Geometry (10th)

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144 terms · Geometry vocabulary and posulates

Absolute Value

The distance from 0 to a number, n, on a number line

Acute Angle

An angle whose degree measure is greater than 0 and less than 90.

Acute Triangle

A triangle that has three acute angles

Additive Inverse

a +0 =a and 0 + a= a

Angle

a set of points that is the union of two rays having the same endpoint

Associative Property of Addition

a property of real numbers that states that the sum or product of a set of numbers is the same, regardless of how the numbers are grouped. (a +b) + c =a+ (b +c)

Associative Property of Multiplication

a property of real numbers that states that the sum or product of a set of numbers is the same, regardless of how the numbers are grouped. (ab)c =a(bc)

Base Angles

The angles whose vertices are the endpoints of the base of the triangle

Bases

The third side of the triangle

Betweenness

Given two numbers, another number is said to be between those two numbers if it is greater than the first but less than the second

Bisector of an Angle

A ray whose endpoint is the vertex of the angle an that divides the angle into two congruents angles

Bisector of a line Segment

any line, or subset of a line, that interests the segment at its midpoint

Closure Property of Addition

the sumber of two real numbers is a real number

Closure property of multiplication

the product of two real numbers is a real number

Collinear set of points

A set of points all of which lie on the same straight line

Commutative Property of addition

a + b = b+a

Commutative Property of Multiplication

ab = ba

Congruent angles

Angles that have the same measure

Coordinate

an ordered pair of numbers that identifies a point on a coordinate plane, written as (x,y)

Definition

a statements of the meaning of a term

Degree

THe unit of measurement for angles,

Distance from a point to a line

the length of the perpendicular from the point to the line

Distributive Property

a property of real numbers that states that the product of the sum or difference of two numbers is the same as the sum or difference of their products

Equilangular triangle

a triagnel that has 3 congruent angles

Equilateral Triangle

A triangle that has 3 congruent sides

Exterior of the angle

the space on the outside of the two rays of an angle

Geometry

is a branch of mathematics that defines and relates the basic properties and measurement of the line segments and angles

half- line/ray

a part of line that consists of a point on the line, called an endpoint, and all the points on one side of the endpoint

Hypotenuse

THe side of the triagnle that is opposite the right angle

Interior of an angle

The area that is in between the two rays

Isosceles Triangle

A triangle that has two congruent sides

Legs

Two congruent sides of an isosceles triangle or the two sides that form a right angle

Line segment

a set of points consisteing of two points on a line, called endpoints, and all the points on the line between the endpoints

Line

An infinite set of points in opposite directions forming a straight path, it has only one dimension, length

Midpoint

A point of the that line segment that divides the segment into tow congruent segments

Multiplication Property of zero

ab= 0 if and only if a=0 and b =0

Muliplicative Identity

a 1 = a and 1 a =a

Multiplicative inverse

a*1/a= 1

Non- collinear set of points

a set of three of more points that do not all lie on the same straight line

number line

a line on which each point represents a real number

obtuse angle

an angle whose degree measure is greater than 90 and less than 180

opposite ray

two rays of the same line with a common endpoint and no other points in common

perpendicular lines

two lines that intersect to form right angles

plane

a set of points that form a flat surface extending indefinitely in all directions

point

an exact location in space, a point has no dimension

polygon

a closed figure in a plane that is the union of the line segments such that the segments intersect only at their endpoints and no segments sharing a common endpoint are collinear

right angle

an angle whose degree measure is 90

right triangle

a triangle that has a right angle

Scalene Triangle

a triangle that has no congruent sides

set

a collection of objects such that it is possible to determine whether a given object belong to a collection or not

straight angle

an angle that is the union of opposite rays and whose degree measure is 180

triangle

a ploygon that has exactly three sides

undefined terms

their meaning is accepted without definition

vertex angle

the angle opposite the base in an isosceles triangle

vertex

the common endpoint of two sides of a polygon, the common endpoint of two rays that form an angle, the common point where two or more edges of a 3-D solid meet

Addition Postulate

If equal quanties ar added to equal quanties, the sums are equal

Axiom

Statements that we accept them without proof

Conjecture

Statements that are likely to be true but not yet been proved true by a deductive proof

Counterexample

An example that refutes or disproves a hypothesis, proposition, or theorem

Deductive Reasoning

Uses the laws of logic to combine definitions and general statements that we know to be true to reach a valid conclusion

Direct proof

A proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved

Division Postulate

If equals are dividied by nonzero equals, the quotients are equal

Equivalence Relation

A relation for which these postulates (reflective, symmetric and transitive) are true

Generalization

A proposition asserting something to be true either of all members of a certain class or of an indefinite part of that class

Indirect proof/Proof by contradiction

A proof that starts with the negation of the statement to be proved and used the laws of logic to show that it is false

Inductive Reasoning

The method of reasoning in which a series of particular examples leads to a conclusion

Multiplication Postulate

If equals are multiplied by equals, the products are equal

Partition Postulate

A whole is equal to the sum of all its parts

Postulate

A statement whose truth is accepted without proof

Powers Postulate

The squares of equal quantities are equal

Proof

Is a valid argument that establishes the truth of a statement

Reflective Property of Equality

A quantity is equal to itself

Roots Postulate

Postivite square roots of postivie equal quantites are equal

Substitution Postulate

A quantity may be substituted for its equal in any statment of equality

Subtraction Postulate

If equal quantities are sbutracted from equal quantities, the differences are equal

Symmetric Property of Equality

An equality may be expressed in either order

Theorem

A statement that is proved by deductive reasoning

Transitive Property of Equality

Quantities equal to the same quantity are equal to each other

Adjactent Angles

Two angles in a plane that share a common side and share a common vertex but have no interior points in common

ASA Triangle Congruence

Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, two angles and the included side of the other

Complementary Angles

Two angles the sum of whose degree measures is 90

Congruent Polygons

Polygons that have the same size and shape

Corresponding Angles

Congruent angles in the same relative postion with in two firgures

Corresponding Sides

Sides that are in the same relative position with in two figures

Linear Pair of Angles

Two adjacent angles whose sum is a straight angle

SAS Triangle Congruence

Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectiviely, two sides and teh included angle of the other

SSS Triangle Congruence

Two triangles are congruent if the three sides of one triangle are congruent respectively to the thtree sides of the other

Supplementary Angles

Two angles the sum of whose degree measure 180

Veritical Angles

The non adjacent angles formed by the interestion of two lines

Altitude of a Triangle

A line segment (of its length) Drawn from a vertex perpendicular to the line containing the oppostite side

Angle Bisector

A segment of ray that divides an angle into two congruent angles

Circumcenter

The point where the three perependicular biserctors of the sides of a triangle interest

Concurrent

When the three lines interesct in one point

Corollary

A theorem that can easily be duduced from another theorem

Equidistant

The same distance apart at every point

Geometric Construction

A drawing of geometric figure done using only a pencil, a compass and a straightedge, or their equivalents

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite these sides are congruent

Median of a Triangle

A line segment that joints any vertex of the triangle to the midpoint of the opposite side

Perpendicular Bisector of a Line Segment

A line, a line segment or a ray that is perpendicular to the line segment at its midpoint

Abscissa

x- coordinate, the distance from the point to the y axis

Axis of Symmetry

The line where the figure could be folded so that parts of the figure on opposite sides of the line would coincide

Composition of Transformations

WHen two transformations are performed one following the other

Coordinate Plane

A plane spanned by the x axis and y axis in which th ecoordinates of a point are its distances from two interesecting perpendicular axes

Dilation

A transformation that makes a object get bigger or smaller

Direct Isometry

A transformation that preserves distance and orientations

Fixed points

A point that does not change after a transformation

Function

A set of ordered pairs in which no two pairs have the same first element

Glide Reflection

A composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of reflections performed in either order

Image

The figure after a transformation occurs

Isometry

A transformation that perserves distance

Line Reflection

The correspondence between the object points and the images points

Line Symmetry

when the figure is its own image under a line reflection

Oppostie Isometry

A transformation that preserves distance but changes the order or orientation from clockwise to counterclockwise or form counterclockwise to clockwise

Ordered Pair

All the points on a plan have coordinates and they are written (x,y)

Ordinate

The y coordinate, the distace from the point to the x axis

Orientation

Location of position relative to the points of the compass

Origin

The point on the coordinate plan which the x axis and the y axis intersect

Point symmetry

The figure is its own image under a reflection in a point

Preimage

The image before the transformation occurs

Quarter Turn

A counterclockwise rotation of 90 about the origin

Rotation

A transformation of a plane about a fixed point P though an angle of d degrees

Transformation

A one-to-one correspondence between two sets of points, S and S^1, such that every point in set S corresponds to one and only one point in the S^1, called its image, and every point in S^1 is the image of one and only one point in S called its preimage

Translation Symmetry

If the image of every point of the figure is a point on the figure

Translation

A transformation of the plane that moves every point in the plane in the same distance in the same direction

X- Axis

The horizontal Line on the coordinate plane

Y-Axis

The vertical line on the coordinate plane

Additon Postulate of Inequalities

1) If equal quantities are added to unequal quantities, then the sums are unequal in the same order. 2) If unequal quantities are added to unequal quantities, then the sum is unequal in the same order

Adjacent Interior Angles

The angle that is next to the angle but on the inside of the triangle

Exterior Angles of a Polygon

an angle that forms a linear pair with one of the interior angles of the polygon

Remote Interior angles/ Non-Adjacent

Angles that are in the triangle, but are not adjacent to the angle

Substitution Postulate of Inequalities

A quanity may be substituted for its equal in any statment of inequality

Subtraction Postulate of Inequalities

If equal quantities are subtracted from unequal quantities, then the differences are unequal in the same order

Transitive Property of Inequality

If a, b, and c are real numbers such that a>b, and b>c, then a>c

Triangle Inequality Theorem

The length of one side of a triangle is less than the sum of the length of the other two sides

Trichotomy Postulate

Given any two quantities, a and b, one and only one of the following is true: a,b or a=b or a>b

Logic

Study of reasoning

Negation

Usually formed by placing the word "not" in the originial statement (~)

Conjunction

A compound statement formed by combining 2 simple statements using the word "and" (^)

Disjunction

A compound statement formed by combining 2 simple statements using the word "or" (v)

Conditional

If a number is a whole number then it is an interger! If --> then statements

Biconditional

If <--> and only if statements

Inverse

Negate both

Converse

Switch

Contrapostive

Switch and negate

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