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: 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: Coinciding Planes: planes with exactly one line in common 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)
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: Vertex of an Angle: the rays that form the 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: Degree: tool used to measure an angle 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
Circle: set of all points equidistant from a given point called the center (does not include the interior) Arc: Chord: a continuous part of a circle 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: Minor Arc: 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.) 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
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: 1 side (also called a Monogon; degenerate case in Euclidean Geometry: a point) Digon: 2 sides (degenerate case in Euclidean Geometry: a line segment) Triangle: 3 sides (also called a Trigon) Quadrilateral: 4 sides (also called a Tetragon) Pentagon: 5 sides Hexagon: 6 sides Heptagon: 7 sides (also called a Septagon) Octagon: 8 sides Nonagon: 9 sides (also called an Enneagon) Decagon: 10 sides Hendecagon: 11 sides (also called an Undecagon) Dodecagon: 12 sides Icosagon: 20 sides Hectogon: 100 sides Chiliagon: 1000 sides Myriagon: 10,000 sides Megagon: 1,000,000 sides Googolgon: googol sides Apeirogon: 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: Concave Curve: a simple closed curve such that the segment connecting any two points in its interior is completely contained in its interior 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
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: Acute Triangle: a triangle with one right angle 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: 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 rotational symmetry of the figure, although multiple degree measures may exist) Point Symmetry: property of a figure indicated by the possession of rotational symmetry