Biconditional

The conjunction of a conditional statement and its converse.

EX. (p<-->q)

Compound Statement

A statement formed by joining two or more statements.

EX. Albany is a city in New York, and is the capital.

Conclusion

In a conditional staement the statement that immediately follows the word then.

EX. If the sun is out then it is day time.

Conditional Statement

A statement that can be written in if-then form

EX. If the sun is out then it is day time.

Conjecture

An educated guess based on known information.

EX. Since the sun is out it must be day time.

Contrapositive

The statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement.

EX. ~q-->~p

Converse

The statement formed by exchanging the hypothesis and conclustion of a condition of a conditional statement.

EX. q-->p

Counterexample

An example used to show that a given statement is not always true.

EX. The sun can be out in the evening

Deductive Reasoning

A system of reasoning that uses facts, rules, definitios or properties to reach a logical conclusion.

Hypothesis

In a conditional statement, the statement that immediately follows the word if.

EX. If the sun is out then it is daytime.

Inverse

The statement formed by negating both the hypothesis and the conclusion of a conditional statement.

EX. ~p-->~q

Negation

If a statement is represented by p then not p is the negation of the statement.

EX. p-->~p

Properties

Equalities for real numbers

Theorum

A statement or conjecture that can be proven true by undefined terms, definitions and postulates.

EX. Theorum 2.8 says If two angles are verticle angles, then they are congruent.

Alternate Exterior Angles

Outside of parallel lines opposite sides of transversal.

EX.[image]

Alternate Interior Angles

Inside of parallel lines opposite of transversal

EX.[image]

Consecutive Interior Angles

Inside of parallel ines same side of transversal.

EX.[image]

Corresponding Angles

One inside, one outside, same side of transversal

EX. [image]

Parallel lines

Coplanar lines that dont intersect

EX. __________

     __________

Parallel planes

Planes that dont intersect.

EX. [image]

Skew Lines

Lines that dont intersect and are not coplanar

EX. [image]

Transversal

 A line that intersects two or more lines in a plane at different points.

EX. [image]

Acute Angles

An angle with a measure of less than 90 degrees.

EX. [image]

Distance

The amount of space from one point to another

EX. point M is 5 inches from point N

Perpendicular Lines

Two or more lines that for a right angle

EX.[image]

Acute Triangle

A triangle in which all of its angles are acute.

EX.[image]

Exterior

A point is in the exterior of an angle if it is neither on the angle nor in the interior of the angle.

EX. [image]

Plane

A basic undefined term of geometry. A flat surface made up of points that has no depth and extends indefinitely in all directions.

EX.[image]

Adjacent Angle

Two angles that lie on the same plane, have a common vertex and a common side, but no common interior points.

EX.[image]

Interior

 A point is in the interior of an angle if it does not lie on the angle itself and it lies on a segment with endpoints that are on the sides of the angles.

EX. [image]

Point

A basic undefined term of geometry. A point is a location. In a figure, points are represnted by a dot.

EX.  [image]

Angle

The intersection of two noncollinear rays at a common endpoint. The rays are called sides and the common endpoint is called the vertex.

EX. [image]

Line

A basic undefined term of geometry. A line is made up of points and has no thickness or width. In a figur, a line is shown with an arrowhead at each other.

EX.[image]

Postulate

A statement that describes a fudamental relationship between the basic terms of geometry.

EX. Postulate 2.8 says that if two angles are verticle, then they are congruent

Angle Bisector

A ray that divides an angle into two congruent angles.

EX.[image]

Line segment

A measurable part of a line that consists of two points, called endpoints, and all of the points between them.

EX.[image]

Ray

A line that has one endpoint and one side thst never ends.

EX. [image]

Collinear

Points that lie on the same line.

EX. [image]

Linear Pair

A pair of adjacent angles whose noncommon sides are opposite rays

EX. [image]

Right Angle

An angle which measure 90 degrees

EX. [image]

Complementary angles

Two angles that have a sum of 90 degrees.

EX. [image]

Midpoint

The point halfway between the endpoints of a segment.

EX. [image]

Segment bisector

A segement, line, or plane that intersects a segment at its midpoint

EX. [image]

Congruent

Having the same measure.

EX. [image]

Obtuse Angle

An angle that is over 90 degrees

EX.[image]

Space

A boundless three-dimentional set of all points

EX. [image]

Coplanar

Points that lie on the same plane.

EX. [image]

Opposite Rays

Two rays with a common end point.

EX. [image]

Supplementary Angles

 Two angles with measures that add up to 180 degrees.

EX.[image]

Degree

A unit of measure used to measure angles and arcs.

EX. Angle A= 80 degrees

Perpendicular Bisector

A line, segment, or ray that passes through the midpoint of a side and is perpendicular to that side.

EX. [image]

Undefined Terms

Words that are not formally explained by means of basic words or consepts

EX. A line or a plane

Vertex

The common endpoint of an angle.

EX. [image]

Vertical Angles

Two nonadjacent angles formed by two intersecting lines.

EX. [image]

Acute Triangle

A triangle with all angles being acute.

EX.[image]

Altitude of a triangle

A segment from a vertex of a triangle to the line containing the opposite side.

EX. [image]

Base angles

Two angles formed by the base and a congruent side

EX. [image]

Centroids

The point of concurrency of the medians of a triangle

EX. [image]

Circumcenter

The point of concurrency of the perpendicular bisectors of a triangle

EX. [image]

Congruence

Mapping for a geometric figure and its image

EX. [image]

Congruent triangles

Triangles that have their corresponding parts congruent

EX. [image]

Coordinate Proofs

A proof that uses figues in the coordinate plane to prove concepts.

EX. [image]

Corollary

Statement that can easily be proven using a theorum

EX. The three angles of a regular triangle equal 180

Equiangular Triangle

A triangle with all angles congruent

EX. [image]

Equilateral Triangle

 Triangle with all sides congruent

EX. [image]

External angle

An angle formed by one side of a shape and the extension of another

EX. [image]

Hypotenuse

The side of the triangle opposite the right angle

EX. [image]

Incenter

The point of concurrency of the angle bisector of a triangle.

EX. [image]

Isosceles Triangle

A triangle with at least 2 sides congruent

EX. [image]

Legs of a right triangle

Always equal 90 degrees

EX. [image]

Median of a triangle

A line segment with end points that are a vertex of a triangle

EX. [image]

Obtuse Triangle

Triangles with an obtuse angle

EX. [image]

Perpendicular bisector

Line, segment, or a ray that passes through the mid point of a triangle

EX. [image]

Point of congruency

The point of intersection of concurrent lines

EX. [image]

Remote interior angle

The angles of a triangle that are not adjacent to a given angle

EX. [image]

Right Triangle

A triangle with a right angle

EX. [image]

Scalene Triangle

A triangle with no 2 sides congruent

EX. [image]

Vertex Angles

The angle opposite the base on an isosceles triangle

EX.  [image]

Included angle

 In a triangle, the angle formed by two sides is the included angle for those two sides.

EX. [image]

Included side

The side of a triangle that is a side of each of the two angles.

EX. [image]

Indirect reasoning

Assumes that the conclusion is false and then shows tha this assumption leads to a contradiction of the hypothesis or some other accepted fact.

EX. X is odd

X is not odd

Indirect proof

Assumes the statement is false. One then uses logical reasoning to deduce that a statement controdicts an assumption. Once the contradiction is obtained, one concludes that the statement is actually true.

EX. 1/2 >Y

1/2 > or equal to Y

Concave

If any point of the lines contain any point in the interior of the polygon

EX. [image]

Convex

If the polygon has no points of a line pass through it

EX. [image]

mid-segment of a triangle

A segment with endpoints that are the midpoints of two sides of a triangle

EX. [image]

Polygon

A closed figure formed by a finite number of coplanar segments called sides

EX. [image]

Proportion

An equation that states two ratios are equal

EX. 2/3=6/9

Regular

A convex polrgon in which all of the sides and angles are congruent

EX. [image]

Scale Factor

The ratio of the lengths of two corresponding sides of two similar polygons or two similar solids.

EX. 6:3=2:1

Similar polygons

Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

EX. [image]

Angle of Depression

The angle between the line of sight and the horizontal when and observer looks upward.

EX.[image]

Angle of elevation

The angle between the line of sight and the horizontal when an observer looks upward

EX.[image]

Altitude of a right triangle

A segment from a vertex of the triangle to the line containing the opposite side and a perpendicular

EX. [image]

Cosine

For an acute angle of a right triangle. the ratio of the measure of a leg adjacent to the acute angle and the hypotenuse.

EX. Cos30=x/15

Geometric Mean

For any positive numbers a and b the positive number x such that a/x=x/b

EX. a/x=x/b

Pythagorean triple

 A group of three whole numbers that satisfies the equation a²⁺b²⁼c²

EX. 3²+4²=5²

Sine

For an acute angle of a right triangle the ratio of the measure of the leg opposite the acute angle to the measure of hte hypotenuse

EX. sin30=x/15

Solving a triangle

Finding the measure of all angles and sides of a triangle.

EX. [image]

Tangent

The ratio of the measure of the leg opposite the acute angle to the measure of the leg adjacent to the acute angle.

EX. tan30=x/12

Trigonometry

The study of the properties of triangles and trigonometric functions and their applications

EX. [image]

Consecutive angles of a quadrilateral

Angles whose verticles are the endpoints of the same side

EX.[image]

Decagon

Shape with 10 sides

EX. [image]

Diagonal

A segment that connects non consecutive verticles of a polygon

EX. [image]

Dodecagon

A shape with 12 sides

EX. [image]

Exterior angles of a polygon

Angles on outside of polygon

EX.[image]

Hexagon

A shape with 6 sides

EX. [image]

Heptagon

A shape with 7 sides

EX. [image]

Interior angles of a polygon

angles on the inside of a polygon

EX. [image]

Isosceles trapazoid

A trapazoid in which the legs are congruent,both pair of base angles are congruent, and diagonals are congruent

EX. [image]

Kite

A quadrilateral with exactly two distinct pairs of adjacent congruent sides

EX. [image]

Median of a trapazoid

In a trapazoid, the segment that joins the median of the legs

EX. [image]

N-gon

A polygon with n sides

EX. [image]

Nonagon

A shape with 9 sides

EX. [image]

Octagon

A shape with 8 sides

EX. [image]

Opposite angles of a quadrilateral

Any 2 angles in a quadrilateral that arent consecutive

EX. [image]

Opposite sides of a quadrilateral

Any 2 sides in a quadrilateral that arent consecutive

EX.[image]

Parallelogram

A quadrilateral with parallel opposite sides

EX. [image]

Pentagon

A shape with 5 sides

EX. [image]

Perimeter

The sum of the lengths of the sides of a polygon

EX. [image]

Quadrilateral

A polygon with 4 sides

EX. [image]

Rectangle

A quadrilateral with four right angles

EX. [image]

Rhombus

A quadrilateral with all four sides congruent

EX. [image]

Square

A polygon  with all four sides congruent

EX. [image]

Trapezoid

A quadrilateral with exactly one pair of parallel sides

EX. [image]

Bases of a trapezoid

The parallel sides of a trapazoid

EX. [image]

Height of a trapezoid

How high the trapazoid is

EX. [image]

Arc

A part of a circle that is defined by 2 endpoint

EX. [image]

Center

Equidistance from a given point

EX. [image]

Central angle

An angle that intersects a circle in 2 points and has its vertex at the center

EX. [image]

Chord

A segment with endpoints that are on a given circle

EX.[image]

Circle

The set of all points in a plane equidistant from a given point

EX. [image]

Circumference

distance around the circle

EX. [image]

Circumscribed

A circle is circumscribed about apolygon if the circle contains all of the verticlas of a polygon

EX. [image]

Diameter

Chord that passes through the circle

EX. [image]

Inscribed

A polygon is inserted into a circle if each of its verticles lines on the circle

EX. [image]

Intercepted

The endpoint of an arc lie on a line all points of the arc except the endpoints

EX. [image]

Major arc

An arc with a measure greater than 180 degrees

EX. [image]

Minor arc

An arc with a measure less than 180 degrees

EX. [image]

Pi

An irrational number represented by the ratio 3.14

EX. [image]

Point of tangency

For a line that intersects a circle in only one point, the point at which they intersect

EX. [image]

Radius

In a circle, any segment with endpoints that are the center of the circle and a point in the circle

EX.[image]

Secant

Any line that intersects a circle in exactly two points

EX. [image]

Semicircle

An arc that measures 180 degrees

EX. [image]

Tangent

A line in the plane of a circle that intersects the circle in exactly one point

EX. [image]

Apothem

A segemtn that is drawn from the center of a regualr polygon perpendicular to a side of the polygon

EX. [image]

Geometric probability

Using the priciples of length and area to find the probability of an event

EX. 30%

Irregular Figure

A figure that cannot be classified as a single polygon

EX. [image]

Irregular Polygon

A polygon that is not regular

EX. [image]

Sector of a circle

A region of a circle bounded by a central angle and its intercepted arc

EX. [image]

Segment of a circle

The region of a circle bounded by an arc and a chord

EX. [image]

Cone

A solid with a circular base, vertex not contained in the same plane as the base, and a lateral surface area composed of all points in the segment

EX. [image]

Congruent solids

Two solids are congruent if the corrisponding angles are congruent, corresponding edges are congruent, and corresponding faces are congruent

EX.[image]

Cylinder

A figure with bases that are formed by congruent circles in parallel planes

EX. [image]

Hemisphere

One of the two congruent parts into which a great circle separates a sphere

EX. [image]

Lateral Area

The area not including the bases

EX. [image]

Lateral Faces

In a prism the faces that arent bases

EX.[image]

Net

A two dimentional figure that when folded forms a three dimentional figure.

EX.[image]

Oblique

3D figures that arent right

EX. [image]

Altitude

A segment perpendicular to the bases with an endpoint in each plane

EX. [image]

Bases

Flat planes that contain points and are perpendicular to the altitude.

EX. [image]

Cube

A 3D figure with 6 congruent bases

EX. [image]

Edges

A shape formed by 2 bases that intersect

EX. [image]

Face

A flat surface on a shape

EX.[image]

Great Circle

For a given sphere, the intersection of the sphere and a plane that contains the center of a sphere

EX. [image]

Height

The length of an altitude in a figure

EX. [image]

Platonic Solids

The five regular polyhedra

EX. [image]

Dodecahedron

A polyhedron with 12 flat faces

EX. [image]

Isosahedron

A polyhedron with 20 flat surfaces

EX.[image]

Hexahedron

A polyhedron 6 flat faces

EX. [image]

Octahedron

A polyhedron with 8 flat surfaces

EX.[image]

Tetrahedron

A polyhedron with 4 trianglular faces

EX. [image]

Polyhedron

Closed 3D figures mad up of flat polygonal regions called faces

EX. [image]

Prism

A solid with two bases, lateral faces and lateral edges

EX. [image]

Pyramid

A solid with all faces except one intersect at a common vertex, the other face is the base

EX. [image]

Regular Prism

A right prism with bases that are regular polygons

EX. [image]

Right Prism

A prism with lateral edges that are also altitudes

EX. [image]

Similar Solids

Solids that have exactly the same shape, but not necessarily the same size

EX. [image]

Slant Height

The distance from any part of a base to the vertex

EX. [image]

Sphere

In space the set of all points that are given distance from a given point

EX. [image]

Surface Area

The sum of the areas of all faces and sides surface of a 3D figure

EX. [image]

Volume

A measure of the amount of space enclosed by a 3D figure

EX. [image]

Center of Rotation

A fixed point around which shapes move in a circular motion to a new position

EX.[image]

Column Matrix

A matrix containing one column often used to represent an ordered pair.

EX. [image]

Center of Dilation

The new point at which shapes move to a dilation

EX.[image]

Columns of a Matrix

The numbers going top to bottom of a matrix

EX. [image]

Composition

Successive reflections in parallel lines

EX. [image]

Dilation

A transformation determined by dividing by the scale factor

EX. [image]

Identity Matrix

A scalar matrix in which all of the diagonal elements are unity

EX.[ 1 0 ]
     [ 0 1 ]

Image

A picture

Isometry

A mapping for which the original figure and its image are congruent

EX. [image]

Mapping

a mathematical relation such that each element of a given set is associated with an element of another set

Pre-Image

Original position of a figure in a plane

EX.[image]

Reflection

A transformation representing a flip of the figure over a point, line, or plane

EX. [image]

Reflection Matrix

A matrix that can be multiplied by the vertex matrix of a figure to find the coordinates of the reflected image

EX. [image]

Rotation

A transformation that turns every point of a preimage through a specified angle and direction about a fixed point, called the center of rotation

EX. [image]

Rotation Matrix

A matrix that can be multiplied by the vertex matrix of a figure to find the coordinates of the rotated image

Row Matrix

The transpose of a row vector is a column vector

EX. [image]

Scalar

A constant multiplied by a vector

EX. [image]

Standard Matrix

When the initial point of a vector is at the origin

EX. [image]

Translation

A transformation that moves all points of a figure the same distance in the same direction

EX. [image]

Translation Matrix

A matrix that can be added to the vertex matrix of a figure to find the coordinates of the translated image

Vertex Matrix