Download Calculus Fall 2010 Lesson 01

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Lesson Plan #013
Class: Geometry
Date: Wednesday October 3rd , 2012
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
Objectiv
Do Now:
We stated that a triangle that has two congruent sides is an isosceles triangle.
Using a ruler draw two isosceles triangles by drawing two sides of equal length
then connecting with a third side. In each isosceles triangle, 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
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. Inductive reasoning is good, but it has its drawbacks. For
example, when you measure with the ruler and protractor, your measurements are approximate.
Also, 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 isosceles
triangles are acute triangles, can you show a counter
example to show my conclusion is false?
Instead of using inductive reasoning, if we instead use definitions and theorems and other statements
assumed to be true to arrive at a true conclusion, this is called 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.
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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
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)
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If for some relation (such as equal to, congruent to, etc.) is reflexive, symmetric, and transitive, then we say
that relation is an equivalence relation.
Determine if congruence of angles is an equivalence relation.
Determine if perpendicularity of lines is an equivalence relation.
Logic Review:
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