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Chapter 1
Vocab
A
net is a two-dimensional diagram that
you can fold to form a three-dimensional
figure.
Vocab Terms:
• POINT- has no dimension (no length, width, or
thickness) represented by a dot
Point C
Vocab Terms:
Vocab Terms:
Vocab Terms:
Vocab Terms:
Postulate 1-1
Postulate 1-2
Postulate 1-3
Postulate 1-4
Congruent Angles - angles that
have the same measure.
Types of Angles:
Two angles are adjacent angles if
they share a common vertex and
side, but have no common interior
points.
1
4
3
2
5
6
Vertical angles – two angles whose sides form
two pairs of opposite rays. Vertical angles are
congruent.
Angles 1 and 3 are vertical angles, so are angles 2
and 4
Linear pair – two adjacent angles whose
uncommon sides are opposite rays.
Angles 5 and 6 are a linear pair and add up to
180°.
Complementary Angles:
Two angles whose sum is 90°.
Note that these two angles
can be “pasted” together
to form a right angle.
Supplementary Angles:
Two angles whose sum is 180°.
Note that these
two angles
can be “pasted”
together
to form a straight
angle.
An angle bisector is a ray that divides an angle
into two congruent angles. Its endpoint is at the
angle vertex.
Midpoint
The midpoint of a segment is the point that divides, or
bisects, the segment into two congruent segments.
Midpoint Formula:
Find the midpoint between
(-5,-3) and (1,-7).
Distance Formula
-computes the
distance between
two points in a
coordinate plane
Classifying Polygons
 Polygon
– a closed plane figure formed by
three or more segments. Each segment is
called a side. Each endpoint of a side is
called a vertex.
 Each
segment intersects exactly two other
segments at their endpoints.
 No two segments with a common endpoint are
collinear.
Classifying Polygons
 You
can classify a polygon by the number
of sides.
Classifying Polygons
A
diagonal is a segment that connects
two nonconsecutive vertices.
Circle
r
C  2 r   d
C  2 r   d
A r
2
units2
units
Square
P = 4s units
s
A = s2 units2
Rectangle
l
w
P = 2 l + 2w
A = lw
units2
units
Triangle
P=a+b+c
a
c
h
A = ½bh
b
units2
units
Chapter 2
Using Inductive Reasoning
 In
the previous examples you used
inductive reasoning to make a
conjecture.
 Conjecture:
an educated guess about
what you think is true based on
observations.
 What
conjecture can you make about
the twenty-first term in R, W, B, R, W, B, ...?
Finding a Counterexample
Not
all conjectures turn out to be true.
You can prove that a conjecture is false
by finding at least one counterexample.
Conditional Statements
 Type
of logical statement
 Has 2 parts- a hypothesis and a
conclusion
 Can be written in “if-then” form- the “if”
part contains the hypothesis and the
“then” part contains the conclusion.
Truth Value
 The
truth value of a conditional is either
true or false.
 To
show it is true, every time the hypothesis
is true, the conclusion must also be true.
 Example:
If you live in the United States, then
you live in North America.
 To
show it is false, find only one case
where the hypothesis is true and the
conclusion is false.
 Example:
If you live in North America, then you
live in the United States.
Negation
The
negation of a statement is the
opposite of the statement.
The symbol “~p” is read “not p”
Example:
 Statement:
“The ocean is green.”
 Negation: “The ocean is not green.”
Related Conditional Statements
Equivalent Statements: when two
statements are both true or both false.
ORIGINAL-
If angle A is 30°,
then angle A is acute.
INVERSE- If angle A is not 30°,
BOTH
then angle A is not acute.
FALSE
CONVERSE- If angle A is acute,
then angle A is 30°.
CONTRAPOSITIVE- If angle A is
not acute, then angle A is not
30°.
BOTH
TRUE
Biconditional Statement:
A
biconditional is a single true
statement that combines a true
conditional and its true converse.
A statement that contains the phrase
“if and only if”.
Conditional: If it is Sunday, then I am watching
football.
Converse: If I am watching football, then it is
Sunday.
It is Sunday, if and only if I am watching
football. (p
q).
Biconditional Statements:
Can
be either true or false.
To be true, BOTH the conditional
and its converse must be true.
A TRUE Biconditional Statement is
true both “forward” and
“backward”.
All definitions can be written as a
biconditional statements.
Identifying a Good Definition
A
good definition is a statement that
can help you identify or classify an
object.
 Uses
clearly understood terms.
 Is precise. Avoid words such as large, sort
of, and almost.
 Is reversible. Can be written as a true
biconditional.
 **One way to show it is NOT a good
definition is to find a counterexample.
Symbolic Notation
Conditional
statement has a
hypothesis and a conclusion.
Written with symbolic notation: p
represents hypothesis, q represents
conclusion.
If p, then q.
Symbolic notation: p  q
p implies q
Converse?
If q, then p.
or
q p
Biconditional?
p if and only if q
or
pq
Inverse?
If not p, then not q.
or
~ p ~ q
Contrapositive?
If not q, then not p.
or
~ q ~ p
Checkpoint:
 Let
p be “You are a pianist” and q be “you are a
musician”.
 Write in words:
pq
q p
pq
~ p ~ q
~ q ~ p
Laws of Logic:
 DEDUCTIVE
REASONING – uses
facts, definitions, and accepted
properties in a logical order to
write a logical argument.
 INDUCTIVE
REASONING – uses
examples and patterns to form a
conjecture.
2 LAWS OF DEDUCTIVE REASONING
 LAW
If
OF DETACHMENT:
pq
is a true conditional
statement and p is true, then q is true.
2ND LAW
 LAW
If
OF SYLLOGISM:
and
pq
qr
conditional statements, then
is true.
are true
pr
Algebraic Properties
Theorem – a conjecture or statement
that you prove true
***When writing a proof for a
theorem, separate the theorem into
a hypothesis and conclusion. The
hypothesis becomes the “Given”
statement and the conclusion is
what you want to prove.
Chapter 3
Parallel Lines – two lines that
are coplanar and do not intersect.
Notation: //
Skew Lines – lines that do not
intersect are not coplanar.
Parallel Planes – two planes that
do not intersect.
TRANSVERSAL – a line that intersects
two or more coplanar lines at
different points.
l
1 2
3 4
5 6
7 8
Notice, angles 3, 4, 5, and 6 are interior angles
(between the lines). Angles 1, 2, 7, and 8 are
exterior angles (outside the lines).
CORRESPONDING ANGLES:
occupy corresponding positions
l
3
1 2
4
5 6
7 8
Corresponding angles lie on the same side of
the transversal, and in corresponding positions
ALTERNATE EXTERIOR ANGLES:
lie outside the two lines on opposite
sides of the transversal.
l
1 2
3 4
5 6
7 8
ALTERNATE INTERIOR ANGLES:
lie between the two lines on
opposite sides of
the transversal.
l
1 2
3 4
5 6
7 8
SAME-SIDE INTERIOR ANGLES:
lie between the two lines on the
same side of the
transversal.
l
1 2
3 4
5 6
7 8
Theorem 3-10: Triangle Sum
Theorem
The sum of the measures of the interior angles of a
triangle is 180°.
mA  mB  mC  180
Theorem 3-11:
Exterior Angle Theorem
1
The measure of an exterior angle of a triangle equals
the sum of the measures of the two remote interior
angles.
m1  mA  mB
SLOPE
Definition
Symbols
The slope, m, of a
line is the ratio of
the vertical change
(rise) to the
horizontal change
(run) between any
two points.
A line contains the
points (x1, y1) and
(x2, y2)
rise yy2  yy1
rise
m 
 2 1
m

run xx2  xx1
run
2
1
Diagram
Equations of Lines:
Y = mx + b
slope
Y-intercept :the y-coordinate
of the point where the line
crosses the y-axis.
SLOPE-INTERCEPT FORM
*only used for nonvertical lines
y  y1  m  x  x1 
Point-Slope Form
Standard Form
Ax  By  C
Checkpoint:
 Find
the slope of a line that passes through
the points (-3, 0) and (4,7).
Checkpoint:
Find all three forms of the equation of the lines
below.
Line p, passes through (0, -3) and (1, -2).
Line r passes through (-6, -1) and (3, 7).
Graphing Lines
1. Identify the form of
the equation.
2. Identify/Graph the yintercept
3. Start at the y-intercept
and use the slope to
graph a couple points
on the line
4. Connect the points
Using Two Points to Write an
Equation
 Write
the equation of
the line using the point(2, -1) in point-slope
form.
 Find
the slope of the line
 Use the given values and
plug into the point-slope
form
 Write
the equation of
the line in slopeintercept form.
Writing Equations of Horizontal and
Vertical Lines
Slopes of Parallel Lines
When two line are parallel, their slopes are the
same.
**If two lines are parallel,
their slopes will be the same,
but they MUST have a
different y-intercept!
**Do all horizontal lines have
the same slope? Explain.
Sage-n-Scribe
Slopes of Perpendicular Lines
If two lines are perpendicular, their slopes are
negative reciprocals.
In a coordinate plane, two nonvertical lines are
perpendicular if and only if the product of their
slopes is -1 (opposite reciprocals).
Vertical and horizontal lines are perpendicular.
Perpendicular Slopes:
3
m2 
4
 Opposite
4
m1  
3
Reciprocals
of each other.
Chapter 4
Congruent figures – have exactly
the same size and shape.
(Congruent Corresponding Parts)
*Corresponding angles
and sides are congruent.
Theorem 4.1: Third Angles Theorem
If two angles of one
triangle are congruent
to two angles of another
triangle, then the third
angles are also congruent.
If A  D and B  F
then C  E
Triangles: figure formed by 3 segments
joining 3 noncollinear points.
CLASSIFICATION BY SIDES
(# of congruent sides)
Scalene Triangle-no congruent sides
Isosceles Triangle-at least 2 congruent sides
Equilateral Triangle-3 congruent sides
CLASSIFICATION BY ANGLES
(all triangles have at least two acute angles, the
third angle is used to classify)
EQUIANGULAR TRIANGLE – 3 congruent angles
(also acute)
 Vertex
– each of the three points joining the sides of
a triangle.
Vertex A
 Adjacent sides – two sides sharing a common
vertex.
Sides AB and AC
 Opposite side – the third side not sharing the vertex
Side BC
Postulate 4-1: Side-Side-Side
(SSS) Postulate
If three sides of one triangle are congruent to
three sides of a second triangle, then the
two triangles are congruent.
Postulate 4-2: Side-Angle-Side
(SAS) Congruence Postulate
If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of second triangle, then the
two triangles are congruent.
Postulate 4-3: Angle-Side-Angle
(ASA) Postulate
If 2 angles and the included side of one
triangle are congruent to 2 angles and the
included side of a second triangle, then the
2 triangles are congruent
Theorem 4-2: Angle-AngleSide (AAS) Theorem
If two angles and a nonincluded side of one triangle
are congruent to two angles and the corresponding
nonincluded side of the second triangle, then the
two triangles are congruent.
Theorem 4.3 Isosceles Triangle
Theorem
If two sides of a triangle
are congruent, then the
angles opposite them
are congruent.
Theorem 4.4 Converse:
If two angles of a
triangle are congruent,
then the sides opposite
them are congruent
Right Triangles
Hypotenuse:
 It
is the side opposite the right angle
in a right triangle.
 It is the longest side in a right triangle
AAS, SSS, SAS, ASA, and…
For RIGHT triangles ONLY:
Theorem 4.6: Hypotenuse-Leg (HL)
Congruence
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of
a second right triangle, then the two triangles
are congruent.
Chapter 5
Midsegment of a triangle is a
segment that connects the
midpoints of two sides of a triangle.
Theorem 5.1:
Midsegment Theorem
If a segment joins the midpoints of two
sides of a triangle, then the segment is
parallel to the third side and is half as long.
x
½x
Perpendicular Bisector: A segment,
ray, line, or plane that is
perpendicular to a segment at its
midpoint.
C
A
B
M
D
Equidistant from two points – distance
from each point is the same
 Theorem
5-4: Angle
Bisector Theorem If a
Theorems:
point is on the bisector
of an angle, then it is
equidistant from the
sides of the angle.
 Theorem 5-5: Converse
B
of the Angle Bisector
D
Theorem If a point in the
interior of an angle is
A
equidistant from the
C
sides of the angle, then
it lies on the bisector of
the angle.
Perpendicular Bisectors of a
Triangle
A
perpendicular bisector of
a side of a triangle is a line
(or ray or segment) that is
perpendicular to a side of
the triangle at the midpoint
of the side.
Using Angle Bisectors of a
Triangle
 An
angle bisector of a
triangle is a bisector of
an angle of a triangle.
 The point of
concurrency of the
angle bisectors is
called the incenter of
the triangle. (Center of
the inscribed circle.)It
always lies inside the
triangle.
Median of a Triangle
A
median of a triangle is a
segment whose endpoints
are a vertex of the triangle
and the midpoint of the
opposite side.
A
median
B
Every triangle has three medians.
C
Point of Concurrency for Medians
The point of
concurrency
for the 3
medians is
called the
centroid.
The centroid is
always on the
inside of the
triangle.
Centroid
Concurrency of Medians Theorem
Theorem 5.8 :
The medians of a
triangle are concurrent
at a point that is two
thirds of the distance
from each vertex to
the midpoint of the
opposite side.
BG = (2/3)BE, AG = (2/3)AF,
and CG = (2/3)CD
E
D
F
Note: The centroid of a triangle can
be used as its balancing point.
Altitude of a Triangle
 The
altitude of a
triangle is the
perpendicular
segment from a
vertex to the
opposite side or to
the line that contains
the opposite side.
 An altitude can lie
inside, on, or outside
the triangle.
Orthocenter of a Triangle
Theorem 5.9: Concurrency
of Altitudes Theorem:
The lines that contain the
altitudes of a triangle are
concurrent.
If segments AH, BH and CH
are the altitudes of triangle
DEF, the lines AH, BH, and
CH intersect at a point H,
the orthocenter.
B
A
C
Summary
Writing an Indirect Proof
Step 1: State as a temporary
assumption the opposite (negation)
of what you want to prove
Step 2: Show that this temporary
assumption leads to a contradiction
Step 3: Conclude that the temporary
assumption must be false and that
what you want to prove must be
true
Triangle Inequality Theorem
Theorem. 5.12: The sum of the lengths of any
two sides of a triangle is greater than the
length of the third side.
Hinge Theorem (SAS Inequality
Theorem)
Theorem 5.13: If two sides of one triangle are
congruent to two sides of another triangle,
and the included angles are not congruent,
then the longer third side is opposite the
larger included angle.
R
RT > VX
100
S
T
V
80
W
X