$ , $A_{3}$ , $A_{4}, \dots$ 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}, \dots

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}

The axiom of Archimedes is a linear axiom.

Remark 1. ¹⁰ To the preceeding ﬁve groups of axioms, we may add the following one, which, although not of a purely geometrical nature, merits particular attention from a theoretical point of view. It may be expressed in the following form:

Axiom of Completeness.¹¹ (Vollständigkeit): To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the ﬁve groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the ﬁve groups of axioms as valid.

This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s theorem by virtue of which, for all sets of points situated upon a straight line between two deﬁnite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the “axiom of completeness.”

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Chapter 1The Five Groups of Axioms