Download Axioms, Definitions and Theorems

Axioms, Definitions and Theorems
(2.3) Incidence Axioms for Geometry
Undefined Terms: point, line, plane, (three-dimensional) space.
Definition (Basic Definitions). A line ` is a subset of a plane P if ` ∩ P = `. A line ` meets a plane
P if ` ∩ P is a single point. A line ` is parallel to a plane P if ` ∩ P = ∅.
Two lines are parallel to each other if they are both subsets of the same plane and do not intersect each
other. If the intersection of two lines is a single point, then those two lines meet at that point.
Two or more lines or planes which meet at the same point are called concurrent lines or planes. Two
points are said to be collinear if they both lie on the same line. A set of points are said to determine
a line if there is a unique line passing containing those points. A set of points are said to determine a
plane if there is a unique plane containing those points.
Axiom (I-1). Each two distinct points determine a line.
←→
←→
←→ ←→
Theorem (1). If C ∈ AB and D ∈ AB and C 6= D then AB = CD.
Axiom (I-2). Three non-collinear points determine a plane.
Axiom (I-3). If two points lie in a plane, then any line containing those two points lies in that plane.
Axiom (I-4). If two distinct planes meet, their intersection is a line.
Axiom (I-5). Space consists of at least four noncoplanar points, and contains three noncollinear points.
Each plane is a set of points of which at least three are noncollinear, and each line is a set of at least
two distinct points.
Theorem (2). If two distinct lines ` and m meet, their intersection is a single point. If a line meets a
plane and is not contained by that plane, their intersection is a point.
(2.4) Distance, Ruler Postulate, Segments, Rays, and Angles
Metric Axioms
Axiom (D-1). Each pair of points A and B is associated with a unique real number, called the distance
from A to B, denoted by AB.
Axiom (D-2). For all points A and B, AB ≥ 0, with equality only when A = B.
Axiom (D-3). For all points A and B, AB = BA.
Other Results
Definition (Definition of Betweenness). For any three points A, B and C, we say that B is between
A and C and we write A − B − C, iff A, B, and C are distinct, collinear points, and AB + BC = AC.
Theorem (1). If A − B − C then C − B − A and neither A − C − B nor B − A − C.
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Definition. If A, B, C, and D are four distinct collinear points, then we write A − B − C − D iff the
composite of all four betweenness relations A − B − C, B − C − D, A − B − D and A − C − D are true.
Theorem (2). If A − B − C, B − C − D, and A − B − D hold, then A − B − C − D is true.
Definition. Let A, B, and C be three distinct point coplanar points. Then:
• The segment from A to B is:
AB = {X : A − X − B, X = A or X = B}
• The ray from A in the direction of B is:
−→
AB = {X : A − X − B, A − B − X, X = A, or X = B}
Note that the definition of the ray depends on the order of A and B.
←→
• The unique line from A to B is written AB
• If A, B and C are not collinear, then the angle ∠ABC is defined as:
−→ −−→
∠ABC = BA ∪ BC
Note that the definition of the angle depends on the order of A, B and C.
−→
−→
←→
Definition. The extension of a segment AB is either the ray AB or the ray BA or the line AB. The
−→
←→
extension of the ray AB is the line AB.
Axiom (D-4: Ruler Postulate). The points of each line ` may be assigned to the entire set of real
numbers x, −∞ < x < ∞, called coordinates, in such a way that:
1. Each point on ` is assigned to a coordinate, and each real number has a point on ` assigned to it
as a coordinate.
2. No two points are assigned to the same coordinate.
3. Any two points on ` may be assigned to the coordinates zero and a positive real number, respectively.
4. If points A and B on ` have coordinates a and b, then AB = |a − b|.
The notation A[a] will be used to mean that the geometric point A has coordinate of a.
Theorem (3). For any line ` and any coordinate system under the ruler postulate, if A[a], B[b] and
C[c] are three points on line `, then A − B − C iff either a < b < c or c < b < a.
−→
−→ −→
Theorem (4). If C ∈ AB and A 6= C, then AB = AC.
Theorem (5: Segment Construction Theorem). If AB and CD are two segments and AB < CD, then
−−→
there exists a unique point E on ray CD such that AB = CE, and C − E − D.
(2.5) Angle Measure and the Protractor Postulate
Axiom (A-1: Existence of Angle Measure). Each angle ∠ABC is associated with a unique real number
between 0 and 180 (not including 0 or 180), called its measure and denoted m∠ABC.
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A Brief Interlude on Separation
Definition. A subset K of space is called convex provided it has the property that for all points A ∈ K
and B ∈ K, the segment joining A and B lies in K (that is, AB ⊂ K).
Axiom (H-1: Plane Separation Postulate). Let ` be any line lying in any plane P. The set of all points
in P not on ` consist of the union of two subsets H1 and H2 of P (that is P − ` = H1 ∪ H2 ) such that:
1. H1 and H2 are convex sets.
2. H1 and H2 have no points in common (H1 ∩ H2 = ∅).
3. If A lies in H1 and B lies in H2 , the line ` intersects the segment AB.
Definition. The sets H1 and H2 guaranteed by axiom (H-1) will be called the two sides of `. If A and
B are points not on `, then the side containing A will be called the A-side of ` (written as H(A, `)) and
the side containing B will be called the B-side of ` (written as H(B, `).)
Back to Angle Measure
←→
Definition. A point D is an interior point of ∠ABC it is on the C side of AB and on the A side of
←→
BC.
Axiom (A-2: Angle Addition Postulate). If D lies in the interior of ∠ABC, then m∠ABD+m∠DBC =
m∠ABC.
Theorem (Converse of A-2). If D lies in the interior of m∠ABD + m∠DBC = m∠ABC, then the
−−→
ray BD passes through an interior point of ∠ABC.
−→ −−→
−−→
Definition (Betweenness for rays). For any three rays BA, BD and BC (i.e. three rays with the same
−−→
−→
−−→
−→ −−→ −−→
end point), we say that the ray BD lies between rays BA and BC, and we write BA − BD − BC, iff
the rays are distinct and m∠ABD + m∠DBC = m∠ABC.
−−→
←→
−→
Axiom (A-3: Protractor Postulate). The sets of rays AX such that X is one of AB, or the ray is AB
may be assigned to the set of real numbers x, 0 ≤ x < 180, called coordinates, in such a way that:
1. Each ray is assigned to a unique coordinate.
2. Every number x, 0 ≤ x < 180 is assigned to a ray.
−→
3. The coordinate of AB is 0.
−→
−−→
4. If rays AC and AD have coordinates c and d, then m∠CAD = |c − d|.
Theorem (1: Angle Construction Theorem). For any two angles ∠ABC and ∠DEF such that m∠ABC <
−−→
−−→ −−→ −→
m∠DEF , there is a unique ray EG such that m∠ABC = m∠GEF and ED − EG − EF
Definition. Two rays are said to be opposite if they are distinct, have the same endpoint, and are
collinear.
Two angles are said to form a linear pair if they have one ray in common and the other rays are
opposites.
Any two angles whose angle measures sum to 180 are called a supplementary pair and any two angles
whose angle measures sum to 90 are called a complementary pair
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Theorem (2). Two angles that are supplementary, or complementary, to the same angle have equal
measures.
Axiom (A-4: Linear Pair Axiom). A linear pair of angles is a supplementary pair.
Definition. A right angle is any angle having measure 90. Two two distinct lines ` and m are said to
be perpendicular, and we write ` ⊥ m, iff they contain the sides of a right angles. Two subsets A and
B of the lines ` and m respectively are said to be perpendicular iff ` ⊥ m.
Definition. Two angles are said to be adjacent if they are distinct angles with a common ray and no
interior points in common.
Theorem (3). Suppose that D is a point on AC so that A − D − C and B is a point not on AC. Then
BD ⊥ AC iff the adjacent angles ∠ADB and ∠BDC have equal measures.
Theorem (4). Given a point A on line `, there exists a unique line m perpendicular to ` that contains
A.
Definition. Angles ∠AEB and ∠CED are said to be vertical iff:
−→
−−→
1. EA and ED are opposite rays.
−−→
−−→
2. EB and EC are opposite rays.
Theorem (5: Vertical Pair Theorem). Vertical angles have equal measures.
(3.1) Triangles and Congruence
Definition. If A, B, C are distinct and non-collinear, then the set:
4ABC = AB ∪ BC ∪ AC
is called a triangle. A, B and C are called the vertices of the triangle, AB, BC and AC are called the
sides of the triangle and ∠ABC, ∠BCA and ∠CAB are called the angles of the triangle. AB is the
opposite side to ∠BCA, BC is the opposite side to ∠CAB and AC is the opposite side to ∠ABC.
Definition. We say that AB ∼
= CD iff AB = CD. We say that ∠ABC ∼
= ∠DEF iff m∠ABC =
∼
∼
∼
∼
m∠DEF . We say that 4ABC = 4DEF iff AB = DE, BC = EF , AC = DF , ∠ABC ∼
= ∠DEF ,
∠BCA ∼
= ∠EF D and ∠CAB ∼
= ∠F DE.
(3.3) SAS, ASA, SSS Congruence and Perpendicular Bisectors
Axiom (C-1: SAS Congruence). If in triangles 4ABC and 4DEF , AB ∼
= DE, ∠ABC ∼
= ∠DEF
∼
∼
and BC = EF , then 4ABC = 4DEF .
Theorem (1: ASA Congruence). If in triangle 4ABC and 4DEF , ∠ABC ∼
= ∠DEF , BC ∼
= EF and
∼
∼
∠BCA = ∠EF D, then 4ABC = 4DEF .
Definition. A triangle is called isosceles if it contains two or more congruent sides.
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Theorem (2: Isosceles Triangle Theorem). If two sides of a triangle are congruent, then the opposite
angles to those sides are congruent. If two angles in a triangle are congruent, then the opposite sides to
those angles are congruent.
Definition. The perpendicular bisector to a line segment AB is the line ` so that ` ∩ AB = {X}, where
AX = XB and ` ⊥ AB.
Theorem (3: Perpendicular Bisector Theorem). The set of all points D so that AD = BD is the
perpendicular bisector to the line segment AB.
Theorem (4: SSS Congruence). If in triangles 4ABC and 4DEF , AB ∼
= DE, BC ∼
= EF and
∼
∼
AC = DF then 4ABC = 4DEF .
Theorem (5: Existence of Perpendicular to an External Point). If ` is a line and X is a point not on
`, then there is a unique line perpendicular to ` which is perpendicular to ` and contains X.
(3.4) Exterior Angle Inequality
Definition. If 4ABC is a triangle and D is a point so that B − C − D, then ∠BCD is called an
exterior angle of 4ABC and the angles ∠BAC, ∠ABC are called the opposite interior angles to the
exterior angle.
Theorem. The measure of any exterior angle is greater than the measure of any opposite interior angle.
Theorem. The sum of the measures of any two angles in a triangle is strictly less than 180.
Theorem (Saccheri-Legendre Theorem). The sum of the measures of any three angles in a triangles is
less than or equal to 180.
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