Algebraic Properties and Proofs
Name __________________________
You have solved algebraic equations for a couple years now, but now it is time to
justify the steps you have practiced and now take without thinking… and acting without
thinking is a dangerous habit!
The following is a list of the reasons one can give for each algebraic step one may
take.
ALGEBRAIC PROPERTIES OF EQUALITY
If a = b, then a + c = b + c
If a = b, then a – c = b – c
ADDITION PROPERTY OF EQUALITY
SUBTRACTION PROPERTY OF
EQUALITY
MULTIPLICATION PROPERTY OF
EQUALITY
DIVISION PROPERTY OF EQUALITY
If a = b, then a · c = b · c
If a = b, then
DISTRIBUTIVE PROPERTY OF
MULTIPLICATION OVER ADDITION or
OVER SUBTRACTION
SUBSTITUTION PROPERTY OF
EQUALITY
REFLEXIVE PROPERTY OF EQUALITY
SYMMETRIC PROPERTY OF
EQUALITY
TRANSITIVE PROPERTY OF
EQUALITY
a
c
=
b
c
a(b + c) = ab + ac
a(b – c) = ab – ac
If a = b, then b can be substituted for
a in any equation or expression
For any real number a, a = a
If a = b, then b = a
If a = b and b = c, then a = c
Complete the following algebraic proofs using the reasons above. If a step requires
simplification by combining like terms, write simplify.
Given:
Prove:
1.
2.
3.
4.
5.
3x + 12 = 8x – 18
x=6
Statements
3x + 12 = 8x – 18
12 = 5x – 18
30 = 5x
6=x
x=6
Reasons
1.
2.
3.
4.
5.
Given:
Prove:
1.
2.
3.
Given:
Prove:
3k + 5 = 17
k=4
Statements
3k + 5 = 17
3k = 12
k=4
1.
2.
3.
– 6a – 5= – 95
a = 15
Statements
Given:
Prove:
Reasons
Reasons
3(5x + 1) = 13x + 5
x=1
Statements
Reasons
Given:
Prove:
7y
y
84
2y
61
29
Statements
Given:
Prove:
4(5 n
n
7)
3n
Reasons
3(4 n
9)
11
Statements
Given:
4.7 2 f
Prove:
y
0.5
Reasons
6 1.6 f
8.3 f
47
616
Statements
Reasons
Geometric Properties
We have discussed the RST (Reflexive, Symmetric, and Transitive) properties of
equality. We could prove that these also apply for congruence… but we won’t. We are just
going to accept it…
I know, you’re disappointed.
PROPERTIES OF CONGRUENCE
REFLEXIVE PROPERTY OF
CONGRUENCE
SYMMETRIC PROPERTY OF
CONGRUENCE
TRANSITIVE PROPERTY OF
CONGRUENCE
For any geometric figure A,
If
If
A
B
A
B
and
, then
B
C
B
DEFINITIONS
POSTULATES
PREVIOUSLY PROVED THEOREMS
ALGEBRAIC PROPERTIES
Elementary Geometric Proofs
Using Definitions
Given:
Prove:
XY
BC
XY
BC
Statements
Given:
Prove:
A
m
Reasons
Z
A
m
Z
Statements
A.
, then
Additional Reasons for Proofs
Reasons
A.
A
A
C
Using the Transitive Property and Substitution
Given:
Prove:
m 1
45
m
45
2
; m 2
m 1
Statements
Reasons
You should be aware that there are many ways to complete a proof. In fact, the following
website has 79 distinct proofs for the most famous of all theorems, the Pythagorean
Theorem.
http://www.cut-the-knot.org/pythagoras/index.shtml
Even the simple proof above could be done in at least two ways. The last statement could
have been justified using SUBSTITUTION or the TRANSITIVE PROPERTY. These
properties are similar, but no the same:
SUBSTITUTION works only on NUMBERS ( = ), while the TRANSITIVE PROPERTY
can be used to describe relationships between FIGURES or NUMBERS ( = or ). Keep
this in mind.
Given:
Prove:
1
2 ;
2
3
1
Statements
3
Reasons
Using Multiple Reasons
Given:
Prove:
m A
90
;
A
Z
Z is a right angle
Statements
Given:
Prove:
m 1
90
;
Reasons
1
2;
2
3
3 is a right angle
Statements
Given:
Prove:
m O
180
; m P
Reasons
m S ;
O
P
S is a straight angle
Statements
Reasons
DEFINITIONS AND POSTULATES REGARDING SEGMENTS
SEGMENT ADDITION POSTULATE
DEFINITION OF SEGMENT
CONGRUENCE
DEFINITION OF A SEGMENT
BISECTOR
DEFINITION OF A MIDPOINT
If C is between A and B,
then AC + CB = AB
If A B C D , then AB = CD
A geometric figure that divides a
segment in to two congruent halves
A point that bisects a segment
DEFINITIONS AND POSTULATES REGARDING ANGLES
ANGLE ADDITION POSTULATE
DEFINITION OF ANGLE
CONGRUENCE
DEFINITION OF AN ANGLE BISECTOR
If C is on the interior of ABD,
then m ABC m CBD m ABD
B , then m A m B
If A
A geometric figure that divides a
angle in to two congruent halves
Proofs with Pictures
It is often much easier to plan and finish a proof if there is a visual aid. Use the
picture to help you plan and finish the proof. Be sure that as you write each statement, you
make the picture match your proof by inserting marks, measures, etc.
Given:
Prove:
of A C and B D ; E D
AE
B
A
E is the m idpoint
EC
E
BE
C
D
Statements
Reasons
Given:
Prove:
O B bisects
AOC ;
O E bisects
DOF ;
AOB
DOE
EOF
BOC
Statements
Reasons
Elementary Geometric Proofs
Segments
R
Given:
Prove:
RT
W Y ; ST
RS
XY
Statements
S
T
WX
W
X
Reasons
Y
Given:
O is the midpoint of
Prove:
1.
NO
OC
OC
OW
OW
3.
4.
OC
Given:
EF
Prove:
EG FH
Statements
5.
NW
GH
E
F
G
H
1.
GH
EG
Given
Reasons
EF = GH
EF + FG = GH + FG
EF + FG = EG;
GH + FG = FH
EG = FH
6.
C
Reasons
OW
4.
W
O
1.
2.
NO
EF
H
;
N
Statements
O is the midpoint of
2.
3.
1.
2.
3.
4.
NW
2.
3.
4.
5.
6.
FH
Flow Proofs
Proofs do not always come in two-column format. Sometimes they are more visual, as you
will see in this example.
Given
Flow Proof
Given: 4 x
Prove: x
5
2
3
4
4x
5
Div. Prop.
of Equality
2
Add. Prop.
of Equality
4x
3
x
3
4
4x
3
Complete the flow chart for the following proof.
Given:
Prove:
AC = CE; AB = DE
C is the midpoint of
A
B
C
D
BD
Given
Segment Addition
Postulate
Substitution
Given
Subtraction Property
of Equality
Definition of
Congruence
Definition of
Midpoint
E