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Geometry Mathematics Glossary Term Adjacent Angles Angle of Depression Angle of Elevation Alternate Exterior Angles Alternate Interior Angles Altitude Apothem Arc Arc Length Definition Two angles that lie in the same plane, have a common vertex and a common side, but no common interior points The angle between the line of sight and the horizontal when an observer looks downward The angle between the line of sight and the horizontal when an observer looks upward A pair of angles formed when parallel lines are cut by a transversal; one angle is on the outside of the parallel lines and the other angle is diagonally on the opposite side of the transversal outside of the parallel lines. Two non-adjacent angles that lie on the opposite sides of a transversal between two parallel lines. In a triangle, a segment from a vertex of the triangle to the line containing the opposite side and perpendicular to that side; in a prism or cylinder, a segment perpendicular to the bases with an endpoint in each plane; in a pyramid or cone, the segment that has the vertex as one endpoint and is perpendicular to the base. A segment that is drawn from the center of a regular polygon perpendicular to the side of the polygon Part of a circle The length in linear measure of an arc of a circle is the product of the ratio circumference of the circle. Axis Base angles Between Biconditional Bisector (of a line segment) Bisector (of an angle) Center Central Angle Centroid Chord Circle ππππ π’ππ ππ π‘βπ πππ 360° and the In a cylinder, the segment with endpoints that are the centers of the bases. In a cone, the segment with endpoints that are the vertex and the center of the base. The two angles formed by the base and two congruent sides of an isosceles triangle. For any two points A and B on a line, there is another point C that lies between A and B if and only if A, B, and C are collinear and AC + CB = AB The combination of a conditional statement and its converse; containing the words βif and only ifβ. A line, segment, ray or plane that intersects a segment at its midpoint. A line or ray that divides the angle into two congruent angles. A point within a circle or sphere that is equidistant from all points on the circle or sphere. An angle whose vertex is the center of the circle and whose sides are radii of the circle. The point of concurrency of the medians of a triangle. A line segment whose endpoints lie on the circle. If a chord is drawn through the center of a circle, it is a diameter. The set of all points in a plane that is equidistant from a fixed point (the center of the circle). Circular Cone Circumcenter Circumference Circumscribed (about) Collinear Compass Complementary Angles Compound Statement Concentric Circles Conclusion Concurrent Lines Conditional Statement Cone Congruence Transformations Congruence Triangles Congruent Congruent Angles Congruent Arcs Congruent Segments Congruent Solids A cone with the following characteristics: 1. The base is a circle and the vertex is the point V 2. The axis is the segment with endpoints that are the vertex and the center of the base 3. The segment that has the vertex as one endpoint and is perpendicular to the base is called the altitude of the cone. The point of concurrency of the perpendicular bisectors of the sides of a triangle. The distance around the circle. Given the radius r of a circle, you can calculate circumference C by using the formula πΆ = 2ππ The circumference of a sphere is the circumference of any great circle of the sphere. A circle drawn around a polygon if the vertices of the polygon are on the circle. A polygon is circumscribed about a circle is all the sides of the polygon are tangent to the circle. Points that lie on the same line A geometric tool used to draw circles and parts of circles, called arcs. Two angles with measures that have a sum of 90. A statement formed by combining two or more statements. Circles that lie in the same plane and have the same center. In a conditional statement, the statement that immediately follows then. Three or more lines that intersect at a common point. The point at which they intersect is the point of concurrency. A statement that can be written in the form βIf p, then q.β p is the hypothesis and q is the conclusion. Symbolically, βif p, then qβ can be written as π β π . A solid with a circular base, a vertex not contained in the same plane as the base, and lateral surface area composed of all points in the segments connecting the vertex to the edge of the base. A mapping for which a geometric figure and its image are congruent. Triangles that have their corresponding parts congruent. Having the same measure. Symbol is β Angles that have the same measure Arcs that have the same measure and are in the same circle or congruent circles Segments that have the same length Two solids are congruent if all of the following conditions are met: 1. The corresponding angles are congruent Conjecture Conjunction Consecutive Angles Consecutive Interior Angles Construction Contrapositive Converse Convex Polygon Coordinate Plane Coordinate Proof Coplanar Figures Corner View Corollary Corresponding Angles Cosine Ratio Counterexample Cross Products Cross Section Cube Cylinder 2. Corresponding edges are congruent 3. Corresponding faces are congruent 4. Volumes are congruent An educated guess based on known information A compound statement formed by joining two or more statements with the word and Angles of a polygon that share a common side. When two parallel lines are cut by a transversal, the angles formed which are on the same side of the transversal and between the parallel lines. A method of creating geometric figures without the benefit of measuring tools. Generally only a pencil, straightedge and a compass are used. In the contrapositive of a conditional statement, the hypothesis and conclusion are both reversed and negated. βIf p, then qβ (π β π) becomes βIf not q, then not pβ (~π β ~π) The contrapositive has the same truth-value as the original statement. In the converse of a conditional statement, the hypothesis and conclusion are reversed. βIf p, then qβ (π β π) becomes βIf q, then pβ (π β π) A polygon where no diagonal contains points outside of the polygon. Formed by two number lines, called the axes, intersecting at right angles. The x-axis is the horizontal axis, and the y-axis is the vertical axis. The two axes meet at the origin (0, 0). The axes divide the plane into four quadrants that are named in counterclockwise starting with Quadrant I in the upper right corner. A figure is drawn on a coordinate plane and the formulas for slope, midpoint, and distance are used to prove properties of the figure. Figures in the same plane The view from a corner of a three-dimensional figure, also called the perspective view. A theorem that can be proved easily using another theorem. Two angles that lie on the same side of a transversal, in corresponding positions with respect to the two lines that the transversal intersects. A trigonometric function abbreviated πππ . In a right triangle, the ratio of the length of the leg adjacent to the ππππππππ‘ π πππ reference angle to the length of the hypotenuse. Symbolically, cos π = βπ¦πππ‘πππ’π π An example showing that a statement is false π π In the proportion = where π β 0 πππ π β 0. The cross products are ππ = ππ π π The proportion is true if and only if the cross products are equal. The intersection of a solid and a plane. A polyhedron with six faces, each of which is a square. A three-dimensional figure with two congruent circular bases that lie in parallel planes. Decagon Deductive Argument Deductive Reasoning Diagonal Diameter Dimension Disjunction Distance (two points on a line) Distance A polygon with ten sides A proof formed by a group of algebraic steps used to solve a problem. A system of reasoning that uses facts, rules, definitions or properties to reach logical conclusions. In a polygon, a segment that connects non-consecutive vertices of the polygon. In a circle, a segment that contains the center of the circle and whose endpoints are on the circle. The term diameter can also mean the length of this segment. In a sphere, a segment that contains the center of the sphere and has endpoints that is on the sphere. The number of coordinates used to express a position; for example, a line has dimension one, a plane dimension two, and space dimension three. The points of n-dimensional space have n coordinates the number of coordinates used to express a position; for example, a line has dimension one, a plane dimension two, and space dimension three. The points of n-dimensional space have n coordinates. A compound statement formed by joining two or more statements with the word or The distance between two points on a line is the absolute value of the difference of the coordinates of the points. The distance between any two points Dodecagon Edge Equiangular Triangle or Polygon A polygon with 12 sides A line segment where the faces of a polyhedron intersect. A triangle or polygon whose angles are all congruent. Equilateral Triangle or Polygon Exterior Angle of a Polygon Extremes A triangle or polygon whose sides are all congruent. Face Flow Proof Formal Proof Formula Geometric Mean Geometry Great Circle . An angle formed when one side of a polygon is extended; the angle is adjacent to an interior angle of the polygon π π In the proportion π = π, a and d are the extremes. The flat surface of a polyhedron Arrows show the logical connections between the statements. A logical argument in which each statement is supported by a statement that is accepted as true. A rule that shows the relationship between two or more quantities π π₯ The number π₯ π π’πβ π‘βππ‘ π₯ = π , π€βπππ π, π πππ π₯ πππ πππ πππ ππ‘ππ£π ππ’πππππ The branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles and figures in space from their defining conditions by means of certain assumed properties of space. The intersection of a sphere with a plane containing the center of the sphere. Hemisphere Hexagon Hypotenuse Hypothesis If and only if If-then statement Incenter Included Angle Included Side Indirect Proof Inductive Reasoning One of the two congruent parts into which a great circle separates a sphere A polygon with six sides In a right triangle, the side that is across or opposite from the right angle. In a conditional statement, the statement that immediately follows the word if In an equivalence statement, the words if and only if may be represented by the short symbol iff. The definition of an equivalence statement is written as follows: π β π = π β π πππ π β π It is a bi-conditional logical connective between statements. A conditional statement in the form βIf p, then qβ, where p and q are statements. The point of concurrency of the angle bisectors of the triangle. In a triangle, the angle formed by two sides is the included angle for those two sides. The side of a triangle that is between two angles and the included side is one of the anglesβ sides. The use of indirect reasoning. A type of reasoning that reaches conclusions based on a pattern of specific examples or past events. Inscribed Angle In a circle, an angle is inscribed if the vertex of the angle is on the circle and the sides of the angle are chords of the circle. Inscribed Arc An arc of a circle having endpoints on the sides of an inscribed angle, and its other points in the interior of the angle. A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle. A polygon is inscribed a circle is the vertices of the polygons are on the circle. The intersection of two or more geometric figures is the set of points the figures have in common. The statement formed by negating both the hypothesis and conclusion of a conditional statement. Symbolically, the βif p, then qβ (π β π) becomes βif not p, then not qβ (~π β ~π) A trapezoid whose nonparallel opposite sides are congruent A triangle that has at least two congruent sides. If there are two congruent sides, they are called legs. The vertex angle is between them. The third side is called the base and the other two angles are called the base Inscribed In Intersection Inverse Isosceles Trapezoid Isosceles Triangle angles. Lateral Area Lateral Edges Lateral Faces Law of Detachment Law of Syllogism In a prism or pyramid, the lateral area is the sum of the lateral faces. In a cylinder or cone, the lateral area is the area of the curved surface. In a prism, it is the intersection of two adjacent lateral faces. In a pyramid, the lateral edges are the edges of the lateral faces that join the vertex to vertices of the base. In a prism, planes that do not include the bases. In a pyramid, the planes that intersect at the vertex. If p -> q is a true conditional and p is true, then q is also true. Given the statements p implies q and q implies r are both true. Then we may write: (p implies q) and (q implies r). Linear Pair Line Segment Line Symmetry Logical Equivalent Major Arc Means Measure of an Arc Median Midpoint Mid-segment Minor Arc Negation Net n-gon Nonagon Oblique Cone Oblique Cylinder Oblique Prism Octagon Orthocenter Orthogonal drawing Paragraph Proof Parallel Lines Parallelogram Parallel Planes Pentagon Perpendicular Bisector If p -> q and q -> r are true conditional then p -> r is true. Two adjacent angles whose non-common sides are opposite rays; the sum of the measures of the angles in a linear pair is 180º A measurable part of a line that consists of two points, called endpoints, and all the points between them. Divides a figure with reflectional symmetry into two congruent halves. Statements that have the same truth-values. An arc that measures 180º or more. π π In a proportion, π = π, b and c are the means. The measure of a minor arc is the measure of its central angle. The measure of a major arc is 360 minus the measure of its relate minor arc. In a triangle, a line segment with endpoints that are a vertex of a triangle and the midpoint of the side opposite the vertex. In a trapezoid, the segment that joins the midpoints of the legs. The point halfway between the endpoints of a segment. A midpoint divides a segment into two congruent segments. A segment with endpoints that are the midpoints of two sides of a triangle. An arc that measures less than 180º. The negation of a statement has the opposite meaning of the original statement. From the definition we have: p implies q = not p or q. A two-dimensional figure that when folded forms the surfaces of a three-dimensional object. An n-gon is a polygon with n sides. A polygon with nine sides A cone that is not a right cone. A cylinder where the segment joining the centers of the bases is not perpendicular to the planes containing the bases. A prism in which the lateral edges are not perpendicular to the bases. A polygon with eight sides The point of concurrency of the lines containing altitudes of the triangle. A two-dimensional top view, front view and a right side view of a three-dimensional figure. The statements and reasons are connected in sentences. Coplanar lines that do not intersect. A quadrilateral with two pairs of parallel sides. Any side of a parallelogram may be called a base. Planes that do not intersect. A polygon with five sides. A line, segment, or ray that passes through the midpoint of a side and is perpendicular to that side. Perpendicular Lines Perspective Drawing Perspective View Pi (π ) Platonic Solids Point of Concurrency Point-Slope Form Point Symmetry Point of Tangency Polygon Polyhedron Postulate Pre-Image Prime Notation Prism Proof Proportion Pyramid Pythagorean Triple Quadrilateral Lines that intersect and form right angles. The symbol β₯ means, βis perpendicular toβ. A drawing that makes a two-dimensional image look like a three-dimensional object. The view of a three-dimensional figure from the corner. An irrational number represented by the ratio of the circumference of a circle to the diameter of the circle. πΆ π= π The five regular polyhedral: tetrahedron, hexahedron, octahedron, dodecahedron, or icosahedron The point of intersection of concurrent lines. For a non-vertical line with slope π and through point (π₯1 , π¦1 ) is π¦ β π¦1 = π(π₯ β π₯1 ) The type of symmetry for which there is a rotation of 180° that maps a figure onto itself. A point where a geometric line, curve, plane or curved surface touches another curve or surface but does not intersect it. A closed plane figure formed by three or more segments. In a convex polygon, no diagonal contains points outside the polygon. In a concave polygon, a diagonal contains points outside the polygon. Closed three-dimensional figure made up of flat polygonal regions. The flat regions formed by the polygons and their interiors are called faces. Pairs of faces intersect in segments called edges. Points where three or more edges intersect are called vertices. Postulate, or axiom, indicates a statement or assumption that is agreed by everyone to be so obvious or selfevident that no proof is necessary; and which can be used to prove other statements or theorems. Neither axioms nor postulates can be proved (within a system) using more basic statements. The position of a figure before a transformation. Sometimes used in transformations to identify image points. In a figure, π β² (ππππ "π πππππ" is the image of X. A solid with the following characteristics: 1. two faces, called bases, are formed by congruent polygons that lie in parallel planes 2. the faces that are not bases, called lateral faces, are formed by parallelograms 3. the intersections of two adjacent lateral faces are called lateral edges and are parallel segments. A valid argument in which all of the premised are true. π π An equation of the form π = π that states that two ratios are equal. A solid with the following characteristics: 1. all of the faces except one face, intersect at a point called the vertex 2. the face that does not contain the vertex is called the base and is a polygonal region 3. the faces meeting at the vertex are called lateral faces and are triangular regions. A group of three whole numbers that satisfies the equation π2 + π 2 = π 2 , where c is the greatest number. A polygon with four sides. Radius Radius of a Regular Polygon Ratio Rectangle Reflectional Symmetry Regular Polygon Regular Polyhedron Regular Prism Regular Pyramid Related Conditionals Relative Error Remote Interior Angles Rhombus Right Cone Right Cylinder Right Prism Rotational Symmetry Same-Side Interior Angles Scale Scale Drawing Scale Factor Secant Sector of a Circle Segment Segment Bisector Semicircle Side-Splitter Theorem In a circle, any segment with endpoints that are the center of the circle and a point on the circle. In a sphere, any segment with endpoints that are the center of the circle and a point on the sphere. The distance from the center to a vertex of the polygon. A comparison of two quantities by division. A quadrilateral with four right angles. A rectangle is also a parallelogram. The type of symmetry for which there is a reflection that maps a figure onto itself. A figure, which does not change upon undergoing a reflection or reflectional symmetry. A polygon that is both equilateral and equiangular. A polyhedron in which all the faces are regular polygons. A right prism with bases that are regular polygons. A pyramid whose base is a regular polygon. Statements such as the converse, inverse and contrapositive that are based on a given conditional statement. The ratio of the half-unit difference in precision to the entire measure, expressed as a percent. The two nonadjacent interior angles corresponding to each exterior angle of a triangle. A quadrilateral with four congruent sides. A rhombus is also a parallelogram A cone with an altitude (axis) containing the center of the base. A cylinder where the segment joining the centers of the bases is its altitude. A prism whose lateral faces are rectangular regions and a lateral edge is an altitude. The type of symmetry for which there is a rotation of 180° or less that maps a figure onto itself. Two angles that lie on the same side of a transversal and between the lines cut by the transversal, in corresponding positions with respect to the two lines that the transversal intersects. The ratio of any length in a scale drawing to the corresponding of the actual length; the lengths may be in different units. A drawing in which all lengths are proportional to corresponding actual lengths. The ratio of corresponding linear measurements of two similar figures. A line, ray, or segment that intersects a circle at two points (i.e. that contains a chord). A secant to a sphere is a line, ray, or segment that intersects a sphere at two points. The region bounded by two radii and their intercepted arc. The part of a line that consists of two points, called endpoints, and all points between them. A line, segment, ray, or plane that intersects a segment at its midpoint. An arc that is one-half of a circle; an arc that is subtended by a diameter. Its measurement is equal to 180° If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides Similar Figures Similar Polygons Similar Solids Sine Ratio Skew Lines Slant Height Slope-Intercept Form Slope of a Line proportionally. Two figures that have the same shape, but not necessarily the same size. Polygons having corresponding angles congruent and the lengths of corresponding sides proportional. The symbol for similarity is ~ Similar solids have the same shape and have all their corresponding dimensions proportional. A trigonometric function abbreviated π ππ. In a right triangle, the ratio of the length of the leg opposite the πππππ ππ‘π π πππ reference angle to the length of the hypotenuse. Symbolically, sin π = βπ¦πππ‘πππ’π π Lines that do not lie in the same plane and do not intersect. In a right cone, the distance from the vertex to the edge of the base. In a regular pyramid, the length of an altitude of a lateral face. The slope-intercept form of a linear equation is π¦ = ππ₯ + π, where π is the slope of the line and π is the yintercept. The ratio of its vertical change in the coordinate plane to the corresponding horizontal change. π¦ βπ¦ If (π₯1 , π¦1 ) and (π₯2 , π¦2 ) are points on a non-vertical line, then the slope is π₯2 βπ₯1 2 Space Sphere Spherical Geometry Square Standard Form of an Equation of a Circle Statement Straightedge Supplementary Angles Surface Area Tangent (to a circle) Tangent Ratio 1 The slope of a horizontal line is 0 The slope of a vertical line is undefined. The set of all points. The set of all points that are a given distance π, π‘βπ πππππ’π , from a given point C, the center. A plane is considered to be the surface of a sphere and a line is considered to be a great circle of the sphere. In spherical geometry, through a point not on a given line there is no line parallel to the given line. A square is a quadrilateral with four congruent sides and four right angles. The standard form of an equation of a circle with center (β, π) and radius π is (π₯ β β)2 + (π¦ β π)2 = π 2 Any sentence that either is true or false, but not both. A ruler with no markings on it. Two angles with measures that have a sum of 180° In a prism, cylinder, pyramid or cone, the surface area is the sum of the lateral areas and the areas of the bases. In a sphere, the surface area is four times the area of a great circle. A line, segment, or ray that intersects the circle in exactly one point. A trigonometric function abbreviated π‘ππ. In a right triangle, the ratio of the length of the opposite leg of the reference angle to the length of the adjacent leg of the reference angle. πππππ ππ‘π π πππ Symbolically, tan π = ππππππππ‘ π πππ Tessellation A tessellation, or tiling, is a repeating pattern of figures that completely covers a plane without gaps or overlap. Theorem Transformation Translation Transversal Trapezoid Triangle Trigonometric Ratio Trigonometry Truth Table Truth-Value Two-Column Proof Undefined Terms Vanishing Point Vertex Angle Vertical Angles Volume A pure tessellation is a tessellation that consists of congruent copies of one figure. A statement or conjecture that can be proven to be true based on postulates, definitions, or other proven theorems. A change in the position, size or shape of a geometric figure. The given figure is called the pre-image and the resulting figure is called the image. A transformation maps a figure onto its image. A translation (slide) is a transformation that moves points the same distance and in the same direction. A line that intersects two or more parallel lines at distinct points. A quadrilateral with exactly one pair of parallel sides, the bases. The nonparallel sides are called the legs of the trapezoid. Each pair of angles adjacent to a base are base angles of the trapezoid. An altitude of a trapezoid is a perpendicular segment from one base to the line containing the other base. Its length is called the height of the trapezoid. A polygon with three sides A ratio of the lengths of sides of a right triangle. See Cosine Ratio, Sine Ratio and Tangent Ratio A branch of mathematics which studies the relationships between sides and angles of a triangle; the six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. A table used as a convenient method for organizing the truth-values of statements. The truth-value of a statement is βtrueβ or βfalseβ according to whether the statement is true or false, respectively. A formal proof that contains statements and reasons organized in two-columns Basic concepts that are described because they cannot be rigorously defined; impossible to define every terms because eventually the definitions would become circular; point, line, and plane are generally taken as undefined terms in geometry. A point in a perspective drawing where parallel lines appear to converge. The angle formed by the congruent sides of an isosceles triangle. Two nonadjacent angles formed by two intersecting lines. Their measurements are congruent. A measure of the amount of space enclosed by a three-dimensional figure.