3.6
What is a Proof?
Pg. 19
Types of Proof
3.6 – What Is A Proof?
Types of Proof
Whenever you buy a new product that
needs to be put together, you are given a
set of directions. The directions are
written in a specific order that must be
followed closely in order to get the desired
finished product. Sometimes they clarify
their directions by explaining why you are
completing each step. This is the same
idea we use in geometry in proofs.
3.31 – ORDERING STATEMENTS
When you write a proof, the statements
must be in a specific order, building off of
each other. You can't just jump to the end
without breaking down each part. To
illustrate this, with your group explain how
to make a peanut butter and jelly
sandwich. Work with your team to include
all steps to make sure the sandwich will be
made correctly.
3.32 – STATEMENTS AND REASONS
When you write a proof in geometry, each
statement you make must have a reason
to support it. This helps people
understand why each statement was
listed. This can be done in a flowchart
proof or a two-column proof. Examine the
two types below. Notice where the
statements and reasons are. Also, notice
how the statements are in a specific order.
4.3 – REASONS
The reasons for certain statements come
from definitions, properties, postulates, and
theorems. Below are some commonly
used reasons.
Name
Addition
Property
Subtraction
Property
Multiplication
Property
Division
Property
Distribution
Property
Property of Equality
If a = b, then
a+c=b+c
If a = b, then
a-c=b-c
If a = b, then
ac = bc
If a = b, then
If a(x + b), then
a/c = b/c
ax + ab
Substitution
Property
If a = b, then b can replace a
Reflexive
Property
a=a
Symmetric
Property
Transitive
Property
If a = b, then b = a
If a = b and b = c, then
a=c
1. Use the property to complete the statement.
3m2
5(20) 100
a. Write the reason for each statement
Statement
If 4(x + 7), then 4x + 28
BD  BD
If 2x + 5 = 9, then 2x = 4
Reason
Distributive Prop.
Reflexive Prop.
Subtraction Prop.
If A  B and B  C , then A  C Transitive Prop.
If x – 7 = 2, then x = 9
Addition Prop.
Symmetric
If 4x = 12, then x = 3
Division Prop.
PROOFS!
All proofs start and end the same. It is very
helpful to have a picture to refer to. We are given
information and then are told to prove something.
Format of Proofs
Given:
Prove:
Statements
1. What you are given
2.
3.
… To Prove
Picture
Reasons
1. Given
2.
3.
…
Complete the logical argument by writing a reason for
each step.
Statements
1.
8x – 34 = 6
Reasons
1.
given
__________________
2.
8x = 40
2.
Addition Prop
__________________
3.
x=5
3.
Division Prop
__________________
Complete the logical argument by writing a reason for
each step.
Statements
Reasons
1.
4x – 7 = 6x + 7
1.
given
__________________
2.
-2x – 7 = 7
2.
__________________
Subtraction Prop
3.
-2x = 14
3.
Addition Prop
__________________
4.
x = -7
4.
Division Prop
__________________
Complete the logical argument by writing a reason for
each step.
Statements
Reasons
1.
5(x – 3) = 4(x + 2)
1.
given
_______________________
2.
5x – 15 = 4x + 8
2.
Distributive Prop
_______________________
3.
4.
x – 15 = 8
x = 23
Subtraction Prop
3. _______________________
4.
Addition Prop
_______________________
4. Solve the equation. Write a reason for
each step 4x – 9 = 2x + 11
Statements
1.
4x – 9 = 2x + 11
Reasons
1.
Given
2. 2x – 9 = 11
2.
Subtraction Prop.
3.
2x = 20
3.
Addition Prop.
4.
x = 10
4.
Division Prop.
3.33 Solve the equation for y. Write a reason
for each step.
12x – 3y = 30
Statements
Reasons
1. 12x – 3y = 30
1. Given
2. –3y = -12x + 30
2. Subtraction Prop.
3.
y = 4x – 10
3. Division Prop.
3.34 – GEOMETRIC PROOF
Complete the proof by writing a reason for each step.
GIVEN: AL = SK
PROVE: AS = LK
C. Given
E. Reflexive
F. Addition Prop.
A. Segment Addition
D. Segment Addition
B. Substitution
given
Def. of Midpoint
given
Segment Addition
Substitution
Substitution
Simplify
given
Def. Segment Bisector
given
Segment Addition
Substitution
Substitution
Simplify
Statements
Reasons
1. Given
2. Def. of angle bisector
1.
2.
mABD  mDBC
3. given
3. mABD  20
4. ABC  ABD  DBC 4. Angle Addition
5. ABC  ABD  ABD 5. Substitution
6. ABC  20  20 6. Substitution
7.
ABC  40
7. Simplify