Download Summary of Introductory Geometry Terminology

MAT104: Fundamentals of Mathematics II
Introductory Geometry Terminology Summary
Section 11-1: Basic Notions
Undefined Terms:
Point; Line; Plane
Collinear Points: points that lie on the same line
Between[-ness]:
exactly what the term implies with regard to points
[Line] Segment: part of a line consisting of (and named by) two endpoints and all points between them
Closed Segment:
segment with two included endpoints
Half-Open Segment: segment with one included endpoint (also called Half-Closed Segment)
Open Segment:
segment with no included endpoints
Ray:
part of a line consisting of one endpoint, a second unique point on the line, all points between
them, and all points that have the aforementioned second point between them and the endpoint
Half-Line: part of a line on one side of a point on the line (does not include the point)
Coplanar: lying on the same plane (points, lines, or parts of lines)
Noncoplanar: not coplanar
Intersecting Lines:
Skew Lines:
lines with exactly one point in common
lines that cannot be contained in a single plane
Concurrent Lines: three or more lines that intersect in the same point
Coinciding Lines: identical lines; lines intersecting at every point
Parallel Lines:
either: a) nonintersecting coplanar lines; or b) two lines that are actually the same
line (any line is parallel to itself; coinciding lines as defined above); symbol is
Intersecting Planes:
planes with exactly one line in common
Coinciding Planes:
identical planes; planes intersecting at every line
Parallel Planes:
either: a) nonintersecting planes; or b) two planes that are actually the same plane
(any plane is parallel to itself; coinciding planes as defined above)
Parallel Line and Plane: a line and plane that either: a) do not intersect in any point; or b) intersect in
every point on the line
Half-Plane:
part of a plane on one side of a line in the plane (does not include the line)
Prof. Fowler
Axiom:
a statement universally known to be factual but not able to be proven
Axiom 1:
Axiom 2:
Axiom 3:
Axiom 4:
Axiom 5:
Axiom 6:
Axiom 7:
There is exactly one line that contains any two distinct points.
If two points lie in a plane, then the line containing the points lies in the plane.
There is exactly one plane that contains any three distinct noncollinear points.
If two distinct planes intersect, then their intersection is a line.
A line and a point not on the line determine a unique plane.
Two distinct parallel lines determine a unique plane.
Two distinct intersecting lines determine a unique plane.
Theorem: statement proven true based on given information, undefined terms, definitions, axioms, and
previously proven theorems
Angle: the union of two rays with a common endpoint and all points between the rays
Sides of an Angle:
the rays that form the angle
Vertex of an Angle:
the common endpoint of the rays that form the angle
Interior of an Angle: all points between the rays that form the angle
Adjacent Angles: angles with a common vertex, a common side, and nonoverlapping interiors
Protractor:
tool used to measure an angle
Degree:
unit used to measure angles; 1/360 of a circle; symbol is
Minute:
1/60 of a degree; symbol is
(
Second:
1/60 of a minute; symbol is
(
Zero Angle:
)
)
angle measuring
; formed by coinciding rays (rays with the same endpoint and going
in the same direction, thus forming a single ray)
Acute Angle: angle measuring between
and
Right Angle: angle measuring exactly
Obtuse Angle:
angle measure between
and
Straight Angle:
angle measuring exactly
; formed by opposite rays (rays with the same
endpoint and going in opposite directions, thus forming a line)
Reflex Angle:
angle measuring between
and
Complementary Angles: degree measures add up to
Supplementary Angles:
degree measures add up to
Prof. Fowler
Circle: set of all points equidistant from a given point called the center (does not include the interior)
Arc:
a continuous part of a circle
Chord:
a segment with both endpoints on a circle
Diameter: 1) a chord containing the center of the circle in which it is constructed
2) the length of the chord containing the center of the circle in which it is constructed
Central Angle:
an angle with its vertex at the center of a circle (Note: the measure of an arc is equal
to the measure of the central angle by which it is determined.)
Minor Arc:
an arc measuring less than
Major Arc:
an arc measuring between
Semicircle:
an arc with its endpoints coinciding with the endpoints of a diameter of the circle
Perpendicular Lines: lines intersecting at a
Line Perpendicular to a Plane:
Dihedral Angle:
and
angle; symbol is
a line not in the plane that is perpendicular to every line in the plane
passing through its point of intersection with the plane
the [smallest] angle formed between two planes
Perpendicular Planes: planes with a dihedral angle measuring
Prof. Fowler
Section 11-3: Curves, Polygons, and Symmetry
Connected Drawing: a figure that can be constructed without lifting the writing implement
Curve: any connected drawing; can contain or consist exclusively of straight segments
Simple Curve: a curve that does not intersect itself
Closed Curve: a curve with the same starting and stopping point
Polygon: a simple closed curve consisting of only line segments called its sides (does not include the
interior); named by the number of sides as listed below
Henagon:
Digon:
Triangle:
Quadrilateral:
Pentagon:
Hexagon:
Heptagon:
Octagon:
Nonagon:
Decagon:
Hendecagon:
Dodecagon:
Icosagon:
Hectogon:
Chiliagon:
Myriagon:
Megagon:
Googolgon:
Apeirogon:
n-gon:
1 side (also called a Monogon; degenerate case in Euclidean Geometry: a point)
2 sides (degenerate case in Euclidean Geometry: a line segment)
3 sides (also called a Trigon)
4 sides (also called a Tetragon)
5 sides
6 sides
7 sides (also called a Septagon)
8 sides
9 sides (also called an Enneagon)
10 sides
11 sides (also called an Undecagon)
12 sides
20 sides
100 sides
1000 sides
10,000 sides
1,000,000 sides
googol sides
infinite number of sides (degenerate case in Euclidean Geometry: a circle)
n sides; replace n with the actual number of sides (e.g., 15-gon)
Convex Curve:
a simple closed curve such that the segment connecting any two points in its interior
is completely contained in its interior
Concave Curve:
a simple closed curve that is not convex
Polygonal Region:
the union of a polygon and its interior
Vertex of a Polygon: the common endpoint of any two consecutive sides of the polygon
[Interior] Angle of a Polygon:
an angle formed by consecutive sides of a polygon with its interior
being inside the polygon
Exterior Angle of a Polygon: an angle formed by one side of a polygon and the extension of a
contiguous side of the polygon with its interior being outside the polygon
Prof. Fowler
Diagonal: any segment connecting nonconsecutive vertices of a polygon
Congruent Figures:
figures with the same size (measure) and shape
Equiangular Polygons:
polygons with congruent interior angles in order
Equilateral Polygons: polygons with congruent sides in order
Regular Polygon: a polygon with all sides congruent in order and all interior angles congruent in order
Right Triangle:
a triangle with one right angle
Acute Triangle:
a triangle with all three angles being acute angles
Obtuse Triangle: a triangle with one obtuse angle
Scalene Triangle: a triangle with no sides (or angles) congruent
Isosceles Triangle:
a triangle with [at least] two sides congruent
Equilateral Triangle: a triangle with all three sides (and angles) congruent
Trapezoid: a quadrilateral with [at least] one pair of parallel sides
Kite:
a quadrilateral with two distinct pairs of adjacent sides congruent
Isosceles Trapezoid: a trapezoid with congruent base angles (angles formed at opposite ends of one of
the parallel sides)
Parallelogram: a quadrilateral with two pairs of parallel sides
Rectangle: a quadrilateral with four right angles
Rhombus: a quadrilateral with all four sides congruent
Square:
a quadrilateral with four right angles and all four sides congruent
Hierarchy among Quadrilaterals: Page 669
Line of Symmetry:
Rotational Symmetry:
a line through which a figure is its own image when folded along it (also called a
reflecting line)
property of a figure indicated by the ability to rotate the figure less than
about its center such that it coincides with itself (also called Turn Symmetry)
(note: the smallest degree measure through which the figure can be turned to
coincide with itself is the properly reported measure of the rotational
symmetry of the figure, although multiple degree measures may exist)
Point Symmetry: property of a figure indicated by the possession of
rotational symmetry
Prof. Fowler