Axioms and Results MATH 4020 and 5020 Our object of study is a universe of undefined objects called points and lines. Points lie on lines (“lie on” is also undefined); equivalently, we say that a line contains a point. Axioms are statements we assume to be true in the universe. Results must be verified. Axiom I-1 Incidence Axiom 1 If P and Q are distinct points, then there is a unique line that contains both P and Q. Axiom I-2 Incidence Axiom 2 Every line contains at least two distinct points. Definition Collinear Three or more points are collinear if there is a single line that contains them all. Axiom I-3 Incidence Axiom 3 There exist three distinct points that are not collinear. Definition Parallel Two distinct lines are parallel if there is no point on them both. Proposition 2.1 Two distinct, non-parallel lines have a unique point in common. Definition Concurrent Three or more lines are concurrent if there is a single point that lies on them all. Proposition 2.2 There exist three distinct lines that are not concurrent. Proposition 2.3 For every line, there is at least one point not on the line. Proposition 2.4 For every point, there is at least one line that does not contain the point. Proposition 2.5 For every point, there are at least two distinct lines through the point. Lemma A1 For each point A, there are points B and C such that A, B, and C are distinct, non-collinear points. Axiom B-1 Betweenness Axiom 1 If A ∗ B ∗ C, then A, B, and C are distinct, collinear points, and C ∗ B ∗ A. 1

Notation ←→ If A and B are distinct points, then AB will denote the unique line that contains A and B. Lemma N1 ←→ ←→ ←→ If A 6= B are points on P Q, then the line AB is the line P Q. Axiom B-2 Betweenness Axiom 2 ←→ If B and D are distinct points, then there are points A, C, and E on BD such that A ∗ B ∗ D, B ∗ C ∗ D, and B ∗ D ∗ E. Axiom B-3 Betweenness Axiom 3 If A, B, and C are distinct, collinear points, then exactly one of the statements B ∗ A ∗ C, A ∗ B ∗ C, and A ∗ C ∗ B holds true. Definition Between If A ∗ B ∗ C, one says that B is between A and C. Definition Segment For every pair of distinct points A and B, define the segment AB to be the set of points consisting of A, of B, and of all points between A and B. Lemma A2 If A and B are distinct points, then AB = BA. Definition Ray −−→ For every pair of distinct points A and B, define the ray AB to be the set consisting of all the points −−→ of AB and of all the points C such that between A ∗ B ∗ C. The ray AB is said to emanate from A. Notation If l is a line, we will also use l to denote the set of points on the line l. Context will be required to distinguish the line from the set of points on the line. (If you are reading the book, note that it uses {l} to denote the set of points on the line l.) Proposition 3.1 −−→ T −−→ −−→ S −−→ ←→ If A and B are distinct points, then AB BA = AB and AB BA = AB. Definition Side Let l be a line and A and B be points not on the line. If A = B or if AB contains no point of l, then one says that A and B are on the same side of l. If A 6= B and AB contains a point on l, then one says that A and B are on opposite sides of l. Axiom B-4 Betweenness Axiom 4 Let l be a line, and let A, B, and C be points not on l. If A and B are on the same side of l and if B and C are on the same side of l, then A and C are on the same side of l. If A and B are on opposite sides of l and if B and C are on opposite sides of l, then A and C are on the same side of l. Corollary to B-4 Let l be a line, and let A, B, and C be points not on l. If A and B are on opposite sides of l and if B and C are on the same side of l, then A and C are on opposite sides of l.

2

Definition Half-plane l Let l be a line and A be a point not on l. The half-plane bounded by l, denoted HA , is defined to be the set of points that are on the same side of l as A. (If the line l is clear from the context, the superscript may be suppressed; that is, the half-plane may be denoted HA .) Lemma A3 l l l Let A be a point not on a line l. If B is a point in HA , then HA = HB . Proposition 3.2 Every line bounds exactly two half-planes, and these half-planes have no point in common. Lemma A4 ←→ ←−→ If X ∗ Y ∗ Z and W is a point not on XZ, then X and Y are on the same side of W Z. If l is a line, A −−→ l is a point on l, and B is a point not on l, then AB \ {A} ⊆ HB . Proposition 3.3 If A ∗ B ∗ C and A ∗ C ∗ D, then B ∗ C ∗ D and A ∗ B ∗ D. Corollary to 3.3 If A ∗ B ∗ C and B ∗ C ∗ D, then A ∗ B ∗ D and A ∗ C ∗ D. Lemma N2 If A ∗ B ∗ C and A ∗ B ∗ D, then C ∗ B ∗ D is not true. Theorem Infinite Points Every line has infinitely many points on it. Proposition 3.4 Line Separation Property Suppose C ∗ A ∗ B, and let l be the line through A, B, and C. If P is a point on l, then either P lies −−→ −→ on AB or on AC. Theorem Pasch’s Theorem If A, B, and C are distinct, noncollinear points and l is a line that intersects AB in a unique point between A and B, then l also intersects AC or BC. If C is not on l, then l does not intersect both AC and BC. Proposition 3.5 S Suppose A ∗ B ∗ C. Then AC = AB BC and B is the only point common to AB and BC. Proposition 3.6 −−→ −−→ −−→ −→ Suppose A ∗ B ∗ C. Then B is the only point common to BA and BC, and AB = AC. Definition Opposite −−→ −→ ←→ ←→ Rays AB and AC are opposite if they are distinct and AB = AC. Definition Angle −−→ −→ An angle with vertex A is the point A along with two distinct, non-opposite rays AB and AC. Such an angle will be denoted ^BAC. (If the rays are clear from context, the angle may be written ^A.)

3

Definition Interior ←→ A point D is in the interior of ^CAB if D is on the same side of AC as B and D is on the same side ←→ of AB as C. Proposition 3.7 ←→ Consider ^CAB and a point D on BC. Then D is in the interior of ^CAB iff B ∗ D ∗ C. Proposition 3.8 −−→ Suppose D is in the interior of ^CAB. Then every point on AD except A is also in the interior of −−→ ^CAB, and no point on the ray opposite to AD is in the interior of ^CAB. Also, if C ∗ A ∗ E, then B is in the interior of ^DAE. Definition Between Rays −−→ −−→ −→ Ray AD is between rays AB and AC if D is in the interior of ^BAC. Theorem Crossbar Theorem −−→ −−→ −→ −−→ If AD is between AB and AC, then AD intersects BC. Definition Triangle If A, B, and C are distinct, noncollinear points, the set of all three points, denoted 4ABC, is called a triangle. Definition Interior of a Triangle The interior of a triangle is the intersection of the interiors of the three angles of the triangle. Definition Exterior A point is in the exterior of a triangle if it is not in the interior and does not lie on a segment of the triangle. Notation If A, B, and C are noncollinear points, write int(^ABC) for the interior of ^ABC and int(4ABC) for the interior of 4ABC. Proposition 3.9 If a ray r emanating from an exterior point of 4ABC intersects AB at a point between A and B, then r also intersects AC or BC. If a ray emanates from an interior point of 4ABC, then it intersects one of the sides; moreover, if it does not pass through a vertex, it intersects only one side. Definition Congruent The triangles 4ABC and 4DEF are congruent if AB ∼ = DE, AC ∼ = DF , BC ∼ = EF , ^BAC ∼ = ^EDF , ∼ ∼ ∼ ^ABC = ^DEF , and ^ACB = ^DF E. In this case, write 4ABC = 4DEF . Notice that the order in which the points appear matters; 4ABC ∼ = 4DEF is a different statement than 4ABC ∼ = 4DF E. Axiom C-1 Congruence Axiom 1 Let A and B be distinct points, and let r be a ray emanating from point A0 . Then there is a unique B 0 on r such that AB ∼ = A0 B 0 . Axiom C-2 Congruence Axiom 2 If AB ∼ = CD and AB ∼ = EF , then CD ∼ = EF . Also, every segment is congruent to itself.

4

Axiom C-3 Congruence Axiom 3 If A ∗ B ∗ C, A0 ∗ B 0 ∗ C 0 , AB ∼ = A0 B 0 , and BC ∼ = B 0 C 0 , then AC ∼ = A0 C 0 . Axiom C-4 Congruence Axiom 4 −−−→ ←−→ −−→ Consider ^BAC and B 0 A0 . Given a side of A0 B 0 , there is a unique ray A0 C 0 in that side such that ^B 0 A0 C 0 ∼ = ^BAC. Axiom C-5 Congruence Axiom 5 If ^ABC ∼ = ^DEF and ^DEF ∼ = ^GHI, then ^ABC ∼ = ^GHI. Also, every angle is congruent to itself. Axiom C-6 SAS If two sides and the included angle of a triangle are respectively congruent to two sides and the included angle of another triangle, then the triangles are congruent. Corollary to SAS ←→ Consider 4ABC and a segment DE congruent to AB. Given a side of DE, there is a unique point F in that side such that 4ABC ∼ = 4DEF . Proposition 3.10 Consider 4ABC. If AB ∼ = AC, then ^ABC ∼ = ^ACB. Proposition 3.11 Segment Subtraction If A ∗ B ∗ C, D ∗ E ∗ F , AB ∼ = DE, and AC ∼ = DF , then BC ∼ = EF . Proposition 3.12 Suppose AC ∼ = DF . Then for any point B between A and C, there is a unique point E between D and F such that AB ∼ = DE. Definition Write AB < CD and CD > AB whenever there is a point E between C and D such that AB ∼ = CE. Proposition 3.13 Segment Ordering Exactly one of AB < CD, AB ∼ = CD, and AB > CD hold. If AB < CD and CD ∼ = EF , then AB < EF . If AB > CD and CD ∼ = EF , then AB > EF . If AB < CD and CD < EF , then AB < EF . Definition Supplementary −−→ −→ Consider two angles of the form ^DAB and ^CAD. If AB and AC are opposite rays, then the angles are called supplementary or supplements of each other. Proposition 3.14 Supplements of congruent angles are congruent. Definition Vertical Suppose A ∗ B ∗ C, D ∗ B ∗ E, and these points are not all collinear. Then ^ABD and ^CBE are called vertical angles. Definition Right An angle is a right angle if it has a supplement to which it is congruent. 5

Proposition 3.15 Vertical angles are congruent to each other. An angle congruent to a right angle is a right angle. Definition Perpendicular Two distinct lines l and m are perpendicular if the following occurs. The lines intersect at a point A, and there are points B on l and C on m such that ^BAC is a right angle. Proposition 3.16 For every line l and every point P , there is a line through P perpendicular to l. Proposition 3.17 ASA Consider 4ABC and 4DEF , and assume ^BAC ∼ = ^EDF , ^ACB ∼ = ^DF E, and AC ∼ = DF . Then ∼ 4ABC = 4DEF . Definition Isosceles A triangle is isosceles if it has two distinct, congruent segments. Proposition 3.18 Converse to Proposition 3.10 If ^ABC ∼ = ^ACB in 4ABC, then AB ∼ = AC, and 4ABC is isosceles. Proposition 3.19 Angle Addition −−→ −−→ −−→ −−→ −−→ −−→ Let BG be between BA and BC, let EH be between ED and EF , let ^CBG ∼ = ^F EH, and let ^GBA ∼ = ^HED. Then ^ABC ∼ = ^DEF . Proposition 3.20 Angle Subtraction −−→ −−→ −−→ −−→ −−→ −−→ Let BG be between BA and BC, EH be between ED and EF , ^CBG ∼ = ^DEF . = ^F EH, and ^ABC ∼ ∼ Then ^GBA = ^HED. Definition −−→ −−→ −−→ Write ^ABC < ^DEF if there is a ray EG between ED and EF such that ^ABC ∼ = ^GEF . Lemma N3 Angle-Version of Proposition 3.12 −−→ −−→ −−→ −−→ −−→ If ^ABC ∼ = ^DEF and BG is between BA and BC, then there is a unique ray EH between ED and −−→ EF such that ^ABG ∼ = ^DEH. Proposition 3.21 Ordering of Angles Exactly one of ^AP B < ^CQD, ^AP B ∼ = ^CQD, and ^AP B > ^CQD hold. If ^AP B < ^CQD and ^CQD ∼ = ^ERF , then ^AP B < ^ERF . If ^AP B > ^CQD and ^CQD ∼ = ^ERF , then ^AP B > ^ERF . If ^AP B < ^CQD and ^CQD < ^ERF , then ^AP B < ^ERF . Proposition 3.22 SSS Consider 4ABC and 4DEF . If AB ∼ = DE, AC ∼ = DF , and BC ∼ = EF , then 4ABC ∼ = 4DEF . Lemma N4 Suppose X, Y , and Z are noncollinear and W is a point in the interior of 4XY Z. Then X and Y are ←−→ on opposite sides of W Z. Proposition 3.23 Euclid’s Fourth Postulate All right angles are congruent to each other.

6

Definition Acute An angle is acute if it is less than a right angle. Definition Obtuse An angle is obtuse if it is greater than a right angle. Definition Transversal A line is a transversal to two other, distinct lines if it intersects the two lines at distinct points. Definition Interior Angles Let t be a transversal to lines l and m, where t meets l at the point B and t meets m at the point E. If A and C are points on l such that A ∗ B ∗ C and D and F are points on m such that D ∗ E ∗ F , then ^ABE, ^BED, ^BEF , and ^CBE are each called an interior angle. Definition Alternate Interior Angles Let t be a transversal to lines l and m, where t meets l at the point B and t meets m at the point E. If A and C are points on l such that A ∗ B ∗ C and D and F are points on m such that D ∗ E ∗ F , then the pair ^ABE and ^BEF and the pair ^BED and ^CBE are called alternate interior angle. Theorem AIA If two lines cut by a transversal have a pair of congruent alternate interior angles with respect to the transversal, then the two lines are parallel. Corollary 1 to AIA Distinct lines perpendicular to the same line are parallel. The perpendicular dropped from a point P not on a line l to l is unique. Definition Foot Consider the perpendicular dropped from a point P not a line l to l. The point of intersection is called the foot of the perpendicular. Corollary 2 Euclid I.31 If l is a line and P is a point not on l, there is a line through P parallel to l. Definition Exterior Angle An angle supplementary to an angle of a triangle is called an exterior angle of the triangle. The other two angles of the triangle are called remote interior angles of the exterior angle. Theorem EA An exterior angle of a triangle is greater than either remote interior angle. Corollary 1 to EA If a triangle has a right or obtuse angle, the other two angles are acute. Proposition 4.1 SAA Consider 4ABC and 4DEF . Suppose AC ∼ = DF , ^BAC ∼ = ^EDF , and ^ABC ∼ = ^DEF . Then ∼ 4ABC = 4DEF . Definition Right Triangle A triangle is a right triangle if one of its angles is a right angle. 7

Definition Hypotenuse A segment opposite a right angle in a triangle is called a hypotenuse. Lemma N5 A right triangle has exactly one hypotenuse. Proposition 4.2 Hypotenuse Leg Criterion Two right triangles are congruent if the hypotenuse and a leg are congruent, respectively, to the hypotenuse and a leg of the other. Definition Midpoint A point C is a midpoint of the segment AB if A ∗ C ∗ B and AC ∼ = CB. Proposition 4.3 Midpoints Every segment has a unique midpoint. Proposition 4.4 Bisectors Every angle has a unique bisector. Every segment has a unique perpendicular bisector. Proposition 4.5 In a triangle, the greatest angle lies opposite the greatest side, and conversely. Proposition 4.6 Consider 4ABC and 4DEF . Suppose that AB ∼ = DE and BC ∼ = EF . Then ^ABC < ^DEF iff AC < DF . Definition Altitude The altitude of the vertex A in 4ABC is the segment AD, where D is the foot of the perpendicular ←→ dropped from A to BC.

8