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Using
Deductive
Reasoning
Using
Deductive
Reasoning
2-3
2-3 to Verify Conjectures
to Verify Conjectures
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Warm Up
Identify each angle pair.
1. 1 and 3
corr. s
2. 3 and 6
alt. int. s
3. 4 and 5
alt. ext. s
4. 6 and 7
same-side int s
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Objective
Apply the Law of Detachment and the
Law of Syllogism in logical reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Vocabulary
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Theorem
Postulate
Paragraph proof
Conjecture
Deductive reasoning
Slope (rate of change)
Parallel lines
Perpendicular lines
Linear equation
Slope-intercept form
Point-slope form
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Theorem: A statement that has been proven.
Postulate: A statement that is accepted as
true, without proof. Also known as an axiom.
Paragraph Proof: A style of proof in which
the statements and reasons are presented in
paragraph form.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Conjecture: A statement that is believed to be
true.
Deductive reasoning: The process of using
logic to draw conclusions from given facts,
definitions, and properties.
Inductive reasoning: The process of
reasoning that a rule or statement is true
because specific cases are true.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Unit 2 Postulates and Theorems
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Vertical Angles Theorem
Corresponding Angles Theorem
Converse of Corresponding Angles Theorem
Alternate Interior Angles Theorem
Converse of Alternate Interior Angles Theorem
Same-Side Interior Angles Postulate
Converse Same-Side Interior Angles Postulate
Transitive Property of Parallel Lines
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Vertical Angles Theorem:
vertical angles are congruent.
Corresponding Angles Theorem:
If two parallel lines are cut by a
transversal, the corresponding angles are
congruent.
Converse of Corresponding Angles Theorem:
If two coplanar lines are cut by a
transversal so that a pair of corresponding
angles are congruent, then the lines are
parallel.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Alternate Interior Angles Theorem:
If two parallel lines are cut by a
transversal, alternate interior angles are
congruent.
Converse of Alternate Interior Angles Theorem:
If a pair of lines are cut by a transversal
and a pair of alternate interior angles are
congruent, then the lines are parallel.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Same-Side Interior Angles Postulate:
If two parallel lines are cut by a transversal
the two pairs of same-side interior angles
are supplementary.
Converse Same-Side Interior Angles Postulate:
If two lines are cut by a transversal so that
a pair of same-side interior angles are
supplementary, then the two lines are
parallel.
Transitive Property of Parallel Lines:
If two lines are parallel to the same line,
then they are parallel to each other.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 1A: Media Application
Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that you can balance an egg on
its end only on the spring equinox. A person
was able to balance an egg on July 8,
September 21, and December 19. Therefore this
myth is false.
Since the conclusion is based on a pattern of
observations, it is a result of inductive reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 1B: Media Application
Is the conclusion a result of inductive or
deductive reasoning?
There is a myth that the Great Wall of China is
the only man-made object visible from the
Moon. The Great Wall is barely visible in
photographs taken from 180 miles above Earth.
The Moon is about 237,000 miles from Earth.
Therefore, the myth cannot be true.
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 1
There is a myth that an eelskin wallet will
demagnetize credit cards because the skin of
the electric eels used to make the wallet holds
an electric charge. However, eelskin products
are not made from electric eels. Therefore, the
myth cannot be true. Is this conclusion a result
of inductive or deductive reasoning?
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
In deductive reasoning, if the given facts are
true and you apply the correct logic, then the
conclusion must be true. The Law of
Detachment is one valid form of deductive
reasoning.
Law of Detachment
If p  q is a true statement and p is true, then
q is true.
The given statement must match the
hypothesis to be valid.
If the statement matches the conclusion, it is
not valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2A: Verifying Conjectures by Using the Law
of Detachment
Determine if the conjecture is valid by the Law
of Detachment.
Given: If the side lengths of a triangle are 5 cm,
12 cm, and 13 cm, then the area of the triangle
is 30 cm2. The area of ∆PQR is 30 cm2.
Conjecture: The side lengths of ∆PQR are 5cm,
12 cm, and 13 cm.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2A: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the
given conditional.
If the side lengths of a triangle are 5 cm, 12 cm,
and 13 cm, then the area of the triangle is 30
cm2.
The given statement “The area of ∆PQR is 30 cm2”
matches the conclusion of a true conditional. But
this does not mean the hypothesis is true. The
dimensions of the triangle could be different. So the
conjecture is not valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2B: Verifying Conjectures by Using the Law
of Detachment
Determine if the conjecture is valid by the Law
of Detachment.
Given: In the World Series, if a team wins four
games, then the team wins the series. The Red
Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World
Series.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 2B: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the given
conditional.
In the World Series, if a team wins four games,
then the team wins the series.
The statement “The Red Sox won four games in the
2004 World Series” matches the hypothesis of a
true conditional. By the Law of Detachment, the Red
Sox won the 2004 World Series. The conjecture is
valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 2
Determine if the conjecture is valid by the Law
of Detachment.
Given: If a student passes his classes, the
student is eligible to play sports. Ramon passed
his classes.
Conjecture: Ramon is eligible to play sports.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 2 Continued
Identify the hypothesis and conclusion in the given
conditional.
If a student passes his classes, then the
student is eligible to play sports.
The statement “Ramon passed his classes” matches
the hypothesis of a true conditional. By the Law of
Detachment, Ramon is eligible to play sports. The
conjecture is valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Another valid form of deductive reasoning is
the Law of Syllogism. It allows you to draw
conclusions from two conditional statements
when the conclusion of one is the hypothesis of
the other.
Law of Syllogism
If p  q and q  r are true statements, then
p  r is a true statement.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3A: Verifying Conjectures by Using the Law
of Syllogism
Determine if the conjecture is valid by the Law
of Syllogism.
Given: If a figure is a kite, then it is a
quadrilateral. If a figure is a quadrilateral, then
it is a polygon.
Conjecture: If a figure is a kite, then it is a
polygon.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3A: Verifying Conjectures by Using the Law
of Syllogism Continued
Let p, q, and r represent the following.
p: A figure is a kite.
q: A figure is a quadrilateral.
r: A figure is a polygon.
You are given that p  q and q  r.
Since q is the conclusion of the first conditional
and the hypothesis of the second conditional, you
can conclude that p  r. The conjecture is valid
by Law of Syllogism.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3B: Verifying Conjectures by Using the Law
of Syllogism
Determine if the conjecture is valid by the Law
of Syllogism.
Given: If a number is divisible by 2, then it is
even. If a number is even, then it is an integer.
Conjecture: If a number is an integer, then it is
divisible by 2.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 3B: Verifying Conjectures by Using the Law
of Syllogism Continued
Let x, y, and z represent the following.
x: A number is divisible by 2.
y: A number is even.
z: A number is an integer.
You are given that x  y and y  z. The Law of
Syllogism cannot be used to deduce that z  x.
The conclusion is not valid.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 3
Determine if the conjecture is valid by the Law
of Syllogism.
Given: If an animal is a mammal, then it has
hair. If an animal is a dog, then it is a mammal.
Conjecture: If an animal is a dog, then it has
hair.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 3 Continued
Let x, y, and z represent the following.
x: An animal is a mammal.
y: An animal has hair.
z: An animal is a dog.
You are given that x  y and z  x.
Since x is the conclusion of the second conditional
and the hypothesis of the first conditional, you can
conclude that z  y. The conjecture is valid by Law
of Syllogism.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Example 4: Applying the Laws of Deductive
Reasoning
Draw a conclusion from the given information.
A. Given: If 2y = 4, then z = –1. If x + 3 = 12,
then 2y = 4. x + 3 = 12
Conclusion: z = –1.
B. If the sum of the measures of two angles is
180°, then the angles are supplementary. If two
angles are supplementary, they are not angles
of a triangle. mA= 135°, and mB= 45°.
Conclusion: A and B are not angles of a triangle.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Check It Out! Example 4
Draw a conclusion from the given information.
Given: If a polygon is a triangle, then it has
three sides.
If a polygon has three sides, then it is not a
quadrilateral. Polygon P is a triangle.
Conclusion: Polygon P is not a quadrilateral.
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Lesson Quiz: Part I
Is the conclusion a result of inductive or
deductive reasoning?
1. At Reagan High School, students must pass
Geometry before they take Algebra 2. Emily is
in Algebra 2, so she must have passed
Geometry.
deductive reasoning
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Lesson Quiz: Part II
Determine if each conjecture is valid?
2. Given: If n is a natural number, then n is an
integer. If n is an integer, then n is a rational
number. 0.875 is a rational number.
Conjecture: 0.875 is a natural number. not valid
3. Given: If an American citizen is at least 18 years
old, then he or she is eligible to vote. Anna is a 20year-old American citizen.
Conjecture: Anna is eligible to vote. valid
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
Lesson Quiz: Part II
4. If <A is acute, then m<A is less than 90⁰.
What conclusion can you make?
5. If a quadrilateral is a square, then it contains
four right angles.
If a quadrilateral contains four right angles,
then it is a rectangle.
What can you conclude?
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
6. If a baseball player is a pitcher, then that player
should not pitch a complete game two days in a
row.
Vladimir Nunez is a pitcher.
On Monday, he pitches a complete game.
What can you conclude?
7. If a number ends in 0, then it is divisible by 10.
If a number is divisible by 10, then it is
divisible by 5.
What can you conclude?
Holt McDougal Geometry
Using Deductive Reasoning
2-3 to Verify Conjectures
8.
The Volga River is in Europe.
If a river is less than 2300 miles long, it is not one of
the world’s ten longest rivers.
If a river is in Europe, then it is less than 2300 miles
long.
What are two conclusions you can make?
Holt McDougal Geometry