Transcript lesson 1 nature and essence of geometry

PROOFS
Geometry
"measuring the
earth“
 is the branch of math
that has to do with
spatial relationships.
A BIT OF HISTORY
Geometry
first
became
associated
with
land
measurement in Egypt. The
Egyptians were obliged to
invent it in order to restore the
landmarks that were destroyed
by the periodic inundation of
Nile River.
Our textbook is based on
Euclidean (or elementary)
geometry. "Euclidean" (or
"elementary") refers to a book
written over 2,000 years ago
called "The Elements" by a
man named Euclid.
Around 300 B.C.
Euclid organized the Greek
knowledge of geometry
into a thirteen –volume
work called ELEMENTS.
His structure and method
influence the way that
geometry is taught today.
Types of Reasoning
1. INDUCTIVE REASONNG
- reaching a conclusion based
on previous observation.
- CONCLUSION IS PROBABLY
TRUE BUT NOT
NECESSARILY TRUE.
EXAMPLES
1² = 1
1≤1
2² = 4
2≤4
CONSIDER THIS EXAMPLE
(½) ²= ¼
½> ¼
The conclusion does not
hold true
2. Deductive Reasoning
reaching
a conclusion
by combining known
truths to create a new
truth
Deductive Reasoning
deductive
reasoning is
certain, provided that
the previously known
truths are in fact true
themselves.
Types of Reasoning
1. INDUCTIVE REASONNG
- reaching a conclusion based
on previous observation.
- CONCLUSION IS PROBABLY
TRUE BUT NOT
NECESSARILY TRUE.
Types of Reasoning
1. DEDUCTIVE REASONNG
- conclusion based on
accepted statements
(postulates, past
theorems, given
information, definition).
Examples of Postulates/axioms
 Properties from algebra
 Linear pair
- SUBSTITUTION
postulate
PROPERTY
 If two // lines are
cut by a
transversal, then
corresponding
angles are equal.
- REFLEXIVE PROPERTY
- SYMMETRIC PROPERTY
- REFLEXIVE PROPERTY
- SUBTRACTION
PROPERTY
- TRANSITIVE PROPERTY
Examples of Definitions
 Midpoint of a
segment
 Bisector of an
angle
 Congruent angles
and Segments
 Perpendicular
lines
 Supplementary
angles
 Complementary angles
 OTHERS
Types of Reasoning
1. DEDUCTIVE REASONNG
- conclusion must be true
if hypothesis are true.
KINDS OF PROOF
DIRECT PROOF
 HAS 5 MAIN PARTS
- THEOREM
- GIVEN
- PROVE STATEMENT
- DIAGRAM
- PROOF with
STATEMENT & REASON
INDIRECT PROOF
 USUALLY IN PARAGRAPH
FORM
 THE OPPOSITE OF THE
STATEMENT TO BE PROVEN
IS ASSUMED TRUE, UNTIL
THIS ASSUMPTION LEADS
TO A CONTRADICTION.
 IT IS USED WHEN A DIRECT
PROOF IS DIFFICULT OR
IMPOSSIBLE.
GIVEN:


k // m:
Transversal t cuts
k and m.
PROVE:
 2 = 3
DIAGRAM/FIGURE
t
k
1
2
3
m
STATEMENT
REASON
k // m: t is a transversal.  Given
 Vertical angles are
  1 = 2
equal
 If two parallel lines are
  1 = 3
cut by a transversal,
then corresponding
angles are equal
 Transitive property
  2 = 3
