Axioms. Postulates, Theorems, Definitions

collinear points

points are collinear if and only if they are on the same line

distance between two points

if you take the absolute value of the difference of the coordinates of the 2 points, then you get the distance between them.

congruent segments

two segments are congruent if and only if they have the same length (measure).

bisect

an object bisects a segment if and only if it divides it into two congruent parts (in half).

midpoint of a line segment

a point is a midpoint if and only if it divides the segment into 2 congruent parts.

segment bisector

a line or part of a line is a segment bisector if and only if it intersects a segment at its midpoint and divides it into 2 congruent parts.

angle

an object is an angle if and only if it is the union of two rays with a common endpoint called the vertex

acute angle

an angle is acute if and only if its measure is greater than 0 and less than 90.

right angle

an angle is a right angle if and only if it measures 90.

obtuse angle

an angle is obtuse if and only if its greater than 90 and less than 180.

straight angle

An angle is straight if and only if it measure 180.

perpendicular lines

lines are perpendicular if and only if they intersect to form right angles.

perpendicular bisector

a line is a perpendicular bisector if and only if its is perpendicular to a line segment at its midpoint.

congruent angles

angles are congruent if and only if they have the same measure.

angle bisector

a ray is an angle bisector if and only if it divides an angle into two congruent parts.

adjacent angles

angles are adjacent if and only if they are an angle pair that has a common vertex, common side, and no interior points in common.

vertical angles

two angles are vertical if and only if they are two nonadjacent angles formed by intersecting lines.

complementary angles

two angles are complementary if and only if the sum of their measures is 90.

supplementary angles

two angles are supplementary if and only if their measures add up to 180

linear pairs

angles are a linear pair if and only if they are two adjacent angles whose non-common sides form a straight line.

perpendicular pairs

angles are perpendicular pair if and only if they are 2 adjacent angles whose non-common sides are perpendicular.

For any two points…

there exists one and only one line containing both points.

the intersection of two lines is…

a point

for any three non-collinear points…

there exists one and only one plane containing all 3 points

the intersection of two planes is

a line

If two points are in a plane

then the line containing them is also in the plane

A line segment has… midpoint

one and only one

segment addition axiom

the whole is equal to the sum of its parts

angle addition axiom

the whole is equal to the sum of its parts.

an angle has …

one and only one bisector.

supplement axiom

the angles of a linear pair are supplementary.

complement axiom

the angles of a perpendicular pair are complementary.

addition property (axiom/postatulate)

a=b, then a+c = b+c

Subtraction property

a=b, then a-c=b-c

multiplication property

a=b, ac=bc

division property

a=b, then a/c=b/c

reflexive property

a=a

symmetric property

a=b, then b=a

transitive property

a=b and b=c, then a=c

substitution property

if AB+CD=10 and CD=5, then AB+5=10

If lines are parallel then corresponding angles are

congruent

If corresponding angles are congruent then lines are

parallel

Euclids Axiom

Through a point not on a given line, there is exactly one parallel/perpendicular line to the given line.

The Midpoint Theorem

If M is the midpoint of segment AB then: AM=1/2AB, AB=2AM

The Angle Bisector Theorem

If ray BD bisects angle ABC then measure of angle ABD = 1/2 the measurement of angle ABC, measurement of angle ABC = 2 measurement of angle ABD

Supplementary theorem

If two angles are supplementary to the same angle then they are congruent. If two angles are supplementary to congruent angles, then they are congruent.

Complementary theorem

If two angles are complementary to the same angle then they are congruent. If two angles are complementary t congruent angles then they r congruent.

All right angles are..

congruent

If two angles are congruent and supplementary then they r

right angles

If 2 lines intersect to form congruent adjacent angles then they r

perpendicular

If 2 lines r perpendicular then they intersect to form

congruent adjacent angles

vertical angles are

congruent

If lines are parallel then alternate interior angles are

congruent

If alternate interior angles are congruent then lines are

parallel

If lines are parallel then same side interior angles are

supplementary

If same side interior angles are supplementary then the lines are

//

If a transversal is perpendicular to one of the two parallel lines then it is

perpendicular to the other line.

If 2 lines r perpendicular to the same line, then

they r parallel to each other.

If a transversal is perpendicular to each of two lines then the lines r

parallel

If two lines are parallel to the same line, then they r

parallel to each other.