Theorems and Postulates

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Oct 10, 2013 (3 years and 8 months ago)

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Theorems and Postulates


Chapter 1


Segment Addition Postulate


A





C


AB + BC = AC



Angle Addition Postulate


C


m

ACB + m

BCD = m

ACD




Chapter 2


Law of Detachment p

q


Law of Syllogism

p

q and q



r

then p


r


Addition Property If

a = b then a+c = b+c


Subtraction Property If a = b then a


c = b


c


Multiplication Property If a = b then ac = bc


Division Property If a = b and c


0 then a


c = b


c


Reflexive Property a = a,

, and

ABC

ABC


Symmetric Property If a = b, then b = a, If

then

If

A

B then




B

A


Transitive Property If a=b and b=c, then a=c, If





If

A


B and

B

C then

A

C


Substitution Property If a=b, then a can be substituted for b in any equation or expression



Right Angle Congruence Theorem


All
right a
ngles are con
gruent



B

A

B

D

Congruent Supplements Theorem


If 2 angles are supplementary to the same angle then






The 2 angles are congruent


Congruent Complements Theorem
-


If 2 angles are complementary to the same angle
then the 2 angles are congruent


Linear Pair Po
stulate


If two angles form a linear pair then they are supplementary.


Vertical Angle Theorem


Vertical angles are congruent




Chapter 3


Parallel Postulate


If there is a line and a point not on the line, then there is exactly one




Line thr
ough the point that is parallel to the given line.


Perpendicular Postulate
-

If there is a line and a point not on the line, then there is exactly





one line through the point that is perpendicular to the given line.


Corresponding Angles Postulate


If parallel lines are cut by a transversal, then the






corresponding angles are congruent.


Alternate Interior Angles
-

If parallel lines are cut by a transversal, then the alternate





interior angles are congruent.


Alternate Exterior An
gles
-

If parallel lines are cut by a transversal, then the alternate





exterior angels are congruent.


Consecutive Interior Angles
-

If parallel lines are cut by a transversal, then the






consecutive interior angles are supplementary.


Perpend
icular Transversal


If a transversal is perpendicular to one of the two parallel





lines, then it is perpendicular to the other.


Corresponding Angles Converse


If two lines are cut by a transversal so







that the corresponding ang
les are









congruent, then the lines are parallel.


Alternate Interior Angles Converse


If two lines are cut be a transversal so that the






alternate interior angles are congruent, then the lines





are parallel.


Consecutive Interior Angl
es Converse


If two lines are cut by a transversal so that the







consecutive interior angles are supplementary







then the lines are parallel.



Alternate Exterior Angles Converse


If two lines are cut be a transversal so that the






alt
ernate
exterior

angles are

congruent, then the






lines ar
e parallel.


Duel Parallel Line Theorem


If two lines are parallel to the same line then the lines are






parallel.


Duel Perpendicular Line Theorem


If two lines are perpendicular to the same

line then






the lines are parallel.


Lines that are parallel have the same slope.


Lines that are perpendicular have slopes that are the negative reciprocal of each other.




Chapter 4


Triangle Sum Theorem


the sum of the three interior angl
es of a triangle equal 180

.


Exterior Angle Theorem


the exterior angle of a triangle is equal to the sum of the 2





remote interior angles.


Third Angle Theorem


If 2 angles of one triangle are congruent to 2 angles of another





triangle th
en the third angles are congruent.


Side


Side


Side (SSS) Congruence


If 3 sides of one triangle are congruent to the







corresponding sides of another triangle, then the







2 triangles are congruent.


Side


Angle


Side (SAS) Congruence


If two sides and the included angle of one







triangle are congruent
to the corresponding







sides and included angle of another triangle,







then the triangles are congruent.


Angle


Side


Angle (ASA) Congruence
-

If two angles and the included side of one






triangle are congruent to the corresponding







angles and included side of another triangle,







then the triangles are congruent.




Angle


Angle


Side (AAS) Congruence
-

If two angles and the nonincluded side of one






triangle are congruent to the corresponding







angles and nonincluded side of another triangle,







then the triangles are congruent.


Base Angle Theorem


If 2 sides of

a triangle are congruent, then the angles opposite





them are congruent.


Converse of the Base Angle Theorem


If 2 angles of a triangle are congruent, then the







sides opposite them are congruent.


If a triangle is equilateral, then it is equia
ngular.


If a triangle is equiangular, then it is equilateral.


Hypotenuse


Leg (HL) congruence


If the hypotenuse and a leg of a right triangle are







congruent to the hypotenuse and corresponding leg







of a second right triangle, then the triang
les are







congruent.