Download Section 1.4 Proving Conjectures: Deductive Reasoning

Section 1.4 Proving Conjectures: Deductive Reasoning
Deductive Reasoning
Using logical reasoning to draw a conclusion from valid assumptions.
Supplementary Angles
Complementary Angles
Angles that add up to 180°.
Angles that add up to 90°.
Consecutive Integers
Consecutive integers can be written as x, x + 1, x + 2, x + 3, …
Even/Odd Integers
Even integers can be expressed as 2x. Odd integers can be expressed as 2x + 1.
Example 1 Venn Diagram
Use a venn diagram to prove the following conjectures deductively:
(A)All natural numbers are whole numbers. All whole numbers are integers.
3 is a natural number. What can be deduced about the number 3?
(B) Mammals have fur. Lions are classified as mammals. What can be deduced
about lions?
Example 2
John discovered a pattern when adding integers.
1 + 2 + 3 + 4 + 5 = 15
(–3)+(–2)+(–1)+0+1=–5
8 + 9 + 10 + 11 + 12 = 50
Using inductive reasoning John conjectured that “ Whenever you add five
consecutive integers the sum is five times the middle number. ”
Prove using deductive reasoning that John’s conjecture is valid.
Example 3
Prove that the sum of a 2–digit number and the number formed by reversing its
digits will always be divisible by 11.
Example 4
Mary claimed that the following process always gives a result of 3.
Choose a number from 1 to 20.
Add 5.
Double the result.
Subtract 4.
Divide the result by 2.
Subtract the number you started with.
Use inductive and deductive reasoning to show that Mary’s claim is valid.
Inductive Reasoning
Give three examples.
Deductive Reasoning
Example 5
The sum of four consecutive integers is equal to the sum of the first and last
integers multiplied by two.
Use inductive and deductive reasoning to show the given statement is valid.
Inductive Reasoning
Give three examples.
Deductive Reasoning
Example 6 Two – Column Proof
Prove that when two straight lines intersect the vertically opposite angles are equal.
A
B
C
D
Statement
E
Justification
Example 7
Prove the following using deductive reasoning.
If the sum of the digits of a two–digit number is divisible by 3 then the original
number is divisible by 3.
Questions
Pages 31 – 33, # 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20