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USING PROPERTIES FROM ALGEBRA
ALGEBRAIC PROPERTIES OF EQUALITY
Let a, b, and c be real numbers.
2.4
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.
REFLECTIVE PROPERTY
SYMMETRIC PROPERTY
For any real number a, a = a.
If a = b, then b = a.
TRANSITIVE PROPERTY
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.
USING PROPERTIES FROM ALGEBRA
PROPERTIES OF EQUALITY
DISTANCE LENGTH
ANGLE MEASURE
REFLECTIVE
For any distance AB,
AB = AB.
For any angle A,
mA = mA .
SYMMETRIC
If AB = CD, then CD =
AB.
If mA = mB , then
mB = mA .
TRANSITIVE
If AB = CD and CD = EF,
then AB = EF.
If mA = mB and ,
mB = mC, then
mA = mC.
2.4
USING PROPERTIES FROM GEOMETRY
PROPERTIES OF CONGRUENCE AS THEOREMS
SEGMENT CONGRUENCE
REFLECTIVE
SYMMETRIC
TRANSITIVE
2.4
ANGLE CONGRUENCE
For any segment AB,
AB  AB.
For any angle A,
A  A .
If AB  CD,
then CD  AB.
If A  B ,
then B  A .
If AB  CD and CD  EF,
then AB  EF.
If A  B and , B 
C, then A  C.
COMBINING ALL ELEMENTS FROM GEOMETRY
GEOMETRIC BOOK OF LAW
LEGAL REASON FOR MAKING A STATEMENT.
DEFINITIONS
All definitions are Biconditional.
THEOREMS
Includes Formulas
POSTULATES
Or Axioms
PROPERTIES
Of Equality & Congruence
LAWS OF LOGIC Inductive & Deductive Reasoning
There are 2 laws of Deductive Reasoning:
1. Law of Detachment
2. Law of Syllogism
2.4
Transitive Property of Angle Congruence
Prove the Transitive Property of Congruence for angles.
SOLUTION
To prove the Transitive Property of Congruence for angles,
begin by drawing three congruent angles. Label the vertices as
A, B, and C.
A
B
GIVEN
B,
C
PROVE
A
C
B
A
C
Transitive Property of Angle Congruence
A
B
GIVEN
B,
C
Statements
A 
B 
1
PROVE
A
C
Reasons
B,
C
1. Given
2
m
A=m
B
2. Definition of congruent angles
3
m
B=m
C
3. Definition of congruent angles
4
m
A=m
C
4. Transitive property of equality
5
A 
C
5. Definition of congruent angles
Using the Transitive Property
This two-column proof uses the Transitive Property.
GIVEN
m
3 = 40°,
PROVE
m
1 = 40°
1
2,
Statements
1
2
m
1
3
3
m
1=m
4
m
1 = 40°
3
Reasons
3 = 40°,
2
3
1
2
2,
Given
Transitive property of Congruence
3
Definition of congruent angles
Substitution property of equality
Proving Theorem 2.3
THEOREM
THEOREM 2.3 Right Angle Congruence Theorem
All right angles are congruent.
You can prove Theorem 2.3 as shown.
GIVEN
1 and
PROVE
1
2 are right angles
2
Proving Theorem 2.3
GIVEN
1 and
PROVE
1
2 are right angles
2
Statements
1 and
1
Reasons
2 are right angles
2
m
1 = 90°, m
3
m
1=m
4
1
2
2
2 = 90°
Given
Definition of right angles
Transitive property of equality
Definition of congruent angles