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Foundations of Geometry
Section 1.1 The basic geometric objects: point, line, plane.
Object
point
line
plane
Symbols
A, B, C, D, ...
AB , CD , l, k, m
ABC , ABD , N , M
Equality
A sB sC sD
l = AB s CD
ABC = ABD = N
Congruence
A yB yC yD
l y AB y CD
ABC y ABD y N
Picture
Properties
• two geometric objects are congruent just in case they are geometrically equivalent as objects, i.e. one
can be obtained from the other by rigid motions using only translations and rotations, but not using
stretches or shrinks
• every geometric object is geometrically equivalent (congruent) to itself
• all points are geometrically equivalent (congruent) but not necessarily equal
• all lines are geometrically equivalent (congruent) but not necessarily equal
• all planes are geometrically equivalent (congruent) but not necessarily equal
• two distinct points determine a unique line
• three noncollinear points determine a unique plane
Examples: Intersections of Geometric Objects
Line l = AC intersects line
m = BD in point B.
Line AE intersects plane N in
point B.
Planes P , O , and N all intersect
in line AB .
Section 1.2 More geometric objects: (line) segment, ray, angle.
Object
segment
___ ___ ___
AB , DA , BC
ray
angle
AB , BA , BC , DC
:ADE,:AEC,:EBC
Equality
___ ___ ___
AB s DA s BC
AB s BA s BC
:ADE s:AEC s
:EBC
Congruence
___ ___ ___
AB “ DA “ BC
AB y BA y BC
:ADE “:AEC “
:EBC
Symbols
Picture
Properties
• segments have two (finite) endpoints and are not in general congruent, since the have a measure
• rays have one (finite) endpoint and one infinite extent
• all rays are geometrically equivalent (congruent) but not necessarily equal
• an angle is the union of two noncollinear rays with the same endpoint
• angles are not in general congruent, since they have a measure
• two distinct points determine a unique seqment
• two distinct points with choice of endpoint determine a unique ray
Examples: Types of Angles
:ABD is obtuse, :DBC is
acute, together they are
supplementary and also a linear
pair, B is the vertex of both
angles
:GCH and :BCD are vertical
angles, and so are :CBE and
:ABF, CH and CF are
opposite rays
:EBF and :FBC are adjacent
and complementary, :EBF and
:EBD are adjacent
Section 1.3 More geometric objects: triangles, quadrilaterals, polygons.
Object
triangle
quadrilateral
polygon
Symbols
6ABC, 6NPM, 6TQU
ABCD, PQRS, MNTU
P1 P2 P3 ... Pn
Picture
Equality
6ABC =6ABC s
6NPM
PQRS = PQRS s MNTU
P1 P2 ... Pn = P1 P2 ... Pn
Congruence
6ABC y6ABC “
6NPM
PQRS y PQRS “ MNTU
P1 P2 ... Pn y P1 P2 ... Pn
Properties
• every polygon has at least three sides and is a closed figure in the plane
• a triangle (3-gon) is the simplest polygon and next simplest is a quadrilateral (4-gon)
• no three vertices of a polygon are collinear
• the segment connecting any two interior points of a convex polygon is completely contained within
the interior of the polygon
• a polygon is concave just in case it is not convex, i.e. it has a "dent"
• a regular polygon has all sides congruent and all angles congruent
Examples: Types of Polygons
a convex, regular polygon, a
hexagon or 6-gon
a concave octagon, an 8-gon, not a concave polygon, an n-gon, not
regular, and not convex
regular, and not convex