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Closure Postulate of Addition
the sum of a + b is a unique real number
Commutative Property of Addition
Associative Property of Addition
Additive Postulate of Zero
Postulate of Additive Inverses
Closure Postulate of Multiplication
The product of ab is a unique real number
Commutative Property of Multiplication
Associative Postulate of Multiplication
Multiplicative Postulate of One
Postulate of Multiplicative Inverses
when a does not = 0, (1/a)*a=1; a*(1/a)=1
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
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.
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.
Angles that are next to each other. They share a vertex and a common side.
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.
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.
when two segments or two angles are congruent, you can flip them over and they will still be congruent
a(b) = (ab)
if two segments or two angles are congruent to the same segment of angle, they are congruent to each other
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.
The points on any line or line segment can be put into one-to-one correspondence with real numbers.
Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.
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)
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
The formula for finding the slope of a line.
Segment Addition Postulate
True or False? A line segment has finite points.
sum of degrees in any triangle
Complementary angles add up to ________ degrees.
Adjacent angles are angles that share a common ______________, but no ______________ points.