
Through any two points there is exactly one line
postulate

through any three noncollinear points there is exactly one plane containing them
postulate

If two points lie in a plane, then the line containing those points lies in the plane
postulate

if two lines intersect, then they intersect in exactly one point
postulate

if two planes intersect, then they intersect in exactly one line
postulate

the points on a line can be put into a onetoone correspondence with the real numbers
ruler postulate

If B is between A and C, then AB+BC=AC
Segment Addition Postulate

given line AB and a point O on line AB, all rays that can be drawn from O can be put into a onetoone correspondence with the real numbers from 0 to 180
protractor postulate help classify angles by their measure

If S is in the interior of <PQR, then m<PQS + m<SQR= m<PQR
< Add. Post.
angle addition postulate

if p > q is a true statement and p is true, then q is true
law of detachment

if p > and q > r are true statements, then p > r is a true statement
law of syllogism

addition property of equality
if a=b, then a+c=b+c

subtraction property of equality
if a=b, then ac =bc

multiplication property of equality
if a=b, then ac=bc

division property of equality
if a=b, and c DOES NOT = 0, then a/c = b/c

reflexive property of equality
a=a
duhhh

symmetric property of equality
if a=b then b=a

transitive property of equality
if a=b and b=c, then a=c

substitution property of equality
if a=b, then b can be substituted for a in any expression

reflexive property of congruence
figure a ~= figure a
line EF =~ line EF
Reflex. Prop. of =~

Symmetric property of congruence
if figure a ~= figure B, then figure B ~= figure A
sym. prop. of ~=

Transitive Property of Congruence
If figure A~= figure B and figure B ~= figure C, than figure A~= figure C
trans. prop. of ~=

linear pair theorem
if two angles form a linear pair, then they are supplementary.
hypothesis: <A and <B form a linear pair
Conclusion: <a and <b are supplementary

congruent supplements theorem
if two angles are supplementary to the same angle *or to two congruent sides), then the two angles are congruent
hypothesis: <1 and <2 are supplementary. <2 and <3 are supplementary
conclusion: <1 ~= <3

right angle congruence theorem
all right angles are congruent
Hypothesis: <a and <b are right angles
conclusionL <a~=<b

congruent complements theorem
if two angles are complementary to the same angle (or to 2 congruent <'s) then the 2 <'s are congruent
Hypothesis: <1 and <2 are complementary. <2 and <3 are complementary
conclusion: <1~=<3

common segments theorem
given collinear points A, B, C, and D arranged, if line AB ~= line CD, then line AC~= line BD
Hypothesis: line AB ~= line CD
Conclusion: line AC ~= line BD

Vertical Angles Theorem
Vertical angles are congruent
Hypothesis: <a and <b are vertical angles
conclusion: <a ~= <b

if two congruent angles are supplementary, then each angle is a right angle
~= <'s suppl. > right <'s
 hypothesis:
 <1~= <2 <1 and <2 are supplementary
conclusion: <1 and <2 are right angles

corresponding angles postulate
if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

alternate interior angles theorem
if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent

alternate exterior angles theorem
if 2  lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

sameside interior angles theorem
if 2  lines are cut by a transversal, then the 2 pairs of sameside interior angles are supplementary

converse of the corresponding angles postulate
if 2 coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the 2 lines = 
Hypothesis: <1 ~= <2
conclusion: m  n

parallel postulate
through a point P not on line l, there is exactly one line parallel to l

converse of the alternate interior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the 2 lines are parallel

converse of the alternate exterior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the 2 lines are parallel

converse of the same side interior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of same side interior angles are supplementary, the the 2 lines are parallel

2 intersecting lines form a linear pair of congruent angles > the lines are perpendicular
theorem
conclusion: l is perpendicular to m

perpendicular transversal theorem
in a plane, if a transversal is perpendicular to one of 2 parallel lines, then it is perpendicular to the other line
hypothesis: transversal perpendicular to one parallel line
conclusion: transversal perpendicular to the other parallel line

2 lines perpendicular to the same line > the 2 lines are 
theorem

parallel lines theorem
in a coordinate plane, 2 nonvertical lines are parallel iff they have the same slope.
any 2 vertical lines are parallel

perpendicular lines theorem
in a coordinate plane, 2 non vertical lines are perpendicular iff the product of their slopes is 1
vertical and horizontal lines are perpendicular

triangle sum theorem
the sum of the angle measures of a triangle is 180 degrees

the acute angles of a right triangle are complementary
corollary

the measure of each angle of an equiangular triangle is 60 degrees
corollary
m<a = m<b = m<c = 60 degrees

exterior angle theorem
the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles

third angles theorem
if 2 angles of one triangle are congruent to 2 angles of another triangle > third pair of angles are congruent

side side side (SSS) congruence postulate
if 3 sides of 1 triangle are congruent to 3 sides of another triangle, then the triangles are congruent

side angle side (SAS) congruence
if 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle > the triangles are congruent

angle side angle (ASA) congruence
2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle > congruent triangles

angle angle side (AAS) congruence theorem
2 angles and non included side of 1 triangle are ~= to the corresponding angles and non included side of another triangle > triangles are congruent

Hypotenuseleg (HL) congruence
the hypotenuse and a leg of a right triangle are ~= to the hypotenuse and a leg of another right triangle > the triangles are congruent

CPCTC
corresponding parts of congruent triangles are congruent

isosceles triangle theorem
2 sides of a triangle are congruent > angles opposite those angles are congruent

equilateral triangle corollary
if a triangle is equilateral > it's equiangular

equiangular triangle corollary
equiangular triangle > equilateral triangle

perpendicular bisector theorem
point is on the perpendicular bisector of a segment > it is equidistant from the endpoints of the segment

converse of the perpendicular bisector theorem
a point is equidistant from the endpoints of a segment > perpendicular bisector of the segment

angle bisector theorem
if a point is on the bisector of an angle > equidistant from the sides of the angle

converse of the angle bisector theorem
if a point in the interior of an angle is equidistant from the sides of the angle > on the bisector of the angle

circumcenter theorem
the circumcenter of a triangle is equidistant from the vertices of the triangle
can be inside the triangle, outside the triangle, or on the triangle

incenter theorem
the incenter of a triangle is equidistant from the sides of the triangle

centroid theorem
the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side

triangle midsegment theorem
a midsegment of a triangle is  to a side of the triangle, and it's length is half the length of that side.

if 2 sides of a triangle are not congruent > larger angle is opposite the longer side
theorem

in triangle, the longer side is oposite the larger angle
theorem

triangle inequality theorem
the sum of any 2 side lengths of a triangle is greater than the 3rd side length

