Download Calculus Fall 2010 Lesson 01

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Lesson Plan #013
Class: Geometry
Date: Friday October 7th, 2016
Topic: Deductive Reasoning in Geometry
Aim: How do we use deductive reasoning to prove statements in geometry?
HW # 013: Page 35 #’s 11, 15, 20, 21
Objectives:
1) Students will be able to prove statements using
the reflexive, symmetric and transitive properties
of equality.
Do Now:
We stated that a triangle that has two
congruent sides is an isosceles triangle.
construct two different isosceles triangles
First Isosceles Triangle
Using your compass and straight-edge,
Second Isosceles Triangle
In each isosceles triangle, use a protractor to measure the base angles. Based on this exercise, what can you
state about the base angles of an isosceles triangle?
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Assignment #1: Let’s go to http://www.mathopenref.com/isosceles.html and
take a look at isosceles triangles and their base
angle measures.
In the Do Now, we looked at some examples and then made a general truth based on those examples. That
type of reasoning is called inductive reasoning. Use inductive reasoning to find the next term in the pattern
Inductive reasoning is good, but it has its drawbacks.
With inductive reasoning, you are making conclusions without examining every possible example. Any
single counterexample is sufficient to show that a general conclusion reached is false.
For example, if I measured all the angles of many
isosceles triangles and found that all the angles I
measured are acute angles, I would conjecture that all
angles of isosceles triangles are acute. Can you show
a counter example to show my conclusion is false?
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Sample Test Questions:
1)
2)
3)
Instead of using inductive reasoning, if we instead use definitions, laws, rules, formulas, theorems and other
statements assumed to be true (postulates) to arrive at a true conclusion, then we are using deductive
reasoning. For example,
is an example of deductive reasoning!
Which Law was used to arrive at the above true conclusion?
Assignment #2: Determine if the following examples of reasoning are inductive or deductive.
Sample Test Question:
3)
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Sample Test Question:
4)
Example #1:A demonstration of proving a statement using deductive reasoning.
For example,
Given: M is the midpoint of AB
Prove: AM=BM
Statements
1) M is the midpoint of AB
Reasons
1.
Given
A midpoint divides a line segment into 2 congruent
segments. (1)
Congruent segments are equal in length. (2)
2)
AM  BM
2.
3)
AM  BM
3.
In the proof above, the premises of the argument must be taken as true statements. In geometry, there are
statements that are made that are neither undefined terms (such as point, line) nor definitions (a triangle is a
polygon that has exactly 3 sides).
Definition: A postulate is a statement whose truth is accepted without proof.
Definition: A theorem is a statement that is proved by deductive reasoning.
Let’s examine some postulates and see how they are used in proofs.
The following 3 equality postulates are also referred to as the properties of equality.
The Reflexive Property of Equality
a  a A quantity is equal to itself
The Symmetric Property of Equality
If a  b , then b  a
The Transitive Property of Equality
If a  b and b  c , then a  c
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Write proofs indicated below:
1.
Statements
1.CD = 2 inches
2. XY= 2 inches
3.CD =XY
Do proofs 2 and 3 on your own.
Reasons
1. Given
2. Given
3.Transitive Property of Equality (1,2)