Postulates, Theorems, Corollaries & Definitions
(Things that can be used in proofs)
Items are organized by type of figure. Please note that addition, subtraction,
division, multiplication and substitution properties can be used in any proof.
Segments
 Segment Addition Postulate 
(part + part = whole)
PQ + QR = PR
 Midpoint Theorem  if M is the midpoint of AB, then AM = MB
 Reflexive Property  AB = AB
 Symmetric Property  if AM = MB, then MB =AM
 Transitive Property  if AB = CD and CD = EF, then AB = EF
 Definition of Congruence  if AB = CD, then AB = CD
 Definition of Bisector  if line l bisects MO at N, then MN = NO or MN=NO
Angles
 Angle Addition Postulate 
(part + part = whole)
mPQR + mRQS = mPQS
 Supplement Theorem  if two angles form a linear pair, then they are
supplementary. Ex:
1 and 2 form a linear pair, so
m1 + m2 = 180
 Reflexive Property  ABC = ABC
 Symmetric Property  if ABC = DEF, then DEF = ABC
 Transitive Property  if ABC = DEF and DEF = GHI, then
ABC = GHI
 Congruent Supplements Theorem  angles supp. to the same angle or to
= angles are =.
Ex:
1 and 2 are supp. and 2 and 3 are supp., therefore 1 = 3
 Congruent Complements Theorem  angles comp. to the same angle or
to = angles are =.
Ex:
4 and 5 are comp. and 5 and 6 are comp., therefore 4 = 6
 Right Angles Theorem  All right angles are =.
 Vertical Angles Theorem  Vertical angles are =.
Ex:
7 = 9 and 8 = 10
 Perpendicular Lines intersect to form 4 right angles.
Ex:
if a  b, then 11, 12, 13 and 14
are all right angles
 Definition of Congruent Angles  if A = B, then mA = mB
 Definition of Angle Bisector 
Ex:
if UV bisects TUW, then
mTUV = mVUW
or TUV = VUW
Use the following picture for all theorems involving parallel lines:
 Corresponding Angles Postulate  If two parallel lines are cut by a
transversal, then corresponding angles are congruent.
Ex: in the picture 1 = 5, 2 = 6, 3 = 7, 4 = 8
 Alternate Interior Angles Theorem  If two parallel lines are cut by a
transversal, then alternate interior angles are congruent.
Ex: in the picture 3 = 6 and 4 = 5
 Consecutive Interior Angles Theorem  If two parallel lines are cut by a
transversal, then consecutive interior angles are supplementary.
Ex: in the picture m3 + m5 = 180 and m4 + m6 =180
 Alternate Exterior Angles Theorem  If two parallel lines are cut by a
transversal, then alternate exterior angles are congruent.
Ex: in the picture 1 = 8 and 2 = 7
 Perpendicular Transversal Theorem  If a line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
Ex:
if a  b and b c, then a  c
The following theorems are all converses of those on the other side of this page.
The same picture is used.
 Corresponding Angles Converse  If two lines are cut by a transversal
and the corresponding angles are congruent, then the lines are parallel.
Ex: in the picture if 1 = 5, or 2 = 6, or 3 = 7, or 4 = 8,
then m
n
 Alternate Interior Angles Converse  If two lines are cut by a transversal
and alternate interior angles are congruent, then the lines are parallel.
Ex: in the picture if 3 = 6 or 4 = 5, then m
n
 Consecutive Interior Angles Converse  If two lines are cut by a
transversal and consecutive interior angles are supplementary, then the
lines are parallel.
Ex: if m3 + m5 = 180 and m4 + m6 =180, then m
n
 Alternate Exterior Angles Converse  If two lines are cut by a transversal
and alternate exterior angles are congruent, then the lines are parallel.
Ex: in the picture if 1 = 8 or 2 = 7, then m
n
 Perpendicular Transversal Converse  In a plane, if two lines are
perpendicular to the same line, then they are parallel.
Ex:
if a  b and a  c, then b
c