Download Chapter 2 2-1 - Fairland Local Schools

•9/21/2010
Chapter 2
Deductive Reasoning
2-1
If-Then Statements;
Converses
Examples
CONDITIONAL STATMENTS
are statements written in ifthen form. The clause following
the “if” is called the
hypothesis and the clause
following “then” is called the
conclusion.
• If it rains after school,
then I will give you a ride
home.
•If you make an A on your
test, then you will get an A
on your report card.
Examples
CONVERSE
is formed by interchanging
the hypothesis and the
conclusion.
•Statement: If p, then q.
•Converse: If q, then p
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Examples
False Converses
•If Bill lives in Texas, then
he lives west of the
Mississippi River.
•If he lives west of the
Mississippi River, then he
lives in Texas
2-2
Biconditional
•A statement that contains
the words “if and only if”
•p if and only if q
•Segments are congruent
if and only if their lengths
are equal.
Addition Property
• If a = b, and c = d,
Properties from Algebra
Subtraction Property
• then a + c = b + d
Multiplication Property
• If a = b, and c = d,
• If a = b,
• then a - c = b - d
• then ca = bc
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Division Property
• If a = b, and c≠ 0
• then a/c = b/c
Reflexive Property
•a= a
Transitive Property
Substitution Property
• If a = b, then either a or b
may be substituted for
the other in any equation
(or inequality)
Symmetric Property
• If a = b, then b = a
Properties of Congruence
• If a = b, and b = c, then a
=c
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Reflexive Property
Symmetric Property
• DE ≅ DE
• If DE ≅ FG, then FG ≅ DE
•∠D ≅ ∠ D
• If ∠ D ≅ ∠ E, then ∠ E ≅ ∠ D
Transitive Property
• If DE ≅ FG, and FG ≅ JK,
then DE ≅ JK
Distributive Property
• a(b + c) = ab + ac
• If∠ D ≅ ∠ E, and ∠ E ≅ ∠ F,
then ∠ D ≅ ∠ F
2-3
Proving Theorems
THEOREM 2-1
Midpoint Theorem
If a point M is the
midpoint of AB, then AM
= ½AB and MB=½AB
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A
M
B
BISECTOR of ANGLE– is
the ray that divides the angle
into two adjacent angles that
have equal measure.
THEOREM 2-2
Angle Bisector Theorem
A•
If BX is the bisector of
∠ABC, then:
m∠ABX = ½m∠ABC and
m∠XBC = ½ m ∠ABC
2-4
Special Pairs of Angles
•X
C
•
B
COMPLEMENTARY
two angles whose
measures have the sum 90º
J
39º
51º
K
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SUPPLEMENTARY
two angles whose measures
have the sum 180º
H
133º
G
VERTICAL ANGLES– two
angles whose sides form
two pairs of opposite rays.
47º
THEOREM 2-3
Vertical angles are
congruent
2-5
Perpendicular Lines
Perpendicular Lines– two
lines that intersect to form
right angles ( 90° angles)
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2-4 THEOREM
If two lines are
perpendicular, then they
form congruent adjacent
angles.
2-5 THEOREM
If two lines form congruent
adjacent angles, then the
lines are perpendicular.
2-6 THEOREM
2-6
If the exterior sides of two
adjacent acute angles are
perpendicular, then the
angles are complementary
Parts of a Proof
1.A diagram that illustrates the given
information
2.A list, in terms of the figure, of what
is given
3.A list, in terms of the figure, of what
you are to prove
4.A series of statements and reasons
that lead from the given information
to the statement that is to be proved
Planning a Proof
2-7 THEOREM
If two angles are supplements
of congruent angles (or of
the same angle), then the
two angles are congruent.
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2-8 THEOREM
If two angles are
complements of congruent
angles (or of the same
angle), then the two angles
are congruent.
THE END
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