moise geo

Trichotomy Property

For every x and every y, one and only one of the following is true: x>y, x=y, x<y

Distance Postulate

to every pair of points, there corresponds a unique positive number.

Ruler Postulate

the points of a line can be placed in correspondence with the real number system such that: 1.) to every point there corresponds a real number 2.) to ever real number there corresponds a point 3.) The distance of any two numbers is the absolute value of their coordinates

Ruler Placement Postulate

given 2 points, P and Q, of a line, the coordinate system can be chosen in such a way that the coordinates of P is 0 and Q is a positive number.

Definition of Between

B is between A and C if A, B, and C are different points and if AB+BC=AC

Line Postulate

For every 2 points there is exactly one line that contains both. (2 points determine a line)

Definition of Segment

the union of points A and B and all points C such that A-C-B

Definition of Ray

the union of segment AB and all points C such that A-B-C

Point-Plotting Theorem

Let there be ray AB and a positive number x. There is exactly one point (P) on the ray such that the distance of AP=x.

Definition of Midpoint

A-M-B and AM=MB

Midpoint Theorem

Every segment has exactly one midpoint

Definition of Bisect

separate into two equal parts

Definition of Bisector

Midpoint of any line, plane, ray, or segment containing the midpoint and not the rest of the segment.

Intersection of Lines Theorem

If 2 different lines intersect, their intersection is exactly 1 point.

Plane Postulate

3 noncollinear points determine a plane.

Space Postulate

4 noncoplanar points determine a space

Flat Plane Postulate

If 2 points of a line are in a plane, the entire line lies in the plane

Theorem 3-2

If a line intersects a plane not containing it, the intersection is exactly one point.

Theorem 3-3

given a line and a point not on it, there is exactly one plane containing both

Theorem 3-4

Given 2 intersecting lines, there is exactly one plane containing both.

Intersection of Planes Postulate

If two different planes intersect, their intersection is a line.

Plane Separation Postulate

Given a line and a plane containing it, the points of a plane not on the line form 2 sets such that

1. each is convex

2. if P is on one side of the line and Q on the other, then line PQ intersects the line.

Space Separation Postulate

Given a plane in space, the points of space not on the plane form 2 sets

Angle

the union of two nonopposite rays with a common endpoint

Interior of angle BAC

set of all points P in the plane such that 1.) P and B on the same side of line AC 2.) P and C on the same side of line AB

Exterior of angle BAC

Set of all points not included in the interior or on the angle

Triangle

union of segments of 3 noncollinear points

Interior of a Triangle

all points that lie on the interior of each angle

Exterior of a Triangle

all points in the plane but not in the interior or on the triangle

Protractor

instrument used to measure angles unit is degrees

Measure

number of degrees in an angle

Angle Measurement Postulate

To every angle BAC there corresponds a real number between 0 and 180

Angle Construction Postulate

For every real number (r) between 0 and 180 there is exactly one ray AP such that the measure of angle PAB = r.

Angle Addition Postulate

If D is in the interior of angle BAC, then the measure of angle BAC= the measure of angle BAD+the measure of angle DAC

Definition of Linear Pair

The two angles formed if AB and AD are opposite rays and AC is any other ray.

Supplementary Angles

angles whose sum is 180

Supplement Postulate

Linear pairs are supplementary.

Acute Angles

an angle whose measure is between 0 and 90

Right Angles

an angle whose measure is exactly 90

Obtuse Angles

an angle whose measure is between 90 and 180

Complementary Angles

angles whose sum is 90

Congruent Angles

two angles with the same measure

Definition of Perpendicular

two rays are perpendicular if they form the sides of a right angle. 2 sets are perpendicular if each is a segment, ray, or line, they intersect, and the lines containing them are perpendicular.

Reflexive

a=a

Symmetric

x=y then y=x

Transitive

if x=y and y=z, then x=z

Right Angle Theorem

any 2 right angles are congruent

Theorem 4-2

If 2 angles form a linear pair and are congruent, then each is a right angle.

Theorem 4-3

If 2 angles are complementary, then each is acute

Theorem 4-5

If 2 angles are both congruent and supplementary, each is a right angle

Supplement Theorem

Supplements of 2 congruent angles are congruent.

Complement Theorem

Complements of 2 congruent angles are congruent

Definition of vertical angles

Two angles are vertical angles if their sides form 2 pairs of opposite rays

Vertical Angle Theorem

Vertical angels are congruent

Theorem 4-9

Perpendicular lines form 4 right angles

Right Angle Lemma

If 2 angles are congruent and one of them is a right angle, then the other is also a right angle.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

SAS Postulate

Every SAS correspondence is a congruence. SAS can be used to prove ASA an SSS.

ASA Theorm

Every ASA correspondence is a congruence

SSS Theorem

Every SSS correspondence is a congruence

Definition of Angle Bisector

If D is in the interior of angle BAC and angle BAD is congruent to angle DAC, then ray AD is a bisector of angle BAC

Angle Bisector Theorem

Every angle has exactly one bisector

Isosceles Triangle

a triangle with at least two congruent sides

Isosceles Triangle Theorem

If a triangle has 2 congruent sides, then the angles opposite them are congruent.

Equilateral Triangle

all three sides are congruent

Equiangular Triangle

all three angles are congruent

Scalene Triangle

no 2 sides are congruent

Quadrilateral

4 sided figure such that 1.) all points are coplanar 2.) no 3 are collinear 3.) segments intersect only at end points

Rectangle

a quadrilateral with four right angles

Square

a rectangle with four congruent sides

Median of a triangle

segment from the vertex to midpoint of the opposite side

Angle Bisector of a triangle

Segment that lies in the ray bisecting the angle and the end points are the vertex and a point on the other side

Existence

"at least one," or just "one"

Uniqueness

"at most one" or just "only one"

Perpendicular Bisector

line perpendicular to a segment at its midpoint

Perpendicular Bisector Theorem

in a plane, the perpendicular bisector is the set of all points of the plane equidistant from the endpoints of the segment

Corollary of the Perpendicular Bisector Theorem

Given segment AB and line L coplanar. If 2 points of L equidistant from A and B, then L is the perpendicular bisector of segment AB

Right Triangle

a triangle with a right angle

Hypotenuse

the side across from the right angle in a triangle

Leg of a triangle

two sides of a right triangle that make up the right angle

Auxiliary Set

supplemental information introduced to complete a proof

Parts Theorem

The whole is greater than its parts 1.) If A-D-B, then AB>AD and AB>BD 2.) If D is in the interior of angle ABC, then angle ABC>angle ABD and angle ABC>angle CBD

Remote Interior Angles

the two angles that do not form a linear pair with an exterior angle.

Exterior Angle

forms a linear pair with an angle of the triangle.

Exterior Angle Theorem

an exterior angle is greater than either remote interior angle

Corollary to Exterior Angle Theorem

if a triangle has one right angle, then the others are acute

AAS Theorem

Every AAS correspondence is a congruence.

Hypotenuse Leg Theorem

Special case of Angle Side Side. If one hypotenuse and one leg of one right triangle is congruent to some other, then the triangles are congruent.

3 Bears Theorem

Longest angle of a triangle is opposite the longest side in a triangle. Shortest angle opposite the shortest side.

Triangle Inequality Theorem

The sum of any 2 sides of a triangle is greater than the length of the third side

1st Minimum Theorem

Shortest segment from a point to a lien is the perpendicular.

Distance From a point to a line

the length of the perpendicular segment to the line.

Hinge Theorem

If 2 sides of one triangle are congruent to 2 sides of another triangle, and the included angle of one triangle is greater than the included angle of the other, then the third side of one triangle is greater then the third side of the other triangle. (Also works with using the third side in the given info instead of the included angles.)

Altitude of a Triangel

a segment from vertex to a line containing the opposite side.

Line and a Plane are Perpendicular

a line and a plane intersect and every line in the plane through the point of intersection is perpendicular to the given line

Basic Theorem on Perpendiculars

If a line is perpendicular to each of the two intersecting lines at their point of intersection, it is perpendicular to the plane containing them.

Theorem 8-3/5

Through a given point of a given line, there is exactly one plane perpendicular to the line at a given point.

Perpendicular Bisecting Plane Theorem

the Perpendicular bisector of a plane is the set of all points equidistant from the endpoints of the segment.

Theorem 8-7

Two lines that are perpendicular to the same plane are coplanar

Theorem 8-8

Through a given point there passes one and only one plane perpendicular to the given line.