Transcript Chapter 2

Chapter 2
Deductive Reasoning
2-1
If-Then Statements;
Converses
CONDITIONAL STATEMENTS
are statements written in ifthen form. The clause following
the “if” is called the
hypothesis and the clause
following “then” is called the
conclusion.
Examples
• 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.
CONVERSE
is formed by interchanging
the hypothesis and the
conclusion.
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
Counterexample
• An example that shows
a statement to be false
• It only takes one
counterexample to
disprove a statement
Biconditional
•A statement that contains
the words “if and only if”
•Segments are congruent
if and only if their lengths
are equal.
2-2
Properties from Algebra
Addition Property
• If a = b, and c = d,
• then a + c = b + d
Subtraction Property
• If a = b, and c = d,
• then a - c = b - d
Multiplication Property
• If a = b,
• then ca = bc
Division Property
• If a = b, and c 0
• then a/c = b/c
Substitution Property
• If a = b, then either a or b
may be substituted for
the other in any equation
(or inequality)
Reflexive Property
•a = a
Symmetric Property
• If a = b, then b = a
Transitive Property
• If a = b, and b = c, then a
=c
Distributive Property
• a(b + c) = ab + ac
Properties of
Congruence
Reflexive Property
• DE  DE
• D   D
Symmetric Property
• If DE  FG, then FG  DE
• If  D   E, then  E   D
Transitive Property
• If DE  FG, and FG  JK,
then DE  JK
• If D   E, and  E   F,
then  D   F
2-3
Proving Theorems
Midpoint of a Segment –
is the point that divides
the segment into two
congruent segments
THEOREM 2-1
Midpoint Theorem
If a point M is the
midpoint of AB, then AM
= ½AB and MB=½AB
BISECTOR of ANGLE– is
the ray that divides the angle
into two adjacent angles that
have equal measure.
THEOREM 2-2
Angle Bisector Theorem
If BX is the bisector of
ABC, then:
mABX = ½mABC and
mXBC = ½ m ABC
A•
B
X
•
C
•
2-4
Special Pairs of Angles
COMPLEMENTARY
two angles whose
measures have the sum 90º
J
39º
51º
K
SUPPLEMENTARY
two angles whose measures
have the sum 180º
H
133º
G
47º
VERTICAL ANGLES– two
angles whose sides form
two pairs of opposite rays.
THEOREM 2-3
Vertical angles are
congruent
2-5
Perpendicular Lines
Perpendicular Lines– two
lines that intersect to form
right angles ( 90° angles)
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
If the exterior sides of two
adjacent acute angles are
perpendicular, then the
angles are complementary
2-6
Planning a Proof
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
2-7 THEOREM
If two angles are supplements
of congruent angles (or of
the same angle), then the
two angles are congruent.
2-8 THEOREM
If two angles are
complements of congruent
angles (or of the same
angle), then the two angles
are congruent.
THE END