Introduction to the Foundations of Geometry

Created by Professor R. L. Moore and Dr. E. I. Deaton

Please send mail if you have any suggestions, discover any mistakes, or would like to contribute to the meager collection of proofs and commentaries currently on file here. All contributors will be credited of course (especially those supplying elegant proofs of Theorem 50), unless they wish to remain anonymous. Thanks!

Undefined Terms

Point

Line

Between

Axioms

Axiom I Every line is a set of points.

Axiom II There exist three points such that no line contains all of them.

Axiom III If A and B are two points, there exists only one line that contains both A and B.

Axiom IV If A and B are two points, there exists a point C such that B is between A and C. (That is written ABC).

Axiom V If ABC, then CBA.

Axiom VI If ABC, then A, B, and C are distinct collinear points.

Axiom VII If ABC is true, then ACB is not true.

Axiom VIII If A, B, and C are three collinear points, one of them is between the other two.

Axiom IX If A, B, and C are three non-collinear points, and D is a point such that ABD and E is point such that BEC, then there exists a point F such that DEF and AFC.

Definitions

Definition 1 The "line AB" is the line that contains the points A and B.

Definition 2 The statement that A, B, and C are collinear points means that one line contains A, B, and C.

Definition 3 The statement that A, B, and C are non-collinear points means that no line contains A, B, and C.

Theorems

Theorem 1 If A and C are two points, there is a point P such that A, C, and P are non-collinear. [Proof]

Theorem 2 If A, B, and C are three non-collinear points and X is a point of AC, then X is the only point on both AC and BX. [Proof]

Theorem 3 If A and C are two points, there exists a point between A and C. [Proof]

Theorem 4 If A, B, and C are three non-collinear points, there do not exist 3 collinear points X, Y, and Z such that AXB, BYC, and CZA.

Theorem 5 If ABC and ACD, then ABD.

Theorem 6 If A and C are two points, there exist three points between A and C.

Theorem 7 If ABC and ACD, then BCD.

Theorem 8 If ABC and BCD, then ABD.

Theorem 9 If ABD and ACD, and B is not C, then either ABC and BCD, or ACB and CBD.

Theorem 10 If ABC and ABD, and C is not D, then either ACD and BCD, or ADC and BDC.

Definition

Definition 4 BACD means BAC, BAD, ACD, and BCD.

Theorem

Theorem 11 If A, B, C, and D are four collinear points, there exist four points, X, Y, A, and W, such that XYZW and each of the points X, Y, Z, and W is one of the points A, B, C, and D.

Definitions

Definition 5 If A is a point not on line L, the "the A side of L" means a point set such that X belongs to it if and only if X does not belong to L and no point of L is between X and A.

Definition 6 The non-A side of L means a point set such that X belongs to it if and only if there is a point of L between X and A.

Axiom

Axiom X If A, B, and C are three non-collinear points and the line L contains a point between A and B, then L either contains A, B, or C, or a point between A and C, or a point between C and B.

Theorems

Theorem 12 If L is a line and A is a point not on L, the A side of L exists and the non-A side of L exists.

Theorem 13 If X and Y are points on the A side of L, then no point of L is between them. (Case I: A, X, Y, non-collinear)

Theorem 14 If X and Y are points on the non-A side of L, then no point of L is between them.

Theorem 15 If X is a point on the A side of L and Y is a point on the non-A side of L, there is a point of L between them.

Theorem 16 No point on the A side of L is on the non-A side of L.

Theorem 17 If L is a line and A is a point not on L and X is a point not on L, then either X belongs to the A side of L or X belongs to the non-A side of L.

Theorem 18 If A, B, and C are three non-collinear points and D is a point such that ABD, and F is a point such that AFC, then there is a point E such that BEC and DEF.

Theorem 19 If L is a line and A is a point not on L, and B is a point on L, then every point between A and B is on the A side of L.

Theorem 20 If A and B are two points on the C side of L, every point between A and B is on the C side of L.

Theorem 21 If L is a line and A is a point not on L and B is a point on L and X is a point such that BAX, then X is on the A side of L.

Definitions

Definition 7 The "segment AB" is the set of all points between A and B.

Definition 8 The "interval AB" is the point set such that a point belongs to it if and only if it is A, or B, or is between A and B.

Definition 9 If A and B are two points, the "ray AB" is the point set such that X belongs to it if and only if either X is A, or X is B, or X is between A and B, or B is between A and X.

Definition 10 If A, B, and C are three non-collinear points, the angle ABC means a point set such that the point X belongs to it if and only if X belongs to the ray BA or the ray BC.

Definition 11 The statement that K is a side of angle ABC means that either K is the ray BA, or K is the ray BC, not both.

Definition 12 A vertex of an angle is the point the sides have in common.

Definition 13 And angle and a supplementary angle are two angles that have a common side and a line contains their other two sides.

Definition 14 "A side of the line L" is the point set M such that for some point B not on L, M is the B side of L, or "A side of L" is the B side for some point B not on L.

Theorems

Theorem 22 Every line has a side.

Theorem 23 If B is on the A side of L, the B side of L is the A side of L, and the non-B side of L is the non-A side of L.

Theorem 24 If B is on the non-A side of L, the B side of L is the non-A side of L, and the non-B side of L is the A side of L.

Theorem 25 Every line has two and only two sides.

Definitions

Definition 15 The interior of an angle ABC is a point set such that X belongs to it if and only if X is on the A side of the line BC and on the C side of the line AB.

Definition 16 The exterior of an angle ABC is a point set such that X belongs to it if and only if it is not true that X is on the A side of BC and on the C side of AB and it is not true that X is on the ray BA or on the ray BC.

Theorems

Theorem 26 If A is a point of the line CD and not on the ray CD, then ACD.

Theorem 27 If X is a point in the exterior of an angle ABC, then X is either on the non-A side of line BC or on the non-C side of line AB.

Theorem 28 If X and Y are points in the exterior of angle ABC, there is a point Z such that no point of the angle or its interior is between X and Z or between Y and Z.

Theorem 29 There is a point Z in the exterior of the angle ABC such that if X and Y are any two points in the exterior of the angle, no point of the angle or its interior is between X and Z or between Y and Z.

Theorem 30 If X is a point in the interior of an angle ABC and Y is a point in the exterior of the angle, then there is a point of the angle between X and Y.

Theorem 31 If A, B, and C are three non-collinear points and X is a point in the interior of the angle ABC, then the ray BX contains a point between C and A.

Definitions

Definition 17 If A, B, and C are three non-collinear points, the triangle ABC is the interval AB, the interval BC, and the interval CA.

Definition 18 The interior of the triangle ABC is a point set such that X belongs to it if and only if X is on the A side of BC, the B side of AC, and the C side of AB.

Definition 19 The exterior of the triangle ABC is a point set such that X belongs to it if and only if it does not belong to the triangle ABC or to the interior of the triangle ABC.

Theorems

Theorem 32 If X is a point in the exterior of triangle ABC, then X is in the exterior of two of the angles ABC, CAB, ACB.

Theorem 33 If X and Y are two points in the interior of the triangle ABC, there is no point of the triangle between them.

Theorem 34 If X is a point in the interior of the triangle ABC and Y is a point in the exterior, there is a point of the triangle between them.

Theorem 35 If X and Y are two points in the exterior of the triangle ABC, there is a point Z such that no point of the triangle or its interior is between X and Z or between Y and Z.

Theorem 36 If A, B, and C are three non-collinear points and Y is on the non-A side of line BC and on the non-B side of line AC, then Y is on the C side of line AB.

Definition

Definition 20 If A, B, and C are three non-collinear points, there exists a point C' such that CBC' and a point A' such that ABA', the vertical angle of the angle ABC is the angle A'BC'.

Theorems

Theorem 37 If ABC is an angle and X is a point in the interior of ABC, then every point of the line BX except B is either in the interior of angle ABC or in the interior of the vertical angle A'BC'.

Theorem 38 If X is a point distinct from C on the ray CD, then the ray CX is the ray CD.

Undefined Term

Congruent

Axioms

Axiom C I If A and B are points and C and D are points, there exists only one point E such that CDE and AB is congruent to DE. (This is written AB = DE).

Axiom C II If AB = CD, and CD = EF, then AB = EF.

Axiom C III If ABC and A'B'C' and AB = A'B' and BC = B'C', then AC = A'C'.

Axiom C IV If A, B, and C are three non-collinear points and A', B', and C' are three non-collinear points and D is a point such that ABD and D' is a point such that A'B'D' and AB = A'B' and BC = B'C' and CA = C'A' and BD = B'D', then CD = C'D'.

Axiom C V If A and C are two points, there exists between A and C a point M such that AM = MC.

Theorems

Theorem 39 If AB is a segment, CD is a segment, and AB = CD, then CD = AB.

Theorem 40 Every segment is congruent to itself.

Theorem 41 If AB = CD, and AB = EF, then CD = EF.

Theorem 42 If O and E are two points and A and B are two points, there is only one point F distinct from O on the ray OE such that AB = OF.

Theorem 43 If AC = A'C' and the point B is between A and C, then there exists between A' and C' a point B" such that AB = A'B' and BC = B'C'.

Theorem 44 If AB is a segment, CD is a segment, and X is a point between C and D such that AB = CX, then there is a point X' between C and D such that AB = DX'.

Definitions

Definition 21 The statement that AB < CD means that there is a point X between C and D such that neither AB = CX or AB = DX.

Definition 22 The statement that AB > CD means that CD < AB.

Theorems

Theorem 45 If ABC, then AB < AC.

Theorem 46 If AB < CD, then there is a point X between C and D such that AB = CX.

Theorem 47 If AB is a segment and CD is a segment, then either AB < CD, AB = CD, or AB > CD.

Theorem 48 If AB is a segment and CD is a segment, then only one of these statements is true: AB < CD, AB = CD, AB > CD.

Theorem 49 If AB <= CD and CD <= EF, then AB <= EF, and if AB < CD or if CD < EF, then AB < EF.

Theorem 50 Axiom C V follows from Axioms I through X and C I through C IV.

Theorem 51 If AB != CD, then CD != AB.

Theorem 52 If ABC and A'B'C" and AB = A'B' and AC = A'C', then BC = B'C'.

Theorem 53 If A, B, and C are three collinear points and AB = BC, then ABC is true.

Definition

Definition 23 The statement that the angle ABC = the angle A'B'C' means that there are two points X and Y, one on the ray BA and one on the ray BC both distinct from B and two points X' and Y', one on the ray B'A' and one on the ray B'C', both distinct from B' such that BX = B'X', BY = B'Y', and XY = X'Y'.

Theorems

Theorem 54 If angle ABC = angle A'B'C', there exists a point X on the ray BA and a point Y on the ray BC, both distinct from B, and a point X' on the ray B'A' and a point Y' on the ray B'C', both distinct from B' such that either BX = B'X' and BY = B'Y' and XY = X'Y', or BX = B'Y' and BY = B'X' and XY = X'Y'.

Theorem 55 If the angle ABC = the angle A'B'C', there exists a point X on the ray BA, a point Y on the ray BC, both distinct from B, a point X' on the ray B'A', a point Y' on the ray B'C', both distinct from B' such that BX = B'X', BY = B'Y', and XY = X'Y'.

Theorem 56 If A, B, and C are three non-collinear points and A', B', and C' are three non-collinear points and AB = A'B' and BC = B'C' and CA = C'A' and D is a point such that ADB and D' is a point such that A'D'B' and AD = A'D', then CD = C'D'.

Theorem 57 If angle ABC = angle A'B'C' and BA = B'A' and BC = B'C', then AC = A'C'.

Theorem 58 Every angle is congruent to itself.

Theorem 59 If angle A = angle B, then angle B = angle A.

Theorem 60 If angle A = angle B and angle B = angle Y, then angle A = angle Y.

Theorem 61 If angle ABC = angle A'B'C', their vertical angles are congruent, and all their supplementary angles are congruent.

Theorem 62 If angle A' is the vertical angle of angle A, then angle A' = angle A.

Theorem 63 If A and C are two points, there do not exist between A and C two points X and Y such that AX = XC and AY = YC.

Theorem 64 If A and B are two points, there do not exist two points C1 and C2 on the same side of AB such that AC1 = AC2 and BC1 = BC2.

Theorem 65 If AMB and AM = MB and P is a point distinct from M such that AP = BP, then AX = XB if and only if X is on the line MP.

Theorem 66 If A, B, and C are three non-collinear points and A', B', and X are three non-collinear points and AB = A'B', there does exist one and only one point C' on the X side of A'B' such that AC = A'C' and BC = B'C'.

Axiom

Axiom C VI If L is a line and S1 and S2 are two point sets and every point of S1 belongs to L and every point of S2 belongs to L, and no point of S1 is a point of S2, and no point of either of these sets is between two points of the other, but each point of L belongs to one of them, then there exists a point O which is between every point of S1 distinct from it and every point of S2 distinct from it. (Or) a point O such that if X is a point of S1 distinct from O and Y is a point of S2 distinct from O, then O is between X and Y.

Theorem

Theorem 67 Axiom C V follows from Axiom C VI.

Definition

Definition 24 A right angle is one that is congruent to one of its supplements.

Theorem

Theorem 68 There is a right angle.

Definition

Definition 25 The statement that CB is perpendicular to AB means angle ABC is a right angle.

Theorems

Theorem 69 If L is a line and A is a point not on L, there is a line through A perpendicular to L.

Theorem 70 If L is a line and B is a point on L, there is only one line through B perpendicular to L.

Theorem 71 If an angle is congruent to a right angle, it is a right angle.

Theorem 72 If L is a line and there is a perpendicular to L at P and if X is another point of L, there is a perpendicular to L at X making an angle congruent to each angle at P.

Definition

Definition 26 The statement that two lines intersect means that they have one point in common.

Theorems

Theorem 73 If two lines intersect, there are right angles on each line and each right angle is congruent to the other.

Theorem 74 All right angles are congruent to each other.

Definitions

Definition 27 The statement that angle ABC is less than angle A'B'C' (angle ABC < angle A'B'C') means that there exists in the interior of the angle A'B'C' a point X such that angle ABC = angle A'B'X' or angle ABC = angle XB'C'.

Definition 28 The statement that angle ABC is greater than angle A'B'C' (angle ABC > angle A'B'C') means that angle A'B'C' < angle ABC.

Theorems

Theorem 75 If the angle ABC < angle A'B'C', there exists in the interior of angle A'B'C' a point X such that angle ABC = angle A'B'X.

Theorem 76 If A and B are angles, then either A < B, A = B, or A > B.

Theorem 77 If angle ABC = angle A'B'C' and X is a point in the interior of angle ABC, then there is a point X' in the interior of angle A'B'C' such that angle ABX = angle A'B'X' and angle CBX = angle C'B'X'.

Theorem 78 If A, B, and Y are angles and A < B and B = Y, then A < Y.

Theorem 79 If A, B, and Y are angles and A < B and B < Y, then A < Y.

Theorem 80 If A and B are angles, then only one of the following statements is true: A < B, A = B, A > B.

Definition

Definition 29 If AB and CD are two segments, the statement that EF is a sum of AB and CD means that there is a point X such that EXF and either AB = EX and CD = XF or AB = XF and CD = EX.

Theorems

Theorem 81 If AB and CD are two segments and EF is a sum of AB and CD, then there is a point X such that EXF and AB = EX and CD = XF.

Theorem 82 If AB and CD are two segments, then any two sums of AB and CD are congruent.

Definition

Definition 30 The statement that angle ABC is a sum of angle DEF and angle GHK means that there is a point X in the interior of the angle ABC such that either angle CBX = angle DEF and angle XBA = angle GHK or angle CBX = angle GHK and angle XBA = angle DEF.

Theorems

Theorem 83 If angle ABC is a sum of angle DEF and angle GHK, there is a point X in the interior of angle ABC such that angle CBX = angle DEF and angle XBA = angle GHK.

Theorem 84 If A and B are angles and A' and B' are angles and A = A' and B = B' and there is a sum of A and B, then there is a sum of A' and B' and every sum of A and B is equal to every sum of A' and B'.

Theorem 85 If ABC is a triangle and F is a point such that ABF, then angle CBF > angle BCA.

Theorem 86 If A, B, and C are three non-collinear points and BC > CA, then angle BAC > angle CBA.

Theorem 87 If ABC is a triangle, then AC + CB > AB.

Theorem 88 If ABC is a triangle and AB = BC, then angle ACB = angle BAC.

Theorem 89 If ABC is a triangle and angle ACB > angle BAC, then AB > BC.

Theorem 90 If AB is perpendicular to BC and E and F are points such that BEFC, then AE < AF and AB < AE.

Theorem 91 If ACB and L is a line perpendicular to line AB at C and O is a point on L not on line AB, there is a point A' on the ray CB such that OA = OA'.

Theorem 92 If angle OFB is a right angle and A is a point such that FAB and CD is a segment such that OA < CD and OB > CD, there is a point X such that AXB and OX = CD.

Theorem 93 Two lines perpendicular to the same line have no point in common.

Definitions

Definition 31 The statement that the triangle ABC = the triangle DEF means that there exist three points X, Y, and Z each of which is one of A, B, and C and three points X', Y', and Z' each of which is one of D, E, and F such that XY = X'Y', YZ = Y'Z', and ZX = Z'X'.

Definition 32 The statement that C is a circle with center at O means C is a point set such that for some segment AB, P belongs to C if and only if OP = AB.

Definition 33 The statement that J is a circle means that there exists a point O and an interval AB such that the point P belongs to J if and only if it is distinct from O and the interval OP = AB. If such a circle exists, O is called the center of J. A radius of J is an interval congruent to an interval with one end point at a center of J and the other end point on J.

Theorem

Theorem 94 If C is a circle and A is a point in the interior of C and B is a point in the exterior of C, then there is a point of C between A and B.

Axiom I	Every line is a set of points.
Axiom II	There exist three points such that no line contains all of them.
Axiom III	If A and B are two points, there exists only one line that contains both A and B.
Axiom IV	If A and B are two points, there exists a point C such that B is between A and C. (That is written ABC).
Axiom V	If ABC, then CBA.
Axiom VI	If ABC, then A, B, and C are distinct collinear points.
Axiom VII	If ABC is true, then ACB is not true.
Axiom VIII	If A, B, and C are three collinear points, one of them is between the other two.
Axiom IX	If A, B, and C are three non-collinear points, and D is a point such that ABD and E is point such that BEC, then there exists a point F such that DEF and AFC.

Definition 1	The "line AB" is the line that contains the points A and B.
Definition 2	The statement that A, B, and C are collinear points means that one line contains A, B, and C.
Definition 3	The statement that A, B, and C are non-collinear points means that no line contains A, B, and C.

Theorem 1	If A and C are two points, there is a point P such that A, C, and P are non-collinear. [Proof]
Theorem 2	If A, B, and C are three non-collinear points and X is a point of AC, then X is the only point on both AC and BX. [Proof]
Theorem 3	If A and C are two points, there exists a point between A and C. [Proof]
Theorem 4	If A, B, and C are three non-collinear points, there do not exist 3 collinear points X, Y, and Z such that AXB, BYC, and CZA.
Theorem 5	If ABC and ACD, then ABD.
Theorem 6	If A and C are two points, there exist three points between A and C.
Theorem 7	If ABC and ACD, then BCD.
Theorem 8	If ABC and BCD, then ABD.
Theorem 9	If ABD and ACD, and B is not C, then either ABC and BCD, or ACB and CBD.
Theorem 10	If ABC and ABD, and C is not D, then either ACD and BCD, or ADC and BDC.

Definition 5	If A is a point not on line L, the "the A side of L" means a point set such that X belongs to it if and only if X does not belong to L and no point of L is between X and A.
Definition 6	The non-A side of L means a point set such that X belongs to it if and only if there is a point of L between X and A.

Theorem 12	If L is a line and A is a point not on L, the A side of L exists and the non-A side of L exists.
Theorem 13	If X and Y are points on the A side of L, then no point of L is between them. (Case I: A, X, Y, non-collinear)
Theorem 14	If X and Y are points on the non-A side of L, then no point of L is between them.
Theorem 15	If X is a point on the A side of L and Y is a point on the non-A side of L, there is a point of L between them.
Theorem 16	No point on the A side of L is on the non-A side of L.
Theorem 17	If L is a line and A is a point not on L and X is a point not on L, then either X belongs to the A side of L or X belongs to the non-A side of L.
Theorem 18	If A, B, and C are three non-collinear points and D is a point such that ABD, and F is a point such that AFC, then there is a point E such that BEC and DEF.
Theorem 19	If L is a line and A is a point not on L, and B is a point on L, then every point between A and B is on the A side of L.
Theorem 20	If A and B are two points on the C side of L, every point between A and B is on the C side of L.
Theorem 21	If L is a line and A is a point not on L and B is a point on L and X is a point such that BAX, then X is on the A side of L.

Definition 7	The "segment AB" is the set of all points between A and B.
Definition 8	The "interval AB" is the point set such that a point belongs to it if and only if it is A, or B, or is between A and B.
Definition 9	If A and B are two points, the "ray AB" is the point set such that X belongs to it if and only if either X is A, or X is B, or X is between A and B, or B is between A and X.
Definition 10	If A, B, and C are three non-collinear points, the angle ABC means a point set such that the point X belongs to it if and only if X belongs to the ray BA or the ray BC.
Definition 11	The statement that K is a side of angle ABC means that either K is the ray BA, or K is the ray BC, not both.
Definition 12	A vertex of an angle is the point the sides have in common.
Definition 13	And angle and a supplementary angle are two angles that have a common side and a line contains their other two sides.
Definition 14	"A side of the line L" is the point set M such that for some point B not on L, M is the B side of L, or "A side of L" is the B side for some point B not on L.

Theorem 22	Every line has a side.
Theorem 23	If B is on the A side of L, the B side of L is the A side of L, and the non-B side of L is the non-A side of L.
Theorem 24	If B is on the non-A side of L, the B side of L is the non-A side of L, and the non-B side of L is the A side of L.
Theorem 25	Every line has two and only two sides.

Definition 15	The interior of an angle ABC is a point set such that X belongs to it if and only if X is on the A side of the line BC and on the C side of the line AB.
Definition 16	The exterior of an angle ABC is a point set such that X belongs to it if and only if it is not true that X is on the A side of BC and on the C side of AB and it is not true that X is on the ray BA or on the ray BC.

Theorem 26	If A is a point of the line CD and not on the ray CD, then ACD.
Theorem 27	If X is a point in the exterior of an angle ABC, then X is either on the non-A side of line BC or on the non-C side of line AB.
Theorem 28	If X and Y are points in the exterior of angle ABC, there is a point Z such that no point of the angle or its interior is between X and Z or between Y and Z.
Theorem 29	There is a point Z in the exterior of the angle ABC such that if X and Y are any two points in the exterior of the angle, no point of the angle or its interior is between X and Z or between Y and Z.
Theorem 30	If X is a point in the interior of an angle ABC and Y is a point in the exterior of the angle, then there is a point of the angle between X and Y.
Theorem 31	If A, B, and C are three non-collinear points and X is a point in the interior of the angle ABC, then the ray BX contains a point between C and A.

Definition 17	If A, B, and C are three non-collinear points, the triangle ABC is the interval AB, the interval BC, and the interval CA.
Definition 18	The interior of the triangle ABC is a point set such that X belongs to it if and only if X is on the A side of BC, the B side of AC, and the C side of AB.
Definition 19	The exterior of the triangle ABC is a point set such that X belongs to it if and only if it does not belong to the triangle ABC or to the interior of the triangle ABC.

Theorem 32	If X is a point in the exterior of triangle ABC, then X is in the exterior of two of the angles ABC, CAB, ACB.
Theorem 33	If X and Y are two points in the interior of the triangle ABC, there is no point of the triangle between them.
Theorem 34	If X is a point in the interior of the triangle ABC and Y is a point in the exterior, there is a point of the triangle between them.
Theorem 35	If X and Y are two points in the exterior of the triangle ABC, there is a point Z such that no point of the triangle or its interior is between X and Z or between Y and Z.
Theorem 36	If A, B, and C are three non-collinear points and Y is on the non-A side of line BC and on the non-B side of line AC, then Y is on the C side of line AB.

Theorem 37	If ABC is an angle and X is a point in the interior of ABC, then every point of the line BX except B is either in the interior of angle ABC or in the interior of the vertical angle A'BC'.
Theorem 38	If X is a point distinct from C on the ray CD, then the ray CX is the ray CD.

Axiom C I	If A and B are points and C and D are points, there exists only one point E such that CDE and AB is congruent to DE. (This is written AB = DE).
Axiom C II	If AB = CD, and CD = EF, then AB = EF.
Axiom C III	If ABC and A'B'C' and AB = A'B' and BC = B'C', then AC = A'C'.
Axiom C IV	If A, B, and C are three non-collinear points and A', B', and C' are three non-collinear points and D is a point such that ABD and D' is a point such that A'B'D' and AB = A'B' and BC = B'C' and CA = C'A' and BD = B'D', then CD = C'D'.
Axiom C V	If A and C are two points, there exists between A and C a point M such that AM = MC.

Theorem 39	If AB is a segment, CD is a segment, and AB = CD, then CD = AB.
Theorem 40	Every segment is congruent to itself.
Theorem 41	If AB = CD, and AB = EF, then CD = EF.
Theorem 42	If O and E are two points and A and B are two points, there is only one point F distinct from O on the ray OE such that AB = OF.
Theorem 43	If AC = A'C' and the point B is between A and C, then there exists between A' and C' a point B" such that AB = A'B' and BC = B'C'.
Theorem 44	If AB is a segment, CD is a segment, and X is a point between C and D such that AB = CX, then there is a point X' between C and D such that AB = DX'.

Definition 21	The statement that AB < CD means that there is a point X between C and D such that neither AB = CX or AB = DX.
Definition 22	The statement that AB > CD means that CD < AB.

Theorem 45	If ABC, then AB < AC.
Theorem 46	If AB < CD, then there is a point X between C and D such that AB = CX.
Theorem 47	If AB is a segment and CD is a segment, then either AB < CD, AB = CD, or AB > CD.
Theorem 48	If AB is a segment and CD is a segment, then only one of these statements is true: AB < CD, AB = CD, AB > CD.
Theorem 49	If AB <= CD and CD <= EF, then AB <= EF, and if AB < CD or if CD < EF, then AB < EF.
Theorem 50	Axiom C V follows from Axioms I through X and C I through C IV.
Theorem 51	If AB != CD, then CD != AB.
Theorem 52	If ABC and A'B'C" and AB = A'B' and AC = A'C', then BC = B'C'.
Theorem 53	If A, B, and C are three collinear points and AB = BC, then ABC is true.

Theorem 54	If angle ABC = angle A'B'C', there exists a point X on the ray BA and a point Y on the ray BC, both distinct from B, and a point X' on the ray B'A' and a point Y' on the ray B'C', both distinct from B' such that either BX = B'X' and BY = B'Y' and XY = X'Y', or BX = B'Y' and BY = B'X' and XY = X'Y'.
Theorem 55	If the angle ABC = the angle A'B'C', there exists a point X on the ray BA, a point Y on the ray BC, both distinct from B, a point X' on the ray B'A', a point Y' on the ray B'C', both distinct from B' such that BX = B'X', BY = B'Y', and XY = X'Y'.
Theorem 56	If A, B, and C are three non-collinear points and A', B', and C' are three non-collinear points and AB = A'B' and BC = B'C' and CA = C'A' and D is a point such that ADB and D' is a point such that A'D'B' and AD = A'D', then CD = C'D'.
Theorem 57	If angle ABC = angle A'B'C' and BA = B'A' and BC = B'C', then AC = A'C'.
Theorem 58	Every angle is congruent to itself.
Theorem 59	If angle A = angle B, then angle B = angle A.
Theorem 60	If angle A = angle B and angle B = angle Y, then angle A = angle Y.
Theorem 61	If angle ABC = angle A'B'C', their vertical angles are congruent, and all their supplementary angles are congruent.
Theorem 62	If angle A' is the vertical angle of angle A, then angle A' = angle A.
Theorem 63	If A and C are two points, there do not exist between A and C two points X and Y such that AX = XC and AY = YC.
Theorem 64	If A and B are two points, there do not exist two points C1 and C2 on the same side of AB such that AC1 = AC2 and BC1 = BC2.
Theorem 65	If AMB and AM = MB and P is a point distinct from M such that AP = BP, then AX = XB if and only if X is on the line MP.
Theorem 66	If A, B, and C are three non-collinear points and A', B', and X are three non-collinear points and AB = A'B', there does exist one and only one point C' on the X side of A'B' such that AC = A'C' and BC = B'C'.

Theorem 69	If L is a line and A is a point not on L, there is a line through A perpendicular to L.
Theorem 70	If L is a line and B is a point on L, there is only one line through B perpendicular to L.
Theorem 71	If an angle is congruent to a right angle, it is a right angle.
Theorem 72	If L is a line and there is a perpendicular to L at P and if X is another point of L, there is a perpendicular to L at X making an angle congruent to each angle at P.

Theorem 73	If two lines intersect, there are right angles on each line and each right angle is congruent to the other.
Theorem 74	All right angles are congruent to each other.

Definition 27	The statement that angle ABC is less than angle A'B'C' (angle ABC < angle A'B'C') means that there exists in the interior of the angle A'B'C' a point X such that angle ABC = angle A'B'X' or angle ABC = angle XB'C'.
Definition 28	The statement that angle ABC is greater than angle A'B'C' (angle ABC > angle A'B'C') means that angle A'B'C' < angle ABC.

Theorem 75	If the angle ABC < angle A'B'C', there exists in the interior of angle A'B'C' a point X such that angle ABC = angle A'B'X.
Theorem 76	If A and B are angles, then either A < B, A = B, or A > B.
Theorem 77	If angle ABC = angle A'B'C' and X is a point in the interior of angle ABC, then there is a point X' in the interior of angle A'B'C' such that angle ABX = angle A'B'X' and angle CBX = angle C'B'X'.
Theorem 78	If A, B, and Y are angles and A < B and B = Y, then A < Y.
Theorem 79	If A, B, and Y are angles and A < B and B < Y, then A < Y.
Theorem 80	If A and B are angles, then only one of the following statements is true: A < B, A = B, A > B.

Theorem 81	If AB and CD are two segments and EF is a sum of AB and CD, then there is a point X such that EXF and AB = EX and CD = XF.
Theorem 82	If AB and CD are two segments, then any two sums of AB and CD are congruent.

Theorem 83	If angle ABC is a sum of angle DEF and angle GHK, there is a point X in the interior of angle ABC such that angle CBX = angle DEF and angle XBA = angle GHK.
Theorem 84	If A and B are angles and A' and B' are angles and A = A' and B = B' and there is a sum of A and B, then there is a sum of A' and B' and every sum of A and B is equal to every sum of A' and B'.
Theorem 85	If ABC is a triangle and F is a point such that ABF, then angle CBF > angle BCA.
Theorem 86	If A, B, and C are three non-collinear points and BC > CA, then angle BAC > angle CBA.
Theorem 87	If ABC is a triangle, then AC + CB > AB.
Theorem 88	If ABC is a triangle and AB = BC, then angle ACB = angle BAC.
Theorem 89	If ABC is a triangle and angle ACB > angle BAC, then AB > BC.
Theorem 90	If AB is perpendicular to BC and E and F are points such that BEFC, then AE < AF and AB < AE.
Theorem 91	If ACB and L is a line perpendicular to line AB at C and O is a point on L not on line AB, there is a point A' on the ray CB such that OA = OA'.
Theorem 92	If angle OFB is a right angle and A is a point such that FAB and CD is a segment such that OA < CD and OB > CD, there is a point X such that AXB and OX = CD.
Theorem 93	Two lines perpendicular to the same line have no point in common.

Definition 31	The statement that the triangle ABC = the triangle DEF means that there exist three points X, Y, and Z each of which is one of A, B, and C and three points X', Y', and Z' each of which is one of D, E, and F such that XY = X'Y', YZ = Y'Z', and ZX = Z'X'.
Definition 32	The statement that C is a circle with center at O means C is a point set such that for some segment AB, P belongs to C if and only if OP = AB.
Definition 33	The statement that J is a circle means that there exists a point O and an interval AB such that the point P belongs to J if and only if it is distinct from O and the interval OP = AB. If such a circle exists, O is called the center of J. A radius of J is an interval congruent to an interval with one end point at a center of J and the other end point on J.