Definition (Axiomatic System). An axiomatic system (a set of axioms and their logical consequences) consists of:

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1 Course Overview Contents 1 AxiomaticSystems Undefined Terms Incidence Distance Betweenness BasicGeometricFigures Congruence of Segments PlaneSeparation Angular Measure Congruence of Angles Congruence of Triangles Geometric Inequalities Real Numbers AlternateParallel Axioms BasicGeometries Absolute Plane Geometry TheEuclidean ParallelAxiom TheHyperbolicParallelAxiom Continuity Principles and Geometric Constructions Axiomatic Systems Definition (Axiomatic System). An axiomatic system (a set of axioms and their logical consequences) consists of: undefined terms - those primitive objects whose definitions are inappropriate. defined terms - convenient expressions for special relationships or objects which occur frequently. axioms - statements which are assumed to be true. theorems - statements which are proven to be true. Course Overview Copyright c 1999 by Jon T. Pitts Page 1 of 15

2 Definition (Model). A model of an axiomatic system is an example consisting of objects which satisfy each of the properties indicated in the axioms. Definition (Properties of Axiomatic Systems). Three qualities of an axiomatic system that we look for are consistency, independence, andcompleteness. An axiomatic system which is not self-contradictory is said to be consistent.q In an axiomatic system {A 1,...,A n }, an axiom A k is independent if it can not be proven to be true from the other axioms {A 1,...,A k 1,A k+1,...,a n }. The entire axiomatic system is independent if each of the axioms in the system is independent. An axiomatic system is complete if all models of the system are isomorphic. This means that two essentially different (non-isomorphic) models cannot be built. 2 Undefined Terms Our geometries will consist of points S, lines L, andplanes P which satisfy various axioms. 3 Incidence Axiom (I-1). Lines and planes are sets of points. Axiom (I-2). For distinct points A and B, there exists a unique line that contains A and B. Axiom (I-3). For three noncollinear points, there exists a unique plane that contains the points. Axiom (I-4). If two distinct points are contained in a plane, then the line determined by these points is a subset of the plane. Axiom (I-5). If two distinct planes intersect, then the intersection of these planes is a line. Axiom (I-6). There are at least three noncollinear points. Each line has at least two points. Every plane has at least three noncollinear points. Notation. If P and Q are different points, then the line containing them is denoted by PQ If P, Q,andRare noncollinear points, then the plane containing them is denoted by PQR Remark. To this point our geometry has the structure [S, L, P]. Axiom (I-6+ (Optional)). There are a least four noncoplanar points. Remark. Unless otherwise stated, we shall not assume that Axiom I-6+ is part of our axiomatic system. Course Overview Copyright c 1999 by Jon T. Pitts Page 2 of 15

3 4 Distance There is a distance function d : S S R such that the following four axioms (D-1, D-2, D-3, D-4) are true. Axiom (D-1). For every pair of points P, Q, d(p, Q) 0. Axiom (D-2). Given points P and Q, d(p, Q) =0if and only if P = Q. Axiom (D-3). d(p, Q) =d(q, P ) for every pair of points P and Q. Notation. We write d(p, Q) as simply PQ. Definition (Coordinate System). Let f : L R be a one-to-one correspondence between a line L and the real numbers. If for all points P, Q of L, wehavepq = f(p) f(q),thenfis a coordinate system for L. For each point P of L, the number x = f(p ) is called the coordinate of P. Axiom (D-4. Ruler Axiom). Every line has a coordinate system. Theorem (Ruler Placement Theorem). Let L be a line, and let P and Q be any two points of L. Then L has a coordinate system in which the coordinate of P is 0 and the coordinate of Q is positive. Remark. To this point our geometry has the structure [S, L, P,d]. 5 Betweenness Definition (Between). We say that point B is between points A and C if and only if A, B, andcare three distinct collinear points and AB + BC = AC. In this case we write A B C. Lemma (Betweenness Lemma). If f : L R is a coordinate system for the line L, then for A, B, C on L, A B C if and only if f(a) f(b) f(c). Theorem (B-1). If A B C,theC B A. Theorem (B-2). Of any three points on a line, exactly one is between the other two. Theorem (B-3). Any four points of a line can be named in an order A, B, C, D, in such a way that A B C D. Theorem (B-4). If A and B are any two points, then Course Overview Copyright c 1999 by Jon T. Pitts Page 3 of 15

4 (1) there is a point C such that A B C, and (2) there is a point D such that A D B. Theorem (B-5). If A B C, thena,b, and C are three different points of the same line. 6 Basic Geometric Figures Definitions (Segments, Rays, Angles, Triangles, Quadrilaterals). (1) Let P and Q be two points. The segment between P and Q is the set whose points are P and Q, together with all points between P and Q. This segment is denoted by PQ. (2) The ray from P through Q is the set of all points R of the line PQ such that P is not between R and Q. This ray is denoted by PQ. The point P is called the endpoint of the ray PQ. (3) An angle BAC is the union of two rays AB and AC which have the same endpoint, but which do not lie on the same line. AB and AC are called the sides of the angle, and the point Q is called the vertex. (4) If A, B, andcare three noncollinear points, then the set AB BC CA is called a triangle, denoted by the symbol ABC. The three segments AB BC CA are called its sides, and the points A, B, andcare called its vertices. (5) If A, B, C, D are four coplanar points no three of which are collinear, and if AB, BC, CD,andDA intersect only at their endpoints, then their union is called a quadrilateral, and is denoted by ΛABCD. Theorem. If A and B are any two points, then AB = BA. Theorem. If C is a point of AB, other than A, then AB = AC. Theorem. If B 1 and C 1 are points of AB and AC, other than A,then BAC = B 1 AC 1. Theorem. If AB = CD, then the points A, B are the same as the points C, D in some order. Theorem. If ABC = DEF, then the points A, B, and C are the same as the points D, E, and F, in some order. Course Overview Copyright c 1999 by Jon T. Pitts Page 4 of 15

5 7 Congruence of Segments Definition (Congruence of Segments). Let AB and CD be segments. If AB = CD,then the segments are congruent, and we write AB = CD. Theorem (C-1). For segments, congruence is an equivalence relation. Theorem (C-2. Segment-Construction Theorem). Give a segment AB and a ray CD, there is exactly one point E of CD such that AB = CE. Theorem (C-3. Segment-Addition Theorem). If A B C, A B C, AB = A B, and BC = B C,thenAC = A C. Theorem (C-4. Segment-Subtraction Theorem). If A B C, A B C, AB = A B, and AC = A C,thenBC = B C. Theorem (C-5). Every segment has exactly one midpoint. 8 Plane Separation Definition (Convex Set). AsetAis convex if for every two points P, Q of A, the entire segment PQis contained in A. Axiom (PS. Plane Separation). Given a line L inaplanee, the set E L is the union of two sets H 1 and H 2 such that: (1) H 1 and H 2 are convex, and (2) if P H 1 and Q H 2,thenPQintersects L. Definition (Half Planes and Edges). Each of the sets H 1 and H 2 is a half plane of E determined by L. L is the edge of H 1 and H 2. Theorem (Half Plane Properties). In a plane containing a line L, the two half planes determined by L are disjoint. Each such half plane H contains at least three noncollinear points and H L is convex. Theorem (Ray Theorem). Let H be a half plane with edge L and let T be a point in H. For each P L, PT is a subset of H L and P is the only point of PT on L. Definition (Interior of an Angle). The interior of BAC is the intersection of the side of AC that contains B and the side of AB that contains C. Course Overview Copyright c 1999 by Jon T. Pitts Page 5 of 15

6 Definition (Interior of a Triangle). The interior of ABC is defined as the intersection of the following three sets: (1) The side of AB that contains C. (2) The side of BC that contains A. (3) The side of CA that contains B. Theorem. The interior of a triangle is always a convex set. Theorem (Postulate of Pasch). Given ABC and a line L in the same plane, if L contains a point between A and C, thenlintersects either AB or BC or both. Theorem (Clubsuit ( )). Given ABC and a line L in the same plane, if L contains no vertex of the triangle, then L cannot intersect all of the three sides. Theorem (Stingray). In ABC,letDbe a point between A and C, and let E be a point such that B and E are on the same side of AC. Then DE intersects either AB or BC. Theorem (Crossbar). If D is in the interior of BAC,then AD intersects BC in a point between B and C. Definition (Convex Quadrilateral). A quadrilateral is called convex if each of its sides lies in one of the planes determined by the opposite side. Theorem (Characterization of Convex Quadrilateral). A quadrilateral is convex if and only if the diagonals intersect each other. 9 Angular Measure Let A denote the set of all angles. There exists a function m : A R such that the following four axioms (M-1, M-2, M-3, M-4) are true. Axiom (M-1). 0 <m ABC < 180 for all angles ABC A. Axiom (M-2. Angle Construction). Let AB be a ray on the edge of the half-plane H. For every real number r between 0 and 180, there is exactly one ray AP, with P in H, such that m PAB = r. Axiom (M-3. Angle Addition). Let BAC A.If D Int( BAC), then m BAC = m BAD + m DAC. Course Overview Copyright c 1999 by Jon T. Pitts Page 6 of 15

7 Definition (Linear Pairs). If AB and AC are opposite rays, and if AD is any third ray, then DAB and DAC form a linear pair. Definition (Supplementary Angles). If m ABC +m DEF = 180, then thetwoangles are supplementary. Axiom (M-4. Supplement Axiom). If two angles form a linear pair, then they are supplementary. 10 Congruence of Angles Definition (Congruence of Angles). If m ABC = m DEF, then the angles are congruent, and we write ABC = DEF. Theorem (C-6). For angles, congruence is an equivalence relation. Theorem (C-7. Angle Construction). Suppose ABC is an angle, B C is a ray, and H is a half plane whose edge contains B C. Then there is exactly one ray B C, with A in H such that ABC = A B C. Theorem (C-8. Angle Addition). Let D be in the interior of BAC, D be in the interior of B A C, BAD = B A D, and DAC = D A C.Then BAC = B A C. Theorem (C-9. Angle Subtraction). Let D be in the interior of BAC, D be in the interior of B A C, BAD = B A D, and BAC = B A C. Then DAC = D A C. Definitions (Special Angles). (1) A right angle is one whose measure is 90. (2) An angle is acute if its measure is less than 90. (3) An angle is obtuse if its measure is greater than 90. (4) Two angle are complementary if the sum of their measures is 90. (5) Two angles form a vertical pair if their sides form pairs of opposite rays. Theorem (Vertical Angle Theorem). If two angles form a vertical pair, then they are congruent. Remark. To this point our geometry has the structure [S, L, P,d,m]. Course Overview Copyright c 1999 by Jon T. Pitts Page 7 of 15

8 11 Congruence of Triangles Definition (Congruence of Triangles). A congruence from ABC to DEF is a oneto-one correspondence ABC DEF from the vertices of ABC onto the vertices of DEF such that AB = DE A = D BC = EF B = E CA = FD C = F. In this case we write ABC = DEF. Two triangles are called congruent if there is a congruence from one triangle to the other. Axiom (SAS. Side-Angle-Side). Given a correspondence between two triangles, if two sides and the included angle of one are congruent to the corresponding parts of the other, then the correspondence is a congruence. Theorem (Isosceles Theorem). In an isosceles triangle, the angles opposite the congruent sides are congruent. Theorem (ASA. Angle-Side-Angle). If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent. Theorem (Converse of Isosceles Theorem). If two angles of a triangle are congruent, then the sides opposite these angles are also congruent. Theorem (Equiangular Triangle Theorem). Every equiangular triangle is equilateral. Theorem (SSS. Side-Side-Side). If each pair of corresponding sides of two triangles are congruent, then the triangles are congruent. Theorem (Angle Bisector Theorem). Every angle has exactly one bisector. Theorem (Existence of Perpendiculars). Given a line and a point not on the line, then there is a line which passes through the given point and is perpendicular to the given line. Course Overview Copyright c 1999 by Jon T. Pitts Page 8 of 15

9 12 Geometric Inequalities Definition (Inequality for Segments). If AB < CD, then we say that AB is smaller than CD, and we write AB < CD. Definition (Inequality for Angles). If m BAC < m B A C, then we say that BAC is smaller than B A C, and we write BAC < B A C. Theorem (Exterior Angle Theorem). In any triangle an exterior angle is greater than either of the remote interior angles. Theorem (Uniqueness of Perpendiculars). The perpendicular to a given line, through a given external point, is unique. Theorem (Inequalities for Triangles). (1) If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the greater angle is opposite the longer side. (2) If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the greater angle. Theorem (Shortness of Perpendiculars). The shortest segment joining a point to a line is the perpendicular segment. Theorem (Triangle Inequality). Given any points A, B, C, thenab + BC AC. Equality holds if and only if either A B C, orbcoincides with either A or C. Theorem (Hinge Theorem). If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and if the included angle of the first triangle is larger than the included angle of the second triangle, then the opposite side of the first triangle is longer than the opposite side of the second triangle. Theorem (Converse of Hinge Theorem). If two triangles have two pairs of corresponding sides congruent, but the third side of one is longer than the corresponding side of the other, then the angle subtending the longer side will be larger than the angle subtending the corresponding other side. Theorem (Angle-Angle-Side (AAS)). If two angles and a side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Theorem (Hypotenuse-Leg Theorem). If the hypotenuse and one leg of a right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. Course Overview Copyright c 1999 by Jon T. Pitts Page 9 of 15

10 13 Real Numbers Axiom (Completeness). Every nonempty subset of R which is bounded above has a least upper bound. Theorem (Alternate Completeness). Every nonempty subset of R which is bounded below has a greatest lower bound. Theorem (Archimedian Principle). For any two positive numbers M and ɛ, there exists a positive integer N such that Nɛ > M. 14 Alternate Parallel Axioms Definition (Parallel Lines). Two line are parallel if they lie in the same plane and they do not intersect. Axiom (Euclidean Parallel Axiom). Give a line L and a point P not on L, there is one and only one line L which contains P and is parallel to L. Axiom (Hyperbolic Parallel Axiom). Give a line L and a point P not on L, there are at least two lines L and L which contains P and are parallel to L. Axiom (Spherical Parallel Axiom). No two lines in the same plane are ever parallel. Remark. At most one of these possible parallel axioms is assumed to be true for any given geometric model. 15 Basic Geometries Definition (Neutral Geometry). A neutral geometry is one which satisfies all of the following axioms: (1) Incidence axioms I-1, I-2, I-3, I-4, I-5, I-6. (2) Distance axioms D-1, D-2, D-3, D-4. (3) Separation axiom PS. (4) Angular measure axioms M-1, M-2, M-3, M-4. (5) Congruence axiom SAS. Course Overview Copyright c 1999 by Jon T. Pitts Page 10 of 15

11 Definition (Euclidean Geometry). A Euclidean geometry is a neutral geometry which satisfies the Euclidean Parallel Axiom. Definition (Hyperbolic Geometry). A hyperbolic geometry is a neutral geometry which satisfies the Hyperbolic Parallel Axiom. Definition (Spherical Geometry). A spherical geometry is ageometry whichsatisfies the the Spherical Parallel Axiom. Remark. Note that a spherical geometry can never be a neutral geometry. 16 Absolute Plane Geometry Remark. Absolute plane geometry is the geometric theory which can be developed for a neutral geometry independent of the whole question of the parallel axiom. Definition (Transversal Lines). If L 1, L 2,andT are three lines in the same plane, and T intersects L 1 and L 2 in two different points P and Q, respectively, then T is a transversal to L 2 and L 2. Definition (Alternate Interior Angles and Corresponding Angles). (1) If T is a transversal to L 1 and L 2, intersecting L 1 and L 2 in P and Q, respectively, and if A and D are points of L 1 and L 2, respectively, lying on opposite sides of T,then AP Q and PQD are alternate interior angles. (2) If x and y are alternate interior angles, and y and z are vertical angles, the x and z are corresponding angles. Theorem (Sufficient Conditions for Parallelism). (1) (Perpendicularity Criterion) If two lines lie in the same plane,and are perpendicular to the same line, then they are parallel. (2) (Existence of Parallel Lines) Given a line and a point not on the line, there is always at least one line which passes through the given point and is parallel to the given line. (3) (Alternate Interior Angle Theorem) Given two lines and a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel. (4) (Corresponding Angle Theorem) Given two lines and a transversal, if a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent and the lines are parallel. Course Overview Copyright c 1999 by Jon T. Pitts Page 11 of 15

12 Remark. No parallel axiom is required for any part of the previous theorem. Theorem (Polygonal Inequality). If A 1,A 2,...,A n are any points (n >1), then A 1 A 2 + A 2 A A n 1 A n A 1 A n. Definition (Rectangles and Saccheri Quadrilaterals). (1) A rectangle is a quadrilateral whose interior angles are right angles. (2) A quadrilateral ΛABCD is Saccheri quadrilateral if A and D are right angles, B and C are on the same side of AD, andab = CD. Theorem (Properties of Saccheri Quadrilaterals). (1) The diagonals of a Saccheri quadrilateral are congruent. (2) The upper base angles of a Saccheri quadrilateral are congruent. (3) The upper base in a Saccheri quadrilateral is congruent to or longer than the lower base. (4) In any Saccheri quadrilateral ΛABCD (with lower base AD), we have m BDC m ABD. Corollary. (1) If ABC has a right angle at A, thenm B +m C 90. (2) Every right triangle has only one right angle; and its other two angles are acute. (3) The hypotenuse of a right triangle is longer than either of the legs. (4) In ABC, letdbe the foot of the perpendicular from B to AC. IfAC is the longest side of ABC,theA D C. Theorem (Sum of Angles of a Triangle). In any triangle ABC, we have m A + m B + m C 180. Course Overview Copyright c 1999 by Jon T. Pitts Page 12 of 15

13 17 The Euclidean Parallel Axiom Theorem (Euclidean Equivalence Theorem). The following six properties are equivalent in any neutral geometry. (1) The geometry is a Euclidean geometry; i.e., it satisfies the Euclidean Parallel Axiom. (2) The sum of the measures of the angles in a triangle is equal to 180. (3) Every Saccheri quadrilateral is a rectangle. (4) There exists a rectangle. (5) There exist noncongruent similar triangles. (6) There are a pair of lines which are everywhere equidistant. Remark. Throughout the remainder of this section, we assume the Euclidean Parallel Axiom (cf. 14). Theorem. Given two lines and a transversal, if the lines are parallel, then each pair of alternate interior angles is congruent. Theorem. Given two lines and a transversal, if the lines are parallel, then each pair of corresponding angles is congruent. Theorem (Sum of Angles of a Triangle). In any triangle ABC, we have m A + m B + m C = 180. Corollary. The acute angles of a right triangle are complementary. Theorem. Every Saccheri quadrilateral is a rectangle. Theorem. For any triangle, the measure of an exterior angle is the sum of the measures of its two remote interior angles. Theorem. In a plane, any two lines parallel to a third line are parallel to each other. Theorem. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Theorem. Every diagonal divides a parallelogram into two congruent triangles. Theorem. In a parallelogram, each pair of opposite sides are congruent. Theorem. The diagonals of a parallelogram bisect each other. Theorem. Every trapezoid is a convex quadrilateral. Course Overview Copyright c 1999 by Jon T. Pitts Page 13 of 15

14 18 The Hyperbolic Parallel Axiom Theorem (Hyperbolic Equivalence Theorem). The following seven properties are equivalent in any neutral geometry. (1) The geometry is a hyperbolic geometry; i.e., it satisfies the Hyperbolic Parallel Axiom. (2) (Existence of Many Parallel Lines) Give a line L and a point P not on L,there exists an infinite number of lines through P in the same plane as L and P which are parallel to L. (3) (Hyperbolic Angle Sum Theorem) The sum of the measures of the angles in a triangle is less than 180. (4) The summit angles of a Saccheri quadrilateral are acute. (5) There do not exist any rectangles. (6) (AAA Congruence Theorem) If the corresponding angles of two triangles are congruent, then the triangles are congruent. (7) There do not exist a pair of lines which are everywhere equidistant. 19 Continuity Principles and Geometric Constructions Definition (Circles). A circle in a plane E with center C E and radius r>0is the set of all points P in E such that CP = r.theinterior (resp. exterior) of this circle is the set of all points Q in E such that CQ < r (resp. CQ > r). Theorem (Elementary Continuity Principle). For a segment in the plane of a circle, if one end point of the segment is in the interior of the circle and the other is in the exterior, then the segment intersects the circle. Theorem (Circular Continuity Principle). For circles C 1 and C 2 in the same plane, if circle C 1 has one point in the interior and one point in the exterior of circle C 2, then the circles intersect in exactly two points. Theorem (Straightedge and Compass Constructions). (1) Copy a segment. (2) Copy an angle. Course Overview Copyright c 1999 by Jon T. Pitts Page 14 of 15

15 (3) Bisect a given segment. (4) Bisect a given angle. (5) Construct a perpendicular to a line at a point on the line. (6) Construct a perpendicular to a line from a point not on the line. Course Overview Copyright c 1999 by Jon T. Pitts Page 15 of 15

An Approach to Geometry (stolen in part from Moise and Downs: Geometry)

An Approach to Geometry (stolen in part from Moise and Downs: Geometry) Undefined terms: point, line, plane The rules, axioms, theorems, etc. of elementary algebra are assumed as prior knowledge, and apply