Geometry Postulates, Theorem, Properties and Definitions

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Ruler Postulate

(1) The points on a line can be paired with real numbers in such a way that any two points can have the coordinates 1 and 0 (2) Once a coordinate system has been chose in this way, the distance between any two points equals the absolute value of the difference of their coordinates

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

Protractor Postulate

On line AB in a given plane, choose any point O between A and B. Consider →OA and →OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: a) →OA is paired with 0, and →OB with 180. b) If →OP is paired with x, and →OQ with y, then m∠POQ = |x - y|.

If point B lies in the interior of ∠AOC then m∠AOB +∠BOC = m∠AOC. If ∠AOC is a straight angle and B is any point not on AC then m∠AOB + ∠BOC=180°

Postulate 5

A line contains at least 2 points; a plane contains at least 3 points not all in one line; space contains at least 4 points not all on 1 plane

Postulate 6

Through any 2 points there is exactly one line

Postulate 7

Through any 3 points there is at least one plane, and through any noncollinear points there is exactly one plane

Postulate 8

If two points are on a plane, then the line that contains the points is in that plane

Postulate 9

If two planes intersect, then their intersection is a line

Theorem 1-1

If 2 lines intersect, then they intersect in exactly one point

Theorem 1-2

Through a line and a point not in the line there is exactly on plane

Theorem 1-3

If two lines intersect then exactly one plane contains the lines

Midpoint Theorem

If M is the midpoint of AB then AM=½AB and MB=½AB

Angle Bisector Theorem

If ray BX is the bisector of ∠ABC then m∠XBC=½m∠ABC and m∠XBC =½m∠ABC

Theorem 2-3

Vertical angles are congruent

Theorem 2-4

If 2 lines are perpendicular, then they form congruent adjacent angles

Theorem 2-5

If two lines form congruent adjacent angles then the lines are perpendicular

Theorem 2-6

If the exterior sides of 2 adjacent acute angles are perpendicular then the angles are complementary

Theorem 2-7

If two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent

Theorem 2-8

If two angles are complements of congruent angles (or of the same angle) then the two angles are congruent

Space

The set of all points

Collinear points

Points all in one line

Intersection

a point where lines intersect

Postulates

or axioms, statements accepted without proof

Congruent

2 objects that have the same size and shape are called congruent

Congruent segments

Segments that have equal lengths

Midpoint

The point that divides the segment into 2 congruent segments

Bisector

Is a line, segment, ray or plane that intersects the segment at its midpoint

angle

A figure formed by two rays with a common endpoint

Congruent angles

Angles that have equal measures

Two common angles in a plane that have a common vertex and a common side, but not common interior points

Bisector of a line

The ray that divides the angle into two congruent adjacent angles

If a=b and c=d then a+c=b+d

Subtraction property

If a=b and c=d then a-c=b-d

is a=b the ca=cb

division property

If a=b and c≠0 the a÷c = b÷c

Substitution property

If a=b then either a or b may be substituted for one another

a=a

if a=b then b=a

Transitive property

If a=b and b=c then a=c

DE≈DE ∠D≈∠D

(POC) Symmetric Property

If DE≈FG then FG≈DE If ∠D≈∠E then ∠E≈∠D

(POC) Transitive Property

If DE≈FG and FG≈JK the DE≈JK. If ∠D≈∠E and∠E≈∠F then ∠D≈∠F

Complementary Angles

Two angles whose sum is 90 degrees

Supplementary angles

2 angles whose sum is 180°

Perpendicular Lines

2 lines that intersect to form a right angle

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