MAT104: Fundamentals of Mathematics II Introductory Terminology Summary

Section 11-1: Basic Notions

Undefined Terms: Point; Line;

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: lines with exactly one point in common

Skew Lines: 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: There is exactly one line that contains any two distinct points. Axiom 2: If two points lie in a plane, then the line containing the points lies in the plane. Axiom 3: There is exactly one plane that contains any three distinct noncollinear points. Axiom 4: If two distinct planes intersect, then their intersection is a line. Axiom 5: A line and a point not on the line determine a unique plane. Axiom 6: Two distinct parallel lines determine a unique plane. Axiom 7: 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 , a common side, and nonoverlapping interiors

Protractor: tool used to measure an angle

Degree: unit used to measure angles; 1/360 of a ; 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 and

Semicircle: an arc with its endpoints coinciding with the endpoints of a diameter of the circle

Perpendicular Lines: lines intersecting at a angle; symbol is

Line Perpendicular to a Plane: a line not in the plane that is perpendicular to every line in the plane passing through its point of intersection with the plane

Dihedral Angle: the [smallest] angle formed between two planes

Perpendicular Planes: planes with a dihedral angle measuring

Prof. Fowler Section 11-3: Curves, , and

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: 1 side (also called a ; degenerate case in : a point) : 2 sides (degenerate case in Euclidean Geometry: a ) : 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 Googolgon: googol sides : infinite number of sides (degenerate case in Euclidean Geometry: a circle) n-gon: 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 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 : 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 : Page 669

Line of Symmetry: a line through which a figure is its own image when folded along it (also called a reflecting line)

Rotational Symmetry: 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 of the figure, although multiple degree measures may exist)

Point Symmetry: property of a figure indicated by the possession of rotational symmetry

Prof. Fowler