Transcript Chapter 2 Geometric Reasoning

Chapter 2
Geometric Reasoning
By April Stephens and Jackie Foley
2-1 Using Inductive Reasoning to
Make Conjectures
• Inductive Reasoning- the process of reasoning
that a rule or statement is true because specific
cases are true
• Conjecture- a statement you believe to be true
based on inductive reasoning
• Counterexample- an example in which your
conjecture is not true
– You must prove your conjecture to show that it is
always true. To prove that it is false, only one
counterexample is needed.
Examples
• Which number comes next in the pattern?
• 1, 2, 4, 8, _
• The answer is 16. You can form a
conjecture based on the pattern that the
numbers double each time. 1x2=2, 2x2=4,
and 4x2=8. 8x2=16, therefore the next
number is 16.
Example 2
• See the chart below. Find a counterexample to
disprove the conjecture that July is the rainiest
month. 45
40
35
30
25
20
15
10
5
0
May
June
July
August
2-2 Conditional Statements
• Conditional Statement- a statement that
can be written in the form “if p, then q.”
• Hypothesis- the part p of a conditional
statement following the word if
• Conclusion- the part q of a conditional
statement following the word then
• Logically Equivalent Statements- related
conditional statements that have the same
truth value
How to Write Statements
• If you have the following statement - the
midpoint M of a segment bisects the segment then you can form a conditional statement by
identifying the different parts.
• The hypothesis of the statement is “M is the
midpoint of a segment.” The conclusion is “M
bisects the segment.”
• You can rewrite this to make the conditional
statement, “If M is the midpoint of a segment,
then M bisects the segment.”
Definitions
• Truth Value- whether the statement is true
or false
– A conditional statement is false only if the
hypothesis is true and the conclusion is false
• Converse- the statement formed by
exchanging the hypothesis and conclusion
• Inverse- the statement formed by negating
the hypothesis and the conclusion
• Contrapositive- the statement formed by
both exchanging and negating the
hypothesis and conclusion
Examples
• Identify the hypothesis and the conclusion of the
following conditional statement.
– If a person is at least 16 years old, then the person
can drive a car.
– Hypothesis: a person is 16 years old. Conclusion:
they can drive a car
• Give the converse and inverse of the following
conditional statement.
– If a patient is ill, then their heart rate is monitored.
– Converse- If a patient’s heart rate is monitored, then
they are ill. Inverse- if a patient is not ill, then their
heart rate is not monitored.
2-3 Using Deductive Reasoning to
Verify Conjectures
• Deductive Reasoning- the process of
using logic to draw conclusions from given
facts, definitions, and properties
– To prove a conjecture is true, you must use
deductive reasoning.
• Law of Detachment- If if p then q is a true
statement and p is true, then q is true.
• Law of Syllogism- If if p then q and if q
then r are true, then if p then r is true.
Examples
• What can you conclude from the following
statements?
• At Bell High School, students must take Biology
before they take Chemistry. Anthony is in
Chemistry.
– Therefore, Anthony has taken Biology.
• The sum of all angle measures in a triangle
equal 180*. Two of the angles equal 50* and
70*.
– Therefore, the third angle has a measure of 60*.
2-4 Biconditional Statements and
Definitions
• Biconditional Statement – a statement that can
be written in the form p if and only if q
– This means that if q is untrue, then p cannot be true
and vice versa.
• Definition – a statement that describes a
mathematical object and can be written as a true
biconditional statement
• Polygon – a closed plane figure formed by three
or more line segments
• Triangle – three-sided polygon
• Quadrilateral – four-sided polygon
Examples
• What is the conditional statement and
converse within the following biconditional
statement?
• A student is a sophomore if and only if
they are in the tenth grade.
– Conditional statement – If a student is a
sophomore, then they are in the tenth grade.
Converse – If a student is in the tenth grade,
then they are a sophomore.
Examples
• What is the converse and the biconditional
statement that can be created from this
condtitional statement?
• If today is Saturday or Sunday, then it is
the weekend.
– Converse – If it is the weekend, then today is
Saturday or Sunday. Biconditional – Today is
Saturday or Sunday if and only if it is the
weekend.
2-5 Algebraic Proof
• Proof – an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true
• Properties of Equality
–
–
–
–
–
–
–
–
Addition – if a = b, then a + c = b + c
Subtraction – if a = b, then a – c = b – c
Multiplication – if a = b, then ac = bc
Division – if a = b and c ≠ 0, then a/c = b/c
Reflexive – a = a
Symmetric – if a = b, then b = a
Transitive – if a = b and b = c, then a = c
Substitution – if a = b, then b can be substituted for a in any
expression
Examples
• Solve the following equations using proofs.
• 2x + 5 = 3x + 2
• First, subtract 2 from each side to get 2x +
3 = 3x. Then subtract 2x from each side to
get x = 3.
• Identify the property that justifies each
statement.
• m<1=m<2, and m<2=m<3, so m<1=m<3.
– This is using the transitive property.
2-6 Geometric Proof
• Theorem – any statement that you can
prove
• Two-column proof – a chart in which you
list the steps of the proof in the left column
and write the matching reason for each
step in the right column
Theorems
• Linear Pair Theorem – if two angles form a
linear pair, then they are supplementary
• Congruent Supplements Theorem – if two
angles are supplementary to the same angle,
then the two angles are congruent
• Right Angle Congruence Theorem – all right
angles are congruent
• Congruent Complements Theorem – if two
angles are complementary to the same angle,
then the two angles are congruent
Examples
• Write a two-column proof using the
following statements and illustration.
A
X
Y
B
• Given – X is midpoint of AY, Y is midpoint
of XB. Prove – AX=YB
– By the definition of midpoint, AX=XY, and
XY=YB. Using the Transitive Property, we can
prove that AX=YB.