Transcript Postulates and Algebraic Proofs

Advanced Geometry
Deductive Reasoning
Lesson 2
Postulates and
Algebraic Proofs
Properties from Algebra
Reflexive Property
ONE TERM
Symmetric Property
TWO TERMS
Transitive Property
THREE TERMS
For every number a, a = a.
**A term is equal to itself.**
For all numbers a and b,
if a = b, then b = a.
**The two sides of an equation can be
switched.**
For all numbers a, b, and c,
if a = b and b = c, then a = c.
**Skip the middle term in the conclusion.**
Properties from Algebra (cont.)
For all numbers a, b, and c,
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 a • c = b • c.
Division Property
if a = b, then a ÷ c = b ÷ c.
You can add, subtract, multiply, or divide
the same term on
both sides of an equation.
Properties from Algebra (cont.)
Substitution Property
For all numbers a and b, if a = b, then
a may be replaced by b in any
equation or expression.
Distributive Property
For all numbers a, b and c,
a(b + c) = ab + ac.
Combine Like Terms
Terms with like variables are combined
without moving anything across the
equal sign.
Examples: Name the property of equality that justifies each
statement.
If 3x + 7 = 12, then 3x = 5.
Subtraction Property
If AB = CD, then AB + EF = CD + EF.
Addition Property
If PQ + RS = 18 and RS = 8,
then PQ + 8 = 18.
Substitution Property
Postulates
accepted to be true without proof
Axiom is another word for postulate.
Theorems
has already been proven to be true
can be used to justify that
other statements are true
The 1st 7 Postulates
• Through any two points there is exactly one line.
Two points determine a line.
• Through
any three points not on the same line
there is exactly one plane.
Three noncollinear points determine a plane.
The 1st 7 Postulates (cont.)
• A line contains at least two points.
• A plane contains at least three points not on
the same line.
The 1st 7 Postulates (cont.)
• If two points lie in a plane, then the entire line
containing those two points lies in that plane.
• If two lines intersect, then their intersection is
exactly one point.
• If two planes intersect, then their intersection is
a line.
Examples: Determine whether each statement is
sometimes, always, or never true.
If plane T contains EF and EF contains point
G, then plane T contains point G.
Always
GH contains three noncollinear points.
Never
If AB and HK are coplanar, then they
intersect.
Example: State the postulate that can be used to
show each statement is true.
E, B, and R are coplanar.
B, D, and W are collinear.
Proof
a logical argument
each statement made is supported by a reason
Reasons:
postulates (axioms)
theorems
definitions
properties
Paragraph Proof
Given: M is the midpoint of PQ
Prove: PM  MQ
From the definition of midpoint of a segment, PM = MQ.
This means that PM and MQ have the same measure.
By the definition of congruence, if two segments have the
same measure, then they are congruent. Thus, PM  MQ.
Two-Column Proof
Given: M is the midpoint of PQ
Prove: PM  MQ
Proof:
Statements
Reasons
a) M is the midpoint of PQ
a) Given
b) PM = MQ
b) Definition of midpoint
c) PM  MQ
c) Definition of congruent
Solve for x. Show every step.
2x + 18 = 6
ALGEBRAIC PROOFS
Example: Write a complete proof for the situation below.
Prove that if AB = CD, AB = 4x + 8 and CD = x + 2, then x = -2.
Example: Write a complete proof for the situation below.
3V
1 2
h

Prove that if V   r h , then
.
 r2
3