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.
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