UNIT 2- REASONING IN GEOMETRY
GQ- What has logic done to improve society? How will the use of logic improve
my success as a student? In what context(s) is proof necessary?
2.1 An Introduction to Proof
What is a Proof?
A proof is convincing argument that something is true. In mathematics, a
proof starts with things that are agreed on (called postulates or axioms).
Formal proofs follow a prescribed form. For Example:
5x + 4 = 24
5x = 20
x=4
Given
Subtraction Property of Equality
Division Property of Equality
Your Try! In your own words, describe what a proof is.
Example 1: Give Proof of a Conjecture
The following table shows the number of points and the corresponding number
of segments:
Points
2
3
4
5
Segments
1
3
6
10
Conjecture: For n points on a circle, the number of segments needed to connect
all points is
n(n  1)
.
2
Proof: For each point, a segment connects it to every other point, giving n – 1
segments. Therefore, the total number of segments should be n(n – 1).
However, AB is the same as BA , so each segment is counted twice. Thus
the total must be divided by 2, giving
n(n  1)
.
2
You Try!
Prove the following conjecture by writing a variable statement and using
algebra. Conjecture: The sum of five consecutive integers is always
divisible by 5.
Use the pattern below to prove the sum of the first 6 even integers and make
a conjecture about the sum of the first 30 positive even numbers. Then,
write a rule for the first n even integers. Try to prove it!
2
2+4
2+4+6
2+4+6+8
2 + 4 + 6 + 8 + 10
=
=
=
=
=
2
6
12
20
30
=
=
=
=
=
2.2 An Introduction to Logic
Drawing Conclusions from Conditionals:
T or F: All grasshoppers are insects.
T or F: All insects are grasshoppers.
If an animal is a _____________________,
then it is a _______________________.
An “if-then” statement is a conditional. The “then” portion of the statement is
dependent on the “if,” or the “if” is a condition of the outcome.
In logical notations conditionals are written as follows:
If p then q
or
p  q (Read as “p implies q”)
In a conditional, the part following the word if is the hypothesis. The part
following the word then is the conclusion.
If a car is a Beetle, then it is a Volkswagen.
Consider the following statement: Ms. Phipps’ car is a Beetle. Where would
this fit into the Venn diagram?
The complete process of drawing the conclusion that Ms. Phipps’ car is a
Volkswagen can be written as a logical argument. This particular argument,
known as syllogism, has three parts:
1. If a car is a Beetle, then it is a Volkswagen.
2. Mrs. Phipps’ car is a Beetle
3. Therefore, Mrs. Phipps’ car is a Volkswagen.
The process of drawing logically certain conclusions by using an argument is
known as deductive reasoning or deduction.
Example 1: Recall the following information:
An equilateral triangle is a triangle with three congruent sides.
An isosceles triangle is a triangle with at least two congruent sides.
a. Draw a Venn diagram that conveys the following information: If a triangle
is equilateral, then the triangle is isosceles. Triangle ABC is equilateral.
b. What conclusion can you draw about triangle ABC?
You Try! Suppose you know that Bill lives in San Francisco. Use logical
reasoning to show that Bill lives in the state of California. Illustrate your
argument with a Venn diagram.
Reversing Conditionals:
When you interchange the hypothesis and the conclusion of a conditional, the
result is a new conditional called the converse of the original conditional.
For example:
Conditional: If a car is Beetle, then it is a Volkswagen.
Converse: If a car is a Volkswagen, then it is a Beetle.
An example which proves that a statement is false is called a counterexample.
Example 2: Write a conditional with the hypothesis “a triangle is equilateral”
and the conclusion “the triangle is isosceles.” Then write the converse of
your conditional. Is the conditional true? Is the converse true?
You Try! Write a conditional with the hypothesis “an animal is a snake” and the
conclusion “the animal is a reptile.” Then write the converse of your
conditional. Is the conditional true? Is the converse true?
Logical Chains
A logical chain results when conditionals are linked together. Consider the
following conditionals:
Given: If there is a parade, then fireworks will go off.
If it is July 4, then American flags are flying.
If flags are flying, then there is a parade.
Prove: If it is July 4, then fireworks will go off.
If-then Transitive Property: Given: If A then B, and if B then C, you can conclude: If
A then C.
2.3 Definitions
Definitions and Venn (Euler) Diagrams
Floppers:
Not Floppers:
Write a definition of a flopper:
Using your definition, identify the floppers in the following illustration.
a.
b.
c.
d.
e.
f.
Write your definition as a conditional:
Write the converse of the conditional:
The original conditional and its converse are true, a property that is true for all
definitions. The true two conditionals can be combined into a compact form by
joining the hypothesis and the conclusion with the phrase, “if and only if” which
is represented by "" . “p if and only if q” or “ p  q .” The “if and only if”
statement is known as a biconditional.
Is it possible to create a single Venn diagram to show that both the statements are
true? What would it look like?
Example 1: The “essence” of a thing…
1. Look at the figure at right. Make up your own name for the object.
Then answer the question, “What must be true of a geometrical figure
in order for it to be a (your name for the object)?”
2. According to your concept of a (your name for the object), which of the
objects below would you consider to be one?
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
3. Write your own definition of a (your name for the object).
Example 2: Use the boxes below to form a definition of adjacent angles.
ADJACENT ANGLES:
1 and 2
2 and 3
1 and 2
1 and 2
2 and 3
4 and 5
(There are others also.)
NOT ADJACENT ANGLES:
1 and 2
1 and 2
1 and 3
ADB and ADC
1 and 2
1. List the adjacent angles in the figure at right.
2. What do adjacent angles have in common? Can they
overlap?
3. Angles that do not overlap have no interior points in
common. Use this fact and your answers from 1 and 2 to fill
in the following definition.
Adjacent angles are angles in a plane that have their __________________ and
one _________________ in common but ______________________.
2.4 Building a System of Geometry Knowledge
Algebraic Properties of Equality:
x–3=5
Given
x–3+3=5+3
Addition Property of Equality
x=8
Simplify
Properties of Equality and Real Numbers
If a = b , then a + c = b + c
Addition Property
Subtraction Property
If a = b, then a – c = b – c
Multiplication Property If a = b, then
Division Property
Substitution Property
If a = b, then
If a = b, then b can replace a in any expression.
Distributive Property
a(b + c) = ab + ac
Linking Steps to Prove a Theorem:
Example: In the figure, AB = CD. Prove that AC = BD.
Overlapping Segments Theorem: Given a segment with points A, B, C, and
D arranged as shown, the following statements are true:
1. If AB = CD, then AC = BD.
2. If AC = BD, then AB = CD.
Two-Column Proofs:
Statements:
AB  CD
AB  BC  BC  CD
AB  BC  AC
BC  CD  BD
AC  BD
Reasons:
Given
Addition Property of Equality
Segment Addition Postulate
Segment Addition Postulate
Substitution Property of Equality
Paragraph Proof:
An alternative to the two-column proof is a paragraph proof. One advantage
to this format is that you have a chance to explain your reasoning. The
following is an example of a paragraph proof for the Overlapping Segments
Theorem:
You are given that AB = CD. Add BC to both sides of the equation, resulting in
AB + BC = BC + CD. In the figure, AB + BC = AC and BC + CD = BD by the
Segment Addition Postulate. The expressions on the left of these equations
match the expression in the previous equations, so you can substitute the
equivalent expressions, AC and BD. The result is AC = BD.
The Equivalence Properties of Equality and Congruence:
Properties of Equality and Congruence
Equality
Congruence
a=a
Reflexive Property
Symmetric Property
If a = b, then b = a
If
If
, then
, then
Transitive Property
If a = b and b = c, then a
=c
If
and
, then
and
, then
.
.
.
If
.
Example: The first stamp pictures below measure 3 centimeters by 6 centimeters.
The second stamp is congruent to the first. The third stamp is congruent to
the second. Prove that the first stamp is congruent to the third.
2.5 Conjectures That Lead to Theorems
A Need for Proof:
For a conjecture to be considered true by mathematicians it must first be
proven deductively.
Example:
1. Draw several pairs of intersecting lines. In the figure
at right 1 and 2 form one pair of vertical angles
and 3 and 4 form another.
2. Measure each pair of vertical angles. What do you
notice?
3. Make a conjecture about vertical angles.
4. What is the relationship between 1 and 3 ? between
2 and 4 ?
5. Complete the following:
m1  m3  __________ m2  m4  __________
6. What property of equality leads to the following
conclusion? m1  m3  m2  m4
7. What property of equality leads to the conclusion that
m1  m2 ?
Inductive and Deductive Reasoning:
Inductive reasoning is the process of forming conjectures that are based on
observations. Conjectures may turn out to be false, so inductive reasoning is
not accepted in mathematical proofs.
Vertical Angles Theorem: If two angles form a pair of vertical angles, then the
vertical angles are congruent.
Example:
Given: 1 and 2 are vertical angles
Prove: 1  2
Statements:
1 and 2 are vertical angles
Reasons:
Activity: Reflections across Parallel Lines
1. Reflect triangle ABC across line l. Label the
image A’B’C’.
2. Reflect triangle A’B’C’ across line m. Label
its image A’’B’’C’’
3. Study the relationship between triangle ABC
and A’’B’’C”. What single transformation
would produce triangle A’’B’’C” from triangle
ABC?
4. Measure the distances AA”, BB”, and CC”.
What do you notice?
5. Measure the distance between line l and m.
How does this distance relate to the distances
you measured in step 4?
6. Do all of the points on the triangle seem to
have moved in the same direction? Explain
your answer.
7. Complete the following theorem.
Theorem: Reflection across two parallel lines is equivalent to a _________________
of __________________ distance between the lines and in a direction
__________________ to the lines.
Proof:
1. Which of the indicated distances are
equal? Why?
2. What is the distance between lines l
and m?
3. What does the expression 2D1 + 2D2
represent in the diagram? How does
it compare to the distance between
lines l and m?
4. Do you think the result from step 5
of the activity is true for any point on
the figure? How could you prove
that each point moves the same
direction?
The Importance of Theorems:
Conjectures are based on a finite set of measurements so they may turn out to be
false. Always ask yourself whether there is a counterexample to your conjecture
which would prove it false. Theorems are different from conjectures in that they
are proven for all cases regardless of measurements. This is why the vertical
angles theorem is true for all possible pairs of vertical angles, no matter what their
measure. You will find this to be true for any theorem in Geometry.
Your Turn! Explain the difference between inductive and deductive reasoning.
Why is deductive reasoning the only type of reasoning allowed in proofs?