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Glossary— Geometry Section 1.1 Term / Definition Chapter One Picture **Undefined Terms – Terms whose general meaning is assumed and whose characteristics are understood only from the postulates or axioms that use them. 1.1 **Point (undefined term) – A location with no size or dimension. It is the most basic building block of geometry. In a two dimensional coordinate system, a point’s location is represented by an ordered pair of numbers (x, y). 1.1 **Line (undefined term) – A straight, continuous arrangement of infinitely many points extending forever in two directions. A line has length, but no width or thickness, so it is one-dimensional. 1.1 **Plane (undefined term) – A flat surface that extends infinitely along its edges. A plane has length and width but no thickness, so it is two-dimensional. 1.1 Definition – A statement that clarifies or explains the meaning of a word or phrase. 1.1 Collinear – On the same line. 1.1 Coplanar – In the same plane. 1.1 Line Segment – Two points and all the points between them that are collinear with the two points. Also called a segment. The measure of a line segment is its length. 1.1 Endpoint – The point at either end of a segment or an arc, or the first point of a ray 1.1 Congruent Segments – Line segments that are identical in shape and size. 1.1 Bisect (segment) – To divide a segment into two congruent segments. 1 Section Term / Definition 1.1 Midpoint – The point on the line segment that is the same distance Picture from both endpoints. The midpoint bisects the segment. 1.1 Ray – A point on a line, and all the points of the line that lie on one side of this point. 1.2 Angle (vertex and sides) – Two noncollinear rays (sides) having a common endpoint (vertex). 1.2 Measure of an Angle –The smallest amount of rotation about the vertex from one ray to the other, measured in degrees. 1.2 Degree – A unit of measure for angles and arcs equivalent to 1/360 of a rotation around a circle. 1.2 Reflex Measure – The largest amount of rotation less that 360° about the vertex from one ray to the other, measured in degrees. 1.2 Protractor – A tool used to measure the size of an angle in degrees. 1.2 Congruent Angles – Angles that are identical in shape and size. 1.2 Angle Bisector – A ray that has its endpoint at the vertex of the angle and that divides the angle into two congruent angles. 1.2 Incoming Angle – The angle formed between the path of an approaching object and the surface from which it rebounds, such as a billiard ball rolling toward a cushion or a ray of light traveling toward a mirror. 1.2 Outgoing Angle – The angle formed between the path of a rebounding object and the surface it collided with, such as a billiard ball bouncing off a cushion or a ray of light reflecting off a mirror. 1.2 Adjacent Angles – Two non-overlapping angles with a common vertex and one common side. 2 Section 1.3 Term / Definition Picture Counterexample – An example that shows a conjecture to be incorrect or a definition to be inadequate. 1.3 Classify and Differentiate – Defining a term by categorizing it, then distinguishing it from other members of the same group, according to choose characteristics. 1.3 Perpendicular – Lines are perpendicular if they meet at 90 degree angles. Line segments and rays are perpendicular if they lie on perpendicular lines. 1.3 Parallel – Planes are parallel if they do not intersect. Figures are parallel if they lie in parallel planes. 1.3 Skew Lines – Lines that are not in the same plane and do not intersect. 1.3 Right Angle – An angle that measures 90 degrees. 1.3 Acute Angle – An angle that measures less than 90 degrees. 1.3 Obtuse Angle – An angle that measures between 90 degrees and 180 degrees. 1.3 Complimentary Angles – A pair of complementary angles has a sum of 90 degrees. 1.3 Supplementary Angles – A pair of supplementary angles has a sum of 180 degrees. 1.3 Vertical Angles – Angles formed by two intersecting lines, they share a common vertex but not a common side. 1.3 Linear Pair – Two angles are a linear pair if they share a vertex and a common side and their noncommon sides form a line. 3 Section Term / Definition 1.4 Polygon (vertex and sides) – A closed figure in a plane, formed Picture by connecting line segments endpoint to endpoint with each segment intersecting exactly two others. 1.4 Triangle – A polygon with 3 sides. 1.4 Quadrilateral – A polygon with 4 sides. 1.4 Pentagon – A polygon with 5 sides 1.4 Hexagon – A polygon with 6 sides. 1.4 Heptagon – A polygon with 7 sides. 1.4 Octagon – A polygon with 8 sides. 1.4 Nonagon – A polygon with 9 sides. 1.4 Decagon – A polygon with 10 sides. 1.4 Undecagon – A polygon with 11 sides. 1.4 Dodecagon – A polygon with 12 sides. 1.4 n-gon – A polygon with n sides. 4 Section 1.4 Term / Definition Picture Consecutive (angles, sides, or vertices of a polygon) – Two angles that share a common side, two sides that share a common vertex, or two vertices that are the endpoints of one side. Consecutive sides are also called adjacent sides. 1.4 Diagonal – A line segment connecting two nonconsecutive vertices of a polygon or polyhedron. 1.4 Convex Polygon – A polygon with no diagonal outside the polygon. 1.4 Concave Polygon – A polygon with at least one diagonal outside the polygon. 1.4 Congruent Polygons – Polygons that are identical in shape and size. 1.4 Perimeter – The length of the boundary of a two-dimensional figure. For a polygon, the perimeter is the sum of the lengths of its sides. 1.4 Equilateral Polygon – Polygons in which all sides have equal lengths. 1.4 Equiangular Polygon – Polygons in which all angles have equal measures. 1.4 Regular Polygon – Polygons that are both equilateral and equiangular. 1.5 Assume – To accept as true without facts or proof. 1.5 Right Triangle – A right triangle has one right angle. 1.5 Acute Triangle – An acute triangle has three acute angles. 5 Section Term / Definition 1.5 Obtuse Triangle – An obtuse triangle has one obtuse angle. 1.5 Scalene Triangle – A triangle with no congruent sides. 1.5 Equilateral Triangle – A triangle with three congruent sides. 1.5 Isosceles Triangle (vertex angle, base angles, and base) – 1.6 1.6 Picture A triangle with at least two congruent sides. The angle between the two sides of equal length is called the vertex angle. The side opposite the vertex angle is called the base. The two angles opposite the two sides of equal length are called the base angles. Trapezoid – A trapezoid is a quadrilateral with exactly one pair of parallel sides. Kite – A quadrilateral with two distinct pairs of consecutive congruent sides. 1.6 Parallelogram – A quadrilateral with two pairs of parallel sides. 1.6 Rhombus - An equilateral parallelogram. 1.6 Rectangle – An equiangular parallelogram. (Each of the 4 angles measures 90 degrees.) 1.6 Square – An equilateral rectangle. An equiangular rhombus. A regular quadrilateral. 1.7 Circle (and its center) – The set of all points in a plane at a given distance from a given point (called the center). 1.7 Radius – A line segment from the center of a circle or sphere to a point on the circle or sphere. Also, the length of that line segment. 6 Section Term / Definition 1.7 Chord – A line segment whose endpoints lie on the circle. 1.7 Diameter – A chord that passes through the center. A diameter is Picture the longest chord of a given circle. 1.7 Tangent Line (and its point of tangency) – A line that intersects a circle only once. The point of intersection of a tangent line and a circle is called the point of tangency. 1.7 Congruent Circles – Circles that are identical in shape and size. 1.7 Concentric Circles – Circles that share the same center. 1.7 Arc (of a circle) – Two points on a circle and the continuous part of the circle between them. 1.7 Semicircle – An arc of a circle whose endpoints are the endpoints of a diameter. 1.7 Minor Arc – An arc of a circle that is less than a semicircle. 1.7 Major Arc – An arc of a circle that is greater than a semicircle. 1.7 Arc Measure – The measure of the central angle that intercepts an arc, measured in degrees. 1.7 Central Angle – An angle whose vertex is the center of a circle and whose sides pass through the endpoints of an arc. 1.8 **Space (undefined term) – Thought of as the set of all points. Space extends infinitely in all directions, so it is three-dimensional. 7 Section Term / Definition 1.8 Isometric Drawing – A drawing of a three-dimensional object that Picture shows three faces in one view. Also called an edge view. 1.8 Solid with Examples of Solids – A three dimensional geometric figure that completely encloses a region of space. Examples: cone, cylinder, hemisphere, polyhedron, prism, pyramid, and sphere 1.8 Net – A two-dimensional pattern that can be folded to form a threedimensional figure. 1.8 Section – The intersection of a solid and a plane. Chapter Two 2.1 Inductive Reasoning - The process of observing data, recognizing patterns, and making generalizations about those patterns. 2.1 Conjecture - A generalization resulting from inductive reasoning. 2.1 Rectangular Numbers - Numbers that can be represented by a rectangular array of dots or squares. 2.2 Function Rule - A rule that, when applied to one set of numbers, generates another set of numbers with at most one output value for each input value. 2.2 Nth Term - The number that a function rule generates as output for a counting number n. 2.2 Function Notation - A convention for expressing a function rule in terms of its input. For example, f (x) is the output of function f whose input is x. 2.2 Linear Function - A function rule that, when applied to consecutive whole numbers, generates a sequence with a constant difference between consecutive terms. 8 Section Term / Definition 2.2 Line of Best Fit - The line that passes closest to all of a given collection of points. 2.4 Deductive Reasoning The process of showing that certain statements follow logically from agreed upon assumptions and proven facts. 2.5 Converse The statement formed by exchanging the “if” (the antecedent) and the “then” (the consequent) of a conditional statement. 2.5 Biconditional Statement – a statement that includes both a conditional statement and its converse, usually written in “if and only if” form 2.6 Transversal - A line that intersects two or more other coplanar lines. 2.6 Corresponding Angles - Two angles formed by a transversal intersecting two lines that lie in the same position relative to the two lines and the transversal. 2.6 Alternate Interior Angles - A pair of angles, formed by a transversal intersecting two lines, that lie between the two lines and are on opposite sides of the transversal. 2.6 Alternate Exterior Angles - A pair of angles, formed by a transversal intersecting two lines, that do not lie between the two lines and are on opposite sides of the transversal. 2.6 Consecutive Interior Angles - A pair of angles, formed by a transversal intersecting two lines, that lie between the two lines and are next to one another. 2.6 Consecutive Exterior Angles - A pair of angles, formed by a transversal intersecting two lines, that do not lie between the two lines and are next to one another. 3.2 Picture Chapter Three Segment Bisector - A line, ray, or segment that passes through the midpoint of a line segment in a plane. 9 Section Term / Definition 3.2 Perpendicular Bisector - A line that divides a line segment into two congruent parts and is perpendicular to the line segment. 3.2 Coincide - To lie exactly on top of each other. 3.2 Equidistant - Equally distant. 3.2 Median - A line segment connecting a vertex of a triangle to the midpoint of the opposite side. 3.2 Midsegment - A line segment connecting the midpoints of two sides. 3.3 Distance - (from a point to a line or plane) The length of the perpendicular line segment from the point to the line or plane. 3.3 Altitude (of a triangle) - A perpendicular segment from a vertex to the base or to the line containing the base. 3.3 Height - The length of an altitude. 3.4 Angle Bisector - (of a triangle) A segment that lies on an angle bisector and that has one endpoint at the vertex and the other on the opposite side of the triangle. 3.7 Concurrent Lines Two or more lines that intersect in a single point. 3.7 Point of Concurrency - The point at which two or more lines, line segments, or rays intersect. 3.7 Incenter - The point of concurrency of a triangle’s three angle bisectors. Picture 10 Section Term / Definition 3.7 Circumcenter - The point of concurrency of a triangle’s three perpendicular bisectors. 3.7 Orthocenter - The point of concurrency of a triangle’s three altitudes (or of the lines containing the altitudes). 3.7 Circumscribed - (about a circle) Having all sides tangent to the circle, such as a triangle circumscribed about a circle. (about a polygon) Passing through each vertex of the polygon, such as a circle circumscribed about a triangle. Inscribed - (in a polygon) Intersecting each side of the polygon exactly once, such as a circle inscribed in a triangle. (in a circle) Having each vertex on the circle, such as a triangle inscribed in a circle. Centroid - The point of concurrency of a triangle’s three medians. 3.7 3.8 3.8 Center of Gravity - The balancing point of an object. 4.1 Chapter Four Paragraph Proofs - A logical explanation presented in the form of a paragraph. 4.1 Auxiliary Line - An extra line or line segment drawn in a figure to help in a proof. 4.2 Leg (of an isosceles triangle) - One of the congruent sides of an isosceles triangle. 4.3 Exterior Angle - An angle that forms a linear pair with one of the interior angles of a polygon. 4.3 Adjacent Interior Angle - The angle of a polygon that forms a linear pair with a given exterior angle of a polygon. Picture 11 Section Term / Definition 4.3 Remote Interior Angles - (of a triangle) The interior angles of a triangle that do not share a vertex with a given exterior angle. 4.4 Included Angle - An angle formed between two consecutive sides of a polygon. 4.5 Included Side - A side of a polygon between two consecutive angles. 4.7 Flowchart Proof - A logical argument presented in the form of a flowchart. 5.3 Picture Chapter Five Kite (vertex angles and non-vertex angles) A quadrilateral with exactly two distinct pairs of congruent consecutive sides. Vertex angles - The angles between the pairs of congruent sides. Nonvertex angles - The two angles between consecutive noncongruent sides of a kite. 5.3 Trapezoid (bases and base angles) - A quadrilateral with exactly one pair of parallel sides. Bases - The two parallel sides. Base angles - A pair of angles with a base of the trapezoid as a common side. 5.3 Isosceles Trapezoid- A trapezoid whose two nonparallel sides are congruent. 5.4 Midsegment (of a trapezoid)- The line segment connecting the midpoints of the two nonparallel sides 5.5 Vector- A quantity that has both magnitude and direction. A vector is represented by an arrow whose length and direction represent the magnitude and direction of the vector. 5.5 Resultant Vector- The result of adding two vectors. Also called a vector sum. 12 Section 5.7 Term / Definition Picture Dart - A concave kite. Chapter Ten (Probability) 10.1 Random Processes - 10.1 Fair - 10.1 Random Numbers - 10.1 Simulation - 10.1 Outcomes - 10.1 Event - 10.1 Simple Event - 10.1 Compound - 10.1 Experimental Probabilities - 10.1 Theoretical Probability - 13 Section Term / Definition 10.1 Geometric Probability 10.2 Tree Diagram 10.2 Independent 10.2 Dependent Events 10.2 Conditional Probability 10.3 Mutually Exclusive 10.3 Intersection 10.3 Union 10.3 Complements 10.4 Expected Value 10.4 Random Value 10.4 Discrete Random Variables Picture 14 Section Term / Definition 10.4 Geometric Random Variables 10.5 Counting Principle 10.5 Permutation 10.5 Factorial 10.6 Combination 10.7 Pascal’s Triangle 10.7 Expansion 10.7 Sample 10.7 Population 7.1 Chapter Seven Image - The result of moving all points of a figure according to a transformation. 7.1 Picture Transformation - A rule that assigns to each point of a figure another point in the plane, called its image. 15 Section Term / Definition 7.1 Rigid Transformation or Isometry - A transformation that preserves size and shape. The image of a figure under an isometry is congruent to the original figure. Also called a rigid transformation. 7.1 Nonrigid Transformation - A transformation that does not preserve the size of the original figure, such as an enlargement or reduction. 7.1 Translation - An isometry in which each point is moved by the same translation vector. 7.1 Translation Vector - A directed line segment from a point to its translated image. 7.1 Rotation - An isometry in which each point is moved by the same angle measure in the same direction along a circular path about a fixed point. 7.1 Angle of Rotation - The angle between a point and its image under a rotation, with its vertex at the center of the rotation and sides that go through the point and its image. 7.1 Reflection / Line of Reflection - An isometry in which every point and its image are on opposite sides and the same distance from a fixed line. Line of reflection - The line across which every point of a figure is moved by a reflection. 7.1 Point Symmetry - The property that a figure coincides with itself under a rotation of 180°. Also called two-fold rotational symmetry. 7.4 Tessellation or Tiling - A repeating pattern of shapes that completely cover a plane without overlaps or gaps. 7.4 Monohedral Tessellation - A tessellation that uses only one shape. 7.4 Regular Tessellation - A tessellation of congruent regular polygons. Picture 16 Section Term / Definition 7.4 Vertex Arrangement or Numerical Name - A notation that uses positive integers and other symbols to describe the arrangement of regular polygons about vertices of a semiregular tessellation. Also called the numerical name. Semiregular Tessellation - A tessellation of two or more different kinds of regular polygons that meet in the same order at each vertex. 7.4 7.4 Picture Archimedean Tilings - A regular or semiregular tessellation. 17