Segment Addition Postulate
If B is between A and C, then AB + BC= AC
Angle Addition Postulate
If point B is in the interior of Angle AOC, then the measurement of Angle AOB + the measurement of Angle BOC = the measurement of Angle AOC. If Angle AOC is a straight angle, and B is any point not on line AC, then the measurement of Angle AOC + the measu
A line contains....
at least two points
A plane contains.....
at least 3 points
space contains
at least 4 points not all in one plane
through any 2 points
there is exactly one line
through any 3 points
there is at least one plane
any three non collinear points contain
exactly one plane
If two points are in a plane
then the line that contains those points is in that plane
If two planes intersect
then their intersection is a line
if two parallel lines are cut by a transversal
then corresponding angles are congruent.
If two lines are cut by a transversal and corresponding angles are congruent
the lines are parallel
SSS Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
ASA Postulate
if two angles and the included side of one triangle are congruent to two angles and the included side of the second, the triangles are congruent
SAS Postulate
if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
If two lines intersect, (1-1)
then they intersect in exactly one point
Through a line and a point not in the line (1-2)
there is exactly one plane
If two lines intersect (1-3)
then exactly one plane contains these lines
Definition of Midpoint (Midpoint Theorem)
If M is the midpoint of line AB then AM=MB
Angle Bisector Theorem
If ray BX is the bisector of angle ABC, Angle ABX= Angle XBC
(cuts the angle in half)
Vertical Angles Theorem
Vertical angles are congruent
Def. of Perpendicular Lines
If two lines are perpendicular they form congruent adjacent angles. Converse= If two lines form congruent adjacent angles, then the lines are perpendicular. If the exterior sides of two adjacent acute angles are perpendicular then the angles are complemen
SCAC
If two angles are supplements of congruent angles (or of the same angle), then the 2 angles are congruent.
CCAC
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
If two parallel planes are cut by a third plane (property of parallel lines
then the lines of intersection are parallel
Alternate interior angles theorem
If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (Converse also)
Same side interior angles theorem
If two parallel lines are cut by a transversal, then same side interior angles are supplementary. (Converse also)
If a transversal is perpendicular to one of two parallel lines, (property of parallel lines)
then it is perpendicular to the other one also
In a plane two lines perpendicular to the same line
are parallel
Through a point outside a line there is exactly,
1. one line parallel to the given line
2. one line perpendicular to the given line
Two lines parallel to a third line (Property of parallel lines)
are parallel to each other
Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180
3rd Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Each angle of an equiangular triangle
has a measure of 60
In a triangle, there can be at most
one right or obtuse angle
The acute angles of a right triangle
are complementary
Remote Interior Angle Theorem
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles
The sum of the measures of the angles of a convex polygon with n sides
Is (n-2)180
The sum of the measures of the exterior angles of any polygon, one angle at each vertex,
is 360
ITT
If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Converse= ITTC)
An equilateral triangle
is also equiangular (Converse)
An equilateral triangle has
three 60 degree angles
The bisector of the vertex angle of an isosceles triangle is
perpendicular to the base at its midpoint
AAS
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
HL
If the hypotenuse and a leg of one triangle are congruent to the corresponding parts of another right triangle , then the triangles are congruent.
If a point lies on the perpendicular bisector of a segment then
the point is equidistant from the endpoints of the segment
(converse)
If a point lies on the bisector of an angle then
the point is equidistant from the sides of the angle (Converse)
Definition of Parallelogram
2 pairs of parallel sides
Properties of a parallelogram
1. Diagonals bisect each other
2. Opposite sides are congruent
3. Opposite Angles are congruent
4. Consecutive angles are supplementary
To prove a parallelogram
One pair of congruent parallel sides
If two lines are parallel then the points on one line
are equidistant from the other line
If three parallel lines cut off congruent segments on one transversal
then they cut off congruent segments on every transversal
Triangle Midline Theorem
A line that passes through the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side The segment that joins the midpoints of two sides of a triangle:
1. Is parallel to the third side
2. Is half as
Definition of a Rectangle
4 right angles
Definition of a Rhombus
4 equal sides
Definition of a Square
4 right angles and 4 equal sides
Properties of a Rectangle
1. Equal diagonals
2. All properties of a Parallelogram
Properties of a Rhombus
1. All properties of a Parallelogram
2. Diagonals bisect angles
3. Diagonals are perpendicular (Only need one pair)
Properties of a Square
1. All properties of a Parallelogram, Rectangle, and Rhombus
Right Triangle Theorem
The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
If an angle of a parallelogram is a right angle,
then the parallelogram is a rectangle.
If two consecutive sides of a parallelogram are congruent,
then the parallelogram is a rhombus
Isosceles Trapezoid Theorem
The base angles of an Isosceles Trapezoid are congruent
Def. of a Trapezoid
One pair of parallel bases.
The median of a trapezoid
1. is parallel to the bases
2. has a length equal to the average of the base lengths
Exterior Angle Inequality Theorem
The exterior angle of a triangle is greater than each remote interior angle.
LSOLA
largest side opposite largest angle. If one side is larger than another IN ONE TRIANGLE, then the angle opposite the largest side is greater than the angle opposite the smaller side. (Converse)
Hinge Theorem (SAS and SSS inequality theorem)
If 2 sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second
The Triangle Inequality Theorem
The sum of any two sides of a triangle is greater than the third
The perpendicular segment from a point to a line is
the shortest segment from the point to the line
The perpendicular segment from a point to a plane
is the shortest from the point to the plane
altitude of a triangle
the perpendicular segment from a vertex to the line containing the opposite side.
Kite
two pairs of congruent sides, but opposite sides are not congruent.
median of a triangle
a segment from a vertex to the midpoint of the opposite side
skew lines
lines that are not coplanar
space
the set of all points
regular polygon
a polygon that is both equiangular and equilateral
parallel lines
coplanar lines that do not intersect
perpendicular bisector
a line (or ray or segment) that is perpendicular to the segment at its midpoint
+ makes bigger theorem
A + B is greater than A
Transitive Property
If A is greater than B and B is greater than C than A is greater than C.
Substitution Property
If A = B and C = B then A = C
Equilateral triangle
a triangle with all equal sides
Right Triangle
A triangle with one right angle
Isosceles Triangle
A triangle with two equal sides
Scalene Triangle
A triangle with no equal sides
Obtuse Triangle
A triangle with one obtuse angle
Complementary Angles
Two angles that add to equal 90 degrees
Supplementary Angles
Two angles that add to equal 180 degrees.
Obtuse Angle
an angle less than 180 but greater than 90 degrees
Acute Angle
an angle greater than 0 but less than 90 degrees
Straight Angle
180 degrees; a straight line
Acute Triangle
three acute angles
To find each individual interior angle of a regular polygon.
(n-2)180
--------------
n
each exterior angle is
supplementary to the interior angle
equiangular
all equal angles
polygon
A closed figure with at least 3 sides. Made up of straight lines
Regular Polygon
A polygon with all equal sides and angles
Congruent
equal
Convex Polygon
A polygon such that no line containing a side of the polygon contains a point in the interior of the polygon
Corresponding Angles
Two angles in corresponding positions relative to two lines
Adjacent Angles
Two angles in a plane that have a common vertex and a common side but no common interior points
Bisector
divides a line, segment, ray, or an angle in half.
Collinear Points
Points all in one line
Coplanar Points
Points all in one plane.
Alternate Interior Angles
Two nonadjacent interior angles on opposite sides of a transversal.
Exterior angle of a triangle
the angle formed when one side of the triangle is extended.
Parallel Lines
Coplanar lines that do not intersect
Parallel planes
planes that do not intersect
Perpendicular Lines
Two lines that intersect to form right angles
ray
A line segment with one starting point and continues on forever in another direction. Part of a line.
plane
undefinable. made up of at least three points. a two dimensional figure that spreads out infinitely. Commonly represented by a 4 sided figure.
line
a figure made up of at least two points that goes on forever in both directions.
Median of a trapezoid
The segment that joins the midpoints of the legs
Midpoint of a segment
the point that divides the segment into 2 congruent segments.
In an isosceles trapezoid diagonals
are equal to each other.
CPCTC
Corresponding parts of congruent triangles are congruent
Reflexive Property
equals itself
AC=AC
Diagonals of a kite are
perpendicular
All right angles
are congruent
Addition Property of equality
Adding the same or something of equal value to both sides of an equation or to a segment, line, ray, and angle. (Multiplication, Subtraction, and Division properties as well).