Acute angle

An angle of less than 90°

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Distance

the extent or amount of space between two things, points, lines, etc.

Perpendicular Lines

 Acute triangle

a triangle whose interior angles are all acute

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Exterior

Plane

a surface generated by a straight line moving at a constant velocity with respect to a fixed point

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Adjacent angle

two angles having the same vertex and having a common side between them[image]

Interior

Point

A dimensionless geometric object having no properties except location

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Angle

the space within two lines or three or more planes diverging from a common point, or within two planes diverging from a common line.

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Line

Postulate

Angle bisector

Line segment

a part cut off from a figure, esp. a circular or spherical one, by a line or plane, as a part of a circular area contained by an arc and its chord or by two parallel lines or planes

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Ray

one of a system of straight lines emanating from a point

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Collinear

Linear pair

two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees

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Right angle

the angle formed by two perpendicular lines that intersect; an angle of 90°

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Complementary   angles

Midpoint

Segment bisector

a line or a ray or a segment that divides a line segment into two equal parts

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Congruent

Obtuse angle

Space

Coplanar

Opposite rays

two rays with a common endpoint that form a straight line

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Supplementary angles

either of two angles that added together produce an angle of 180°

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Degree

the 360th part of a complete angle or turn, often represented by the sign°, as in 45°, which is read as 45 degrees

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Perpendicular Bisector

crosses the line segment at right angles (90°).

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Undefined terms

Vertex

Vertical angles

one of two opposite and equal angles formed by the intersection of two lines

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Biconditional

asserting that the existence or occurrence of one thing or event depends on, and is dependent on, the existence or occurrence of another, as

“A if and only if B.”

Conclusion

Conditional statement

is one that can be put in the form if A, then B where A is called the premise (or antecedent) and B is called the conclusion (or consequent).

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Conjecture

Contrapositive

Converse

"if" and "then" parts of a sentence are switched

If two numbers are both even, then their sum is even" is a true statement. The converse would be "If the sum of two numbers is even, then the numbers are even,"

Counterexample

an example that refutes an assertion or claim

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Deductive reasoning

Flow Proof

method of presentation that helps you to clearly and easily display each step in your argument

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Hypothesis

is the most basic of all the control flow statements. It tells your program to execute a certain section of code only if a particular test evaluates to true.

>>"if the breaks are on, then the car will slow down"

Inductive reasoning

also known as induction or inductive logic , is a type of reasoning that involves moving from a set of specific facts to a general conclusion

>> All of the ice we have examined so far is cold.  Therefore, all ice is cold

Inverse

Negation

Paragraph Proof

Proof

a sequence of steps, statements, or demonstrations that leads to a valid conclusion

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Properties

Statement

Theorem

a theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas

>> If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congurent

Two-column Proof

consists of a list of statements, and the reasons that we know those statements are true

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alternate exterior angles

When two lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but outside the two lines are called Alternate Exterior Angles.

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Alternate interior angles

When two lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles

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Consecutive interior angles

When two lines are crossed by another line (which is called the Transversal), the pairs of angles on one side of the transversal but inside the two lines

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Corresponding angles

When two lines are crossed by another line (which is called the Transversal), the angles in matching corners

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a point the same distance from two or more other points

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Parallel lines

 lines that never intersect

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Parallel planes

planes are parallel if they do not intersect

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are two lines that do not intersect but are not parallel

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Transversal

A line that cuts across two or more (usually parallel) lines

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Included angle

Angle made by two lines with a common vertex

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Included side

The common leg of two angles

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Hypotenuse/Leg

CPCTC

Corresponding parts of congruent triangles are congruent

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SAS

Side-Angle-Side

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SSS

[image]Side-Side-Side

ASA

Angle-Side-Angle

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Angle-Angle-Side

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Indirect reasoning

there are situations where it is not possible to directly prove a statement. But we can sometimes prove that it cannot be false

>> you and a friend are going to a Matchbox 20 Concert. When you arrive you find that you and your friend and two co-workers are the only ones there.You are pretty convinced by now that the concert is not going to be tonight.

Indirect Proof

 we assume that the statement we wish to prove true is actually false

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Overlapping Triangles

triangles that share a side or angle

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Common internal tangent

A common internal tangent of two circles is a tangent of both circles that intersects the segment joining the centers of two circles

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Extremes

the first or the last term, as of a proportion or series.

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Means

either the second or third term in a proportion of four terms

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Proportion

a statement that two ratios are equal

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Ratio

a quotient of 2 numbers

>> 2:1

Scale factor

size change magnitude

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>> sf: x2

Similar

two figures that have a similarity transformation mapping one onto the other;

>> written F~G

Polygons

a union of 3 or more segments where each segment intersects 2 other segments, one at each endpoint

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Concave

Convex

Mid-segment of a triangle

is a segment joining the midpoints of two sides of a triangle

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Law of Sines

Opposite / Hypotenus

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>> sin(A = a/c

Law of Cosines

Adjecent / Hypotenuse

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Cos(A = a/b

Unit Circle

is a circle with a radius of one

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Diagonal

 is a line joining two nonconsecutive vertices of a polygon or polyhedron.

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Isosceles trapezoid

 a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid

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a quadrilateral that has two distinct pairs of consecutive equilateral sides

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Median (midsegment) of a trapezoid

The midsegment of a trepzoid is parallel to both bases, has length equal to the average of the length of the bases

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Parallelogram

 parallelogram is a quadrilateral with two pairs of parallel sides

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Rectangle

a quadrilateral whose angles are all right angles

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Rhombus

a parallelogram with four equilateral sides

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Square

an equilateral and equianglular quadrilateral

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Trapezoid

a quadrilateral that has at least one pair of parallel sides

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Consecutive angles of a quadrilateral

are any two angles whose vertices are the endpoints of the same side

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a ten-sided polygon

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Dodecagon

a twelve-sided polygon

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Exterior angles of a polygon

At any given vertex, the interior angle is supplementary to an exterior angle

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Hexagon

a six-sided polygon

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Heptagon

a seven-sided polygon

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Interior angles of a polygon

n-2(180)/n

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n-gon

a polygon with n sides

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nonagon

a nine-sided polygon

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octagon

an eight-sided polygon

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opposite angles of a quadrilateral

equal in size

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parallelogram

a quadrilateral with both pairs of opposite sides parallel

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a five-sided polygon

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perimeter

sum of all the sides

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quadrilateral

a four-sided polygon

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bases of a trapezoid

are parallel

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height of a trapezoid

The height of the trapezoid is the perpendicular distance between the bases, here symbolized by h

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Undecagon

an eleven-sided polygon

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Arc

a path from one node in a network to another; doesn't have to be straight & can be more than 1 line between 2 nodes; part of a circle

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Center

Central angle

an angle whose vertex is the center of the circle

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Chord

a segment whose endpoints are on a circle

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Circle

the set of points on a plane at a certain distance (radius) from a certain point (center); a polygon with infinite sides[image]

Circumference

the perimeter of a circle

>> 2(3.14)r or (3.14)d

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Circumscribed

the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon

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Common external tangent

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles

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Concentric circles

circles that share the same center, but have different radii

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Diameter

the segment whose endpoints are points on a circle (or sphere) that contains the center of the circle as its midpoint; the length of that segment

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Externally tangent circles

Two circles, neither of which is inside the other, that have a single point in common

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Inscribed angles

is formed when two secant lines of a circle ( when one secant line and one tangent line of that circle) intersect on the circle

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Inscribed figure

an inscribe planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid

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Intercepted arc

 the arc that is formed when segments intersect portions of a circle and create arcs

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Internally tangent circles

Two circles, one of which is inside the other, that have a single point in common

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Major Arc

Minor Arc

is an arc of a circle having measure less than or equal to 180 degrees

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Pi

is a  mathematical constant whose value is the ratio of any circles's circumference to its diameter

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Point of Tangency

the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point

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Radius

the segment whose endpoints are any point on a circle or sphere and its center; the length of that segment

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Secant

a line that intersects the circle in two points

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Semicircle

an arc whose central angle is a right angle

Tangent

the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point

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Apothem

a regular polygon is a line segment from the center to the midpoint of one of its sides

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Geometric probability

Problems of the following type, and their solution techniques, were first studied in the nineteenth century, and the general topic became known as geometric probability.

•  What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines?

• What is the mean length of a random chord of a unit circle?

• What is the chance that three random points in the plane form an acute (rather than obtuse) triangle?

• What is the mean area of the polygonal regions formed when randomly-oriented lines are spread over the plane?

Irregular figure

a irregular figure is not a particular shape it may have two or more figures in to make a irregular figure.

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Irregular polygon

any polygon that is not a regular polygon

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Sector of a circle

a line that intersects the circle in two points

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Segment of a circle

is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord

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Altitude of a 3D Figure

the length of a segment whose endpoints are the vertex and a point on the plane of the base that is perpendicular to the plane of the base

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Bases

the side of an isoscoles triangle whose endpoints are the vertices of the base angles

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Cone

the surface of a conic solid whose base is a circle; see right cone

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Congruent solids

Cube

is a  three-dimensional solid object bounded by six square faces, sides, with three meeting at each vertex

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Cylinder

the surface of a cylindric solid whose base is a circle; see right cylinder

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Edges

a segment that helps to make up a face

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Face

a polygonal region of a surface

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Great Circle

the circle formed by the intersection of a sphere and the plane that contains its center and that divides the sphere into two hemispheres

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Height

the measurement of vertical distance

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Hemisphere

half of a sphere

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Lateral area

the area of the lateral surface of a solid

Lateral faces

the faces of the lateral surface of a prism, or a face of a pyramid that is not a base

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Nets

a 2-D figure that can be folded on its segments or curved on its boundaries to form a 3-D figure

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Oblique (cone, cylinder, prism)

a non-right prism or cylinder or cone

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Platonic solids

a  convex polyhedron that is regular, in the sense of a regular polygon

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polyhedron with 12 flat faces

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Hexahedron

a polyhedron with 6 faces

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Icosahedron

a polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices

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Octahedron

a polyhedron with 8 faces

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Tetrahedron

a polyhedron composed of 4 triangular faces

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Polyhedron

a three-dimensional surface which is the union of polygonal regions and has no holes

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Prism

the surface of a cylindric solid whose base is a polygon

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the surface of a conic solid whose base is a polygon

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Right (cone, cylinder, prism)

prism: - a prism whose direction of sliding is perpendicular to the plane of the base

cone: a cone whose axis is perpendicular to the plane containing its base

cylinder: a cylinder whose direction of sliding is perpendicular to the plane of the base

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

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Slant height

the length of a lateral edge of a conic solid

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Sphere

the set of points in space equidistant from a certain point

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Surface area

Volume

the amount of space a 3-D object can hold

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Center of dilation

 a fixed point in the plane about which all points are expanded or contracted

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Center of Rotation

 the point where the two intersecting lines of a rotation meet

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Column matrix

an ordered list of numbers written in a column

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Composition

Dilation

a change in size

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Identity matrix

unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere

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Image

Isometry

a transformation that keeps the same size and shape of a figure but moves it to a new location

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Mapping

making a transformation

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Pre-image

Reflection

is a mapping that from an Euclidean space to itself that is an  isometry with a  hyperplane as set of fixed points

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Reflection matrix

1   0

0  -1

Rotation

the composite of two reflections over intersecting lines

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Rotation matrix

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Row matrix

multiplied by a rectangular matrix, the row matrix must be on the left

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Scalar

arrangement of pixels

1  0

0   1

Translation

the composite of two reflections over parallel lines

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Translation matrix

Vertex matrix

For example, if you want

to reflect the triangle over the x-axis, you would multiply the vertex matrix