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Closure Postulate of Addition

the sum of a + b is a unique real number

a+b=b+a

Commutative Property of Addition

(a+b)+c=a+(b+c)

Associative Property of Addition

a+0=a; 0+a=a

Additive Postulate of Zero

a+-a=0; -a+a=0

Postulate of Additive Inverses

Closure Postulate of Multiplication

The product of ab is a unique real number

ab=ba

Commutative Property of Multiplication

a(bc)= (ab)c

Associative Postulate of Multiplication

a*1=a; 1*a=a

Multiplicative Postulate of One

Postulate of Multiplicative Inverses

when a does not = 0, (1/a)*a=1; a*(1/a)=1

a(bc)=ab*ac

Distributive Property

a=a

Reflexive Property of Equality

Symmetric Property of Equality

For all real numbers x and y, if x = y, then y = x.

Transitive Property of Equality

For all real numbers x, y, and z , if x = y and y = z, then x = z.

Postulate of Comparison

one and only one of the following statements are true: a<b, a=b, or a>b.

Transitive Postulate of Inequality

if a<b and b<c, then a<c

Additive Postulate of Inequality

if a<b, then a+c<b+c

Multiplicative Postulate of Inequality

If a<b and 0<c, then ac<bc; if a<b and c<0, then bc<ac

Addition Property of Equality

if a=b, then a+c=b+c and c+a=c+b

Subtraction Property of Equality

if a=b, then a-c=b-c and c-a=c-b

Multiplicative Property of Equality

if a=b, then ac=bc and ca=ba

Division Property of Equality

if a=b and c does not equal 0, then a/c=b/c

Subtraction Property of Inequality

if a<b, then a-c<b-c

Division Property of Inequality

if a<b and c does not equal 0, then a/c<b/c

Substitution property

if a=b, "a" may be replaced by "b" and vice versa in any equation or inequality

Zero Product Property

If ab=0, then a=0 or b=0

Segment Addition Postulate

If B is between A and C, then AB + BC= AC

Definition of a midpoint

The midpoint of a segment is the point that divides the segment into two congruent segments.

Definition of a bisector

A bisector of a segment is a line, segment, ray, or plane whose intersection with segment AB is the midpoint of segment AB.

Congruent Angles

Angles with equal measures are..

Angle Addition Postulate

D is in the interior of angle ABC and only if m<ABD + m<DBC = m<ABC.

Adjacent Angles

Angles that are next to each other. They share a vertex and a common side.

midpoint theorem

if M is the midpoint of AB, then AM is congruent to MB

angle bisector theorem

if a point is on the bisector of an angle, then it is equidistant from the other two sides of the angle

Definition of a Right Angle

An angle is a right angle if and only if the angle has a measurement of 90 degrees

Definition of Perpendicular lines

Two lines are perpendicular if and only if the two lines meet to form congruent adjacent angles

Definition of Complementary Angles

Two angles are complementary if and only if their sum= 90 degrees

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Consecutive Interior Angles Postulate

When two parallel lines are cut by a transversal, the two interior angles on the same side are supplementary.

Consecutive Exterior Angles Postulate

When two parallel lines are cut by a transversal, the two exterior angles on the same side are supplementary.

Vertical angles theorem

If two angles are vertical angles, then they are congruent.

CPCTC

Corresponding parts of congruent triangles are congruent.

Definition of Midpoint

If a midpoint lies on a line, then both halves of the line ending at the midpoint are congruent to each other.

Definition of segment bisector

The two sides of the line are congruent to each other when cut by a segment bisector.

Definition of angle bisector

The two sides of the angle are congruent to each other when cut by an angle bisector.

Symmetric Property

when two segments or two angles are congruent, you can flip them over and they will still be congruent

Substitution property

a(b) = (ab)

Transitive Property

if two segments or two angles are congruent to the same segment of angle, they are congruent to each other

Reflexive Property

Anything equals itself; a shared piece.

SSS triangle congruence postulate

If the 3 sides of one triangle are congruent to the 3 sides of another triange then the triangles are cogruent.

SAS triangle congruence postulate

If two angles and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

ASA triangle congruence postulate

Two triangles are congruent if two angles and the included side of one triangle are congruent to the two angles and the included side of the other triangle.

AAS Triangle Congruence Theorem

Two triangles can be proved congruent with this proof if they have two congruent angles and a congruent side in that order or reverse.

Definition of Right Angle

A right angle has 90 degrees.

Ruler Postulate

The points on any line or line segment can be put into one-to-one correspondence with real numbers.

Protractor Postulate

Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.

Hypotenuse-Leg Theorem

If the hypotenuse and a corresponding side of two triangles are congruent, then the triangles are congruent.

Notebook Paper Theorem

If a transversal cuts two parallel lines and one of the angles measures 90 degrees, then the rest of the angles measure 90 degrees.

Base Angles Theorem

If 2 sides of a triangle are congruent then the angles opposite are congruent (used with isosceles)

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

Slope Formula

The formula for finding the slope of a line.

a=b, b=a

Symmetric Property

ab+bc=ac

Segment Addition Postulate

False

True or False? A line segment has finite points.

180 degrees

sum of degrees in any triangle

90 degrees.

Complementary angles add up to ________ degrees.

ray, interior

Adjacent angles are angles that share a common ______________, but no ______________ points.