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Math 467
Modern Geometry
Fall 1999
Course Overview
Contents
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Axiomatic Systems . . . . . . . . . . . . . . . . .
Undefined Terms . . . . . . . . . . . . . . . . . .
Incidence . . . . . . . . . . . . . . . . . . . . . .
Distance . . . . . . . . . . . . . . . . . . . . . . .
Betweenness . . . . . . . . . . . . . . . . . . . .
Basic Geometric Figures . . . . . . . . . . . . . .
Congruence of Segments . . . . . . . . . . . . . .
Plane Separation . . . . . . . . . . . . . . . . . .
Angular Measure . . . . . . . . . . . . . . . . . .
Congruence of Angles . . . . . . . . . . . . . . .
Congruence of Triangles . . . . . . . . . . . . . .
Geometric Inequalities . . . . . . . . . . . . . . .
Real Numbers . . . . . . . . . . . . . . . . . . . .
Alternate Parallel Axioms . . . . . . . . . . . . . .
Basic Geometries . . . . . . . . . . . . . . . . . .
Absolute Plane Geometry . . . . . . . . . . . . . .
The Euclidean Parallel Axiom . . . . . . . . . . .
The Hyperbolic Parallel Axiom . . . . . . . . . . .
Continuity Principles and Geometric Constructions
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1 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
c 1999 by Jon T. Pitts
Copyright Page 1 of 15
Math 467
Modern Geometry
Fall 1999
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, and completeness.
• An axiomatic system which is not self-contradictory is said to be consistent.Q
• In an axiomatic system {A1 , . . . , An }, an axiom Ak is independent if it can not be
proven to be true from the other axioms {A1 , . . . , Ak−1 , Ak+1 , . . . , An }. 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 , and planes 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 P Q
←−→
If P , Q, and R are noncollinear points, then the plane containing them is denoted by P QR
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
c 1999 by Jon T. Pitts
Copyright Page 2 of 15
Math 467
Modern Geometry
Fall 1999
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) = 0 if 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 P Q.
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, we have P Q = |f (P ) −
f (Q)|, then f is 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, and C are 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, the C − 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
c 1999 by Jon T. Pitts
Copyright Page 3 of 15
Math 467
Modern Geometry
Fall 1999
(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, then A, 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 P Q.
←→
(2) The ray from P through Q is the set of all points R of the line P Q such that P is not
−→
between R and Q. This ray is denoted by P Q. The point P is called the endpoint of
−→
the ray P Q.
−→
−→
(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, and C are three noncollinear points, then the set AB ∪ BC ∪ CA is called
a triangle, denoted by the symbol 4ABC. The three segments AB ∪ BC ∪ CA are
called its sides, and the points A, B, and C are called its vertices.
(5) If A, B, C, D are four coplanar points no three of which are collinear, and if AB, BC,
CD, and DA intersect only at their endpoints, then their union is called a quadrilateral,
and is denoted by ABCD.
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 B1 and C1 are points of AB and AC, other than A, then \BAC = \B1 AC1 .
Theorem. If AB = CD, then the points A, B are the same as the points C, D in some
order.
Theorem. If 4ABC = 4DEF , then the points A, B, and C are the same as the points
D, E, and F , in some order.
Course Overview
c 1999 by Jon T. Pitts
Copyright Page 4 of 15
Math 467
Modern Geometry
Fall 1999
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, A0 − B 0 − C 0 , AB ∼
= A0 B 0 ,
and BC ∼
= B 0 C 0 , then AC ∼
= A0 C 0 .
Theorem (C-4. Segment-Subtraction Theorem). If A−B−C, A0 −B 0 −C 0 , AB ∼
= A0 B 0 ,
0
0
0
0
∼
∼
and AC = A C , then BC = B C .
Theorem (C-5). Every segment has exactly one midpoint.
8 Plane Separation
Definition (Convex Set). A set A is convex if for every two points P , Q of A, the entire
segment P Q is contained in A.
Axiom (PS. Plane Separation). Given a line L in a plane E, the set E − L is the union of
two sets H1 and H2 such that:
(1) H1 and H2 are convex, and
(2) if P ∈ H1 and Q ∈ H2 , then P Q intersects L.
Definition (Half Planes and Edges). Each of the sets H1 and H2 is a half plane of E
determined by L. L is the edge of H1 and H2 .
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, P T is a subset of H ∪ L and P is the only point of P T 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
c 1999 by Jon T. Pitts
Copyright Page 5 of 15
Math 467
Modern Geometry
Fall 1999
Definition (Interior of a Triangle). The interior of 4ABC 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 4ABC and a line L in the same plane, if L contains a point between A and C, then L intersects either AB or BC or both.
Theorem (Clubsuit (♣)). Given 4ABC 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 4ABC, let D be 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\P AB = r.
Axiom (M-3. Angle Addition). Let \BAC ∈ A. If D ∈ Int(\BAC), then m\BAC =
m\BAD + m\DAC.
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c 1999 by Jon T. Pitts
Copyright Page 6 of 15
Math 467
Modern Geometry
Fall 1999
−→
−→
−→
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 the two angles
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 0 C 0 is a ray, and H
−−→
−−→
is a half plane whose edge contains B 0 C 0 . Then there is exactly one ray B 0 C 0 , with A0 in
H such that \ABC ∼
= \A0 B 0 C 0 .
Theorem (C-8. Angle Addition). Let D be in the interior of \BAC, D 0 be in the interior
of \B 0 A0 C 0 , \BAD ∼
= \B 0 A0 D 0 , and \DAC ∼
= \D 0A0 C 0 . Then \BAC ∼
= \B 0 A0 C 0 .
Theorem (C-9. Angle Subtraction). Let D be in the interior of \BAC, D0 be in the
interior of \B 0 A0 C 0 , \BAD ∼
= \B 0 A0 D 0 , and \BAC ∼
= \B 0 A0 C 0 . Then \DAC ∼
=
0 0 0
\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
c 1999 by Jon T. Pitts
Copyright Page 7 of 15
Math 467
Modern Geometry
Fall 1999
11 Congruence of Triangles
Definition (Congruence of Triangles). A congruence from 4ABC to 4DEF is a oneto-one correspondence ABC ↔ DEF from the vertices of 4ABC onto the vertices of
4DEF such that
AB ∼
= DE
BC ∼
= EF
CA ∼
= FD
\A ∼= \D
\B ∼= \E
\C ∼= \F.
In this case we write 4ABC ∼
= 4DEF . 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
c 1999 by Jon T. Pitts
Copyright Page 8 of 15
Math 467
Modern Geometry
Fall 1999
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 0 A0 C 0 , then we say that \BAC
is smaller than \B 0 A0 C 0 , and we write \BAC < \B 0 A0 C 0 .
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, then AB + BC ≥ AC.
Equality holds if and only if either A − B − C, or B coincides 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
c 1999 by Jon T. Pitts
Copyright Page 9 of 15
Math 467
Modern Geometry
Fall 1999
13 Real Numbers
Axiom (Completeness). Every nonempty subset of R which is bounded above has a least
upper bound.
Theorem (Alternate Completeness). Every
bounded below has a greatest lower bound.
nonempty
subset
of
R
which
is
Theorem (Archimedian Principle). For any two positive numbers M and , there exists
a positive integer N such that N > 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 L0 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 L0 and L00 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
c 1999 by Jon T. Pitts
Copyright Page 10 of 15
Math 467
Modern Geometry
Fall 1999
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 a geometry which satisfies 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 L1 , L2 , and T are three lines in the same plane, and T
intersects L1 and L2 in two different points P and Q, respectively, then T is a transversal
to L2 and L2 .
Definition (Alternate Interior Angles and Corresponding Angles).
(1) If T is a transversal to L1 and L2 , intersecting L1 and L2 in P and Q, respectively, and
if A and D are points of L1 and L2 , respectively, lying on opposite sides of T , then
\AP Q and \P QD 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
c 1999 by Jon T. Pitts
Copyright Page 11 of 15
Math 467
Modern Geometry
Fall 1999
Remark. No parallel axiom is required for any part of the previous theorem.
Theorem (Polygonal Inequality). If A1 , A2 , . . . , An are any points (n > 1), then
A1 A2 + A2 A3 + . . . An−1 An ≥ A1 An .
Definition (Rectangles and Saccheri Quadrilaterals).
(1) A rectangle is a quadrilateral whose interior angles are right angles.
(2) A quadrilateral ABCD is Saccheri quadrilateral if \A and \D are right angles, B
and C are on the same side of AD, and AB = 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 ABCD (with lower base AD), we have m\BDC ≥
m\ABD.
Corollary.
(1) If 4ABC has a right angle at A, then m\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 4ABC, let D be the foot of the perpendicular from B to AC. If AC is the longest
side of 4ABC, the A − D − C.
Theorem (Sum of Angles of a Triangle). In any triangle ABC, we have
m\A + m\B + m\C ≤ 180.
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c 1999 by Jon T. Pitts
Copyright Page 12 of 15
Math 467
Modern Geometry
Fall 1999
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.
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c 1999 by Jon T. Pitts
Copyright Page 13 of 15
Math 467
Modern Geometry
Fall 1999
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 > 0 is the set
of all points P in E such that CP = r. The interior (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 C1 and C2 in the same plane, if
circle C1 has one point in the interior and one point in the exterior of circle C2 , then the
circles intersect in exactly two points.
Theorem (Straightedge and Compass Constructions).
(1) Copy a segment.
(2) Copy an angle.
Course Overview
c 1999 by Jon T. Pitts
Copyright Page 14 of 15
Math 467
Modern Geometry
Fall 1999
(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
c 1999 by Jon T. Pitts
Copyright Page 15 of 15