Download Write paragraph proofs

Lesson 1.6
Paragraph Proofs
Objective:
Write paragraph proofs
Why are we doing this?
Although most proofs we do in this class are twocolumn, you also need to be familiar with paragraph
proofs.
Paragraph proofs are useful to know because they
help us to think logically through a problem, and put
a solution in a form that everyone can understand
and follow.
We are going to see how to write a paragraph proof,
as well as how to show that a conclusion cannot be
proved.
Example #1
Given: <x = 37 ½°
<y = 37° 30’
Prove: x  y
x
y
Proof:
Since 30’ = ½° we know that 37° 30’ = 37 ½°.
Therefore (  ) x  y.
w5 or Q.E.D
W5 = which was what was wanted
Q.E.D. = Quod Erat Demonstrandum which means
“Which was to be Demonstrated”
Example #2
Given: Diagram Shown
Prove: DBC  E
D
A
(2x)° x°
B
60°
C
E
Proof:
According to the diagram, <ABC is a straight angle.
Therefore, 2x + x = 180
3x = 180
x = 60
Since <DBC = 60° and <E = 60°, the angles are
congruent. Q.E.D
One last thing to keep in mind…
Not all proofs can be proved. If this happens, we use
what’s called a counter-example. We assume that
the original statement is true, and then use a specific
example to show that it is not possible.
Remember, it only takes one false example to
disprove a statement!
Example #3
Given: <1 is acute
<2 is acute
Prove: 1  2
1
Proof:
Since <1 is acute, let it be 50°, and since <2 is acute,
let it be 30°. Therefore, by counter-example, it
cannot be proved that 1  2. Q.E.D
2
Homework
Lesson 1.6 Worksheet
Lesson 1.7/1.8
Deductive Structure and Statements of
Logic
Objective:
Recognize that geometry is based on deductive structure,
identify undefined terms, postulates, and definitions,
understand the characteristics of theorems, recognize
conditional statements, recognize the negation of a
statement, the converse, inverse, and contrapositive, and
draw conclusions using the chain rule.
Definitions
Def. Deductive Structure is a system of thought in
which conclusions are justified by means of
previously assumed or proved statements.
Note: every deductive structure contains 4 elements:
1. Undefined terms
2. Assumptions known as postulates
3. Definitions
4. Theorems and other conclusions
Definitions
Def. A Postulate is an unproven assumption (In
other words, it is so obvious, it does not need to be
proved)
Def. A Definition states the meaning of a term or
idea.
Note: Definitions are reversible!
Example:
Midpoint   Segments
Reversed Definition:  Segments  Midpoint
Original Definition:
Conditional Statements
All definitions are stated in a specific form:
“If p, then q”
This type of sentence is called a Conditional
Statement (or an Implication)
The “if” part = the hypothesis
The “then” part = the conclusion
We write this mathematically as:
p  q .
Conditional Statement Example:
Write the following statement in its conditional form:
“Two straight angles are congruent”
The Converse
The converse of
p  q is: q  p
To write the converse of a statement, you reverse
parts p and q.
Important Note!
Because definitions can be reversed, the conditional
statement (the original) and the converse will
always be true. This is not always the case for
theorems and postulates!
Converse Example:
Conditional Statement:
“If it is raining, then worms come out.”
Converse:
If worms come out, then it is raining
Negation
The negation of any statement p is the statement
“not p.”
The symbol for “not p” is “~p”
Ex. If p = It is raining then ~p = _____________
Converse, Inverse, and
Contrapositive
Every Conditional statement If p then q,
has 3 other statements:
1. Converse: If q, then p
2. Inverse: If ~p, then ~q
3. Contrapositive: If ~q, then ~p
The AZ Example:
Write each form of the conditional and decide whether the
statement is true or false.
Conditional Statement:
“If you live in Phoenix, then you live in AZ.”
Converse: If you live in AZ, then you live in
Phoenix.
Inverse: If you do not live in Phoenix, then you
do not live in AZ.
Contrapositive: If you do not live in AZ, then you do
not live in Phoenix
Theorem 3
If a conditional statement is true, then the
contrapositive of the statement is also true.
Note:
Often times mini Venn Diagrams are useful in
determining whether or not a conditional statement
and its converse, inverse, or contrapositive are
logically equivalent.
Try making Venn Diagrams for each example written
on the last slide.
Chains of Reasoning
Many proofs we do involve a series of steps that
follow a logical form. Often times it looks
something like this:
If p  q, q  r, and r  s, then p  s
This is called the chain rule, and a series of conditional
statements is known as a chain of reasoning.
Example:
If you study hard, then you will earn a good grade, and if you
earn a good grade, then your family will be happy.
We can conclude: If you study hard, your family will be happy.

Example
Draw a conclusion from the following statements:
If gremlins grow grapes, then elves eat earthworms.
If trolls don’t tell tales, then wizards weave willows.
If trolls tell tales, then elves don’t eat earthworms.
Hint:
Rewrite these statements using symbols, then
rearrange the statements and use contrapositives
to match the symbols!
Homework
Lesson 1.7/1.8 Worksheet