Download Definitions, Postulates, Properties and Theorems – and the Pictures

Date _________
Period_________
A Word Bank of Possible Reasons in Proofs
Definitions, Postulates, Properties and
Theorems – and the Pictures
Statement in Proof (Example)
A
AB  XY
ABX & ABY are right angles Definition of perpendicular
lines
X
mABX = 90°
mABY = 90°
Y
B
BD bisects ABC
Definition of
right angles
A
D
ABD  CBD
or
B
Reason Given in Proof
1
2
C
Definition of an angle
bisector
1  2
M is the midpoint of AB
M
B
A
AM  MB
Definition of a midpoint
m1  m2  180
Angle addition
(postulate)
(just the picture)
1
2
(just the picture)
1  3
1
or
2
4
3
2   4
Vertical angles theorem
(VAC)
m1  m2  m3 and
m2  m3  m4
so
m1  m4
1  2
2  3
so… 1  3
Any time you “plug in” angle measures or
side lengths for other angles or side lengths
you are using Substitution. This only works
with =, not .
The Transitive property is like substitution
but only when it fits the pattern like the one
shown (can use with .)
Any time 2 triangles share a side.
C
(both white and grey
triangles share side CD)
CD  CD
Substitution
Transitive Property
Reflexive Property
D
AB  YZ
so…
YZ  AB
DE = LM
so…
DE + XY = LM + XY
You can flip sides of an equation..
You can add a length to an equation just like
you can add a number to an equation in
algebra
Symmetric Property
Addition Property
(added XY to both sides)
You can also subtract a length from an
equation the same way…
DE + XY = LM + XY
so…
DE = LM
Subtraction Property
(subtracted XY from both sides)
6x  30
so…
x5
There are also multiplication and division
properties, but they don’t work the same
way for lengths. Use them for algebra.
Division
Property
(just the picture)
A
B
C
AB  BC  AC
(the two pieces of the segment
make the entire segment)
A
(just the picture)
m1  m2  mABC
D
1
B
Segment Addition
(postulate)
or
mABD  mDBC  mABC
2
C
Angle Addition Postulate
Y
B
A
C
SSS Postulate
ABC  XYZ
SAS Postulate
ABC  XYZ
ASA Postulate
ABC  XYZ
AAS Theorem
ABC  XYZ
HL Theorem
(Hypotenuse-Leg)
Z
X
B
ABC  XYZ
Y
A
X
C
Z
B
Y
A
C
X
Z
B
Y
A
C
X
B
Z
Y
C
A
Z
X
Once you say that
ABC  XYZ
by any of the 5 methods,
then…
AB  XY
A  X
B
Y
A
C
X
Z
B  Y & BC  YZ
C  Z
AC  XZ
CPCTC
Corresponding Parts of
Congruent Triangles are
Congruent
2
m1  m2  m3  180
1
Triangle Sum Theorem
(3 angles add up to 180°)
3
2
1
1  2
If two angles of one
triangle are congruent to
two angles of another,
then the 3rd angles are
congruent too.
C
A  B
Isosceles Triangle Theorem
XZ  YZ
Converse of the Isosceles
Triangle Theorem
1  2
Corresponding Angles
Postulate
3  4
Alternate Interior Angles
Theorem
5  6
Alternate Exterior Angles
Theorem
B
A
Z
X
Y
(lines are
parallel in)
this and the
next 4)
1
2
3
4
5
6
7
m7  m8  180
8
Same Side Interior Angles
Theorem
9
m9  m10  180
10
Same Side Exterior Angles
Theorem
l  m and l  n
l
m
n
mn
If two lines are
perpendicular to the same
line, then those two lines
are parallel