2.6 Prove Statements about Segments and Angles Objectives: Assignment:

2.6 Prove Statements about Segments and
Angles
Objectives:
1. To write proofs using
geometric theorems
Assignment:
• P. 116-119: 1-4, 1619, 22-24
• Challenge Problems
Objective 1
A Brief History of Math
Over thousands of
years the Babylonians
and Egyptians
discovered many
geometric principles
and developed a
collection of “rule-ofthumb” procedures for
doing practical
geometry.
A Brief History of Math
The result of trial and error,
these procedures were
used to compute simple
areas and volumes. The
procedures were used in
surveying to reestablish
land boundaries after
floods, and they were
practical instructions for
building canals and tombs.
A Brief History of Math
By 600 B.C. a prosperous
new civilization had
begun to grow in the
trading towns along the
coast of Asia Minor and
later in Greece, Sicily,
and Italy. People had
free time to discuss
and debate issues of
government and law.
A Brief History of Math
This led to an insistence on
reasons to support
statements made in debate.
Mathematicians began to use
logical reasoning to deduce
mathematical ideas.
A Brief History of Math
Greek mathematician Thales of
Miletus made a number of
valuable geometric conjectures.
Unlike most other
mathematicians before him,
Thales supported his
discoveries with logical
reasoning.
A Brief History of Math
Over the next 300 years, the
process of supporting
mathematical conjectures with
logical arguments became
more and more refined. Other
Greek mathematicians,
including Thales’ most famous
student, Pythagoras, began
linking together chains of
logical reasoning.
A Brief History of Math
Later students of mathematics at
Plato’s Academy linked even
longer chains of geometric
properties together by deductive
reasoning. Euclid, in his
famous work about geometry
and number theory, Elements,
established a single chain of
deductive arguments for most of
the geometry then known.
A Brief History of Math
Euclid started from a collection of
statements that he regarded as
obviously true (postulates). He
then systematically demonstrated
that one after another geometric
discovery followed logically from
his postulates and his previously
verified conjectures (theorems).
In doing this, Euclid created a
deductive system.
Text by Michael Serra
Thanks a lot, Euclid!
So it’s the development of civilization in
general and specifically a series of clever
ancient Greeks who are to be thanked (or
blamed) for the insistence on reason and
proof in mathematics.
When in Greece…
Recall that inductive reasoning leads to
conjectures in mathematics which must be
proven with deductive reasoning. In a
mathematical proof, every statement must
be the consequence of other previously
accepted or proven statements.
Premises in Geometric Arguments
The following is a list of premises that can be
used in geometric proofs:
1. Definitions and undefined terms
2. Properties of algebra, equality, and
congruence
3. Postulates of geometry
4. Previously accepted or proven geometric
conjectures (theorems)
Amazing
Usually we have to
prove a conditional
statement. Think of
this proof as a
maze, where the
hypothesis is the
starting point and
the conclusion is
the ending.
p
q
Amazing
Your job in
constructing the
proof is to link p to q
using definitions,
properties,
postulates, and
previously proven
theorems.
p
q
Un-Amazing
With proofs, sometimes
this is the case:
And so is this:
Exercise 1
Construct a two-column proof of:
If m1 = m3, then mDBC = mEBA.
Exercise 1
Given: m1 = m3
Prove: mDBC = mEBA
Statements
Reasons
1. m1 = m3
1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC
3.Angle Addition Postulate
4. m3 + m2 = mEBA
4.Angle Addition Postulate
5. mDBC = mEBA
5.Substitution Property
Notice in a two-column proof, you first
list what you are given (hypothesis) and
what you are to prove (conclusion).
The proof itself resembles a Tchart with numbered statements
on the left and numbered reasons
for those statements on the right.
Before you begin your proof,
it is wise to try to map out the
maze from p to q.
Two-Column Proof
Generic Two-Column Proof
Given: ____________
Prove: ____________
Insert illustration here
Statements
Reasons
1.
1.
2.
2.
3.
3.
Proof Activity
In this activity, you and your group members
will have a couple of two-column proofs to
assemble. For the first one, you only have
to put the reasons in the correct order. For
the second one, you will have to put both
the statements and reasons in the correct
order.
2.6 Prove Statements about Segments and
Angles
Objectives:
1. To write proofs using
geometric theorems
Assignment:
• P. 116-119: 1-4, 1619, 22-24
• Challenge Problems