INTRODUCTION
TO REAL ANALYSIS
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus
Department of Mathematics
Trinity University
San Antonio, Texas, USA
wtrench@trinity.edu
This book has been judged to meet the evaluation criteria set by
the
6.5 Trapezoids and Kites
Trapezoid: a quadrilateral with exactly one pair of
parallel sides.
Bases: the parallel sides of a trapezoid. Each side has a
pair of base angles.
Legs: The nonparallel sides of a trapezoid.
Base
Leg
Base Angles
Leg
Base
Isoscel
6.4 Rhombuses, Rectangles, and Squares
There are three special types of parallelograms.
Rhombus: a parallelogram with four congruent sides.
Rectangle: a parallelogram with four right angles.
Square: a parallelogram with four congruent sides and
four ri
6.2 PROPERTIES OF PARALLELOGRAMS
Parallelogram: a quadrilateral with both pairs of
opposite sides parallel.
Q
R
PQ RS and QR SP
P
S
Theorems About Parallelograms:
Theorem 6.2: If a quadrilateral is a parallelogram,
then its opposite sides are congruent.
Q
6.3 PROVING QUADRILATERALS ARE
PARALLELOGRAMS
Theorem 6.6: If both pairs of opposite sides of a
quadrilateral are congruent, then the quadrilateral is
a parallelogram. A
B
D
C
ABCD is a parallelogram.
Theorem 6.7: If both pairs of opposite angles of a
qua
4.6 Isosceles, Equilateral, and Right
Triangles
Recall: Isosceles Triangles Have at least two
congruent sides.
Vertex Angle
leg
leg
Base Angles
Base
Theorem 4.6: Base Angles Theorem
If two sides of a triangle are congruent, then the
angles opposite them a
4.5 Using Congruent Triangles
in Proofs
What postulate can be used to prove that
the following triangles are congruent?
B
A
Z
C
X
Y
What are the other corresponding parts?
Are they congruent? How do you know that?
When you already know that two triangle
4.2 Congruence and Triangles
Figures are congruent if they have exactly the
same size and shape.
When two figures are congruent there is a
correspondence between their angles and sides such that
corresponding angles are congruent and corresponding
sides
4.1 Triangles ()
n
A figure formed by three segments joining
three noncollinear points.
May be classified by its sides and by its angles.
n Classifying
Triangles
By Side: Equilateral all sides
Isosceles at least two sides
Scalene no congruent sides
By An
3.7 Perpendicular Lines in the Coordinate Plane
Postulate 18 Slopes of Perpendicular Lines:
In a coordinate plane, two nonvertical lines
are perpendicular if and only if the product of their slopes
is -1.
Vertical and horizontal lines are perpendicular.
Algebra I Review
What is slope?
Formula for the slope of a line:
Find the slope of the line from the following
graph.
y
x
Find the slope of the line that passes through the
points (0,6) and (5,2).
Recall:
The equation of a line written in slopeint
3.5 Using Properties of Parallel Lines
Theorems About Parallel and Perpendicular lines
Theorem 3.11:
If two lines are parallel to the same line, then they are
parallel to each other.
p
q
r
If p ll q and q ll r, then p ll r.
Theorem 3.12:
In a plane, if
3.4 PROVING LINES ARE PARALLEL
Recall: What is the converse of a statement?
When trying to prove that two lines are
parallel we use the converse of the postulates
and theorems we learned in 3.3.
Postulate 16: Corresponding Angles Converse
If two lines are
You have learned that when a transversal cuts
through two or more lines special angle pairs are
formed.
When the lines being cut by the transversal are
parallel these angle pairs have special relationships.
Postulate 15: Corresponding Angles Postulate
Flow Proofs
n
Another type of proof that
uses arrows to show the flow
of the logical argument.
n
Each reason in a flow proof is
written below the statement it
justifies.
Given: 5 and 6 are a linear pair.
6 and 7 are a linear pair.
5
6
7
Prove: 5 7
5 and 6
3.1 Lines and Angles
Parallel Lines Coplanar lines that
do not intersect.
Parallel:
Skew lines Lines that lie in
different planes (noncoplanar). They
are not parallel and do not intersect.
E
B
A
D
C
AB and CD are parallel lines.
CD and BE are skew lin
2.6 PROVING STATEMENTS
ABOUT ANGLES
Properties of Segment Congruence
Reflexive: For any angle A,
A
Symmetric:
If
A.
A B, then
B A.
Transitive: If
A B and
then
A C.
Given:
Prove:
A
A
B and
B
A
2.
3.
4.
5.
A
B
m
A
m
B
m
A
A
B and
C
=m B
=m C
=m C
C
C
Rea
2.5 PROVING STATEMENTS
ABOUT SEGMENTS
Theorem: A conjecture that is proven. A true
statement that follows as a result of other
true statements. Theorems must be proved.
Types of proofs:
Two-column proof: has numbered
statements and reasons that show the
l
2.4 Reasoning with
Proper3es from Algebra
Algebraic Proper3es of Equality
Addi3on Property: If a = b, then a+c = b+c.
Subtrac3on Property: If a = b, then a-
c = b-
c.
Mul3plica3on Property: If a=b, then ac=bc.
2.3 Deductive Reasoning
Deductive
Reasoning: a
process of reasoning logically
from given facts to a
conclusion. If the facts are
true, deductive reasoning
always produces a valid
conclusion.
You
ask your parents if you
can go out with your friends.
They
Recall: A definition uses known words to
describe a new word.
Perpendicular Lines (
T
): Two lines that
intersect to form a right angle.
n
m
m
A line perpendicular to a plane: A line
that intersects the plane in a point and is
perpendicular to every lin
2.1 Conditional Statements
n
n
Conditional Statements: If-then
statements.
If Rauls major is bagpipe,
then he attends CarnegieMellon University.
If you are not completely
satisfied, then your money will
be refunded.
Two parts to each statement:
1. Hypothe
1.7 Introduction to Perimeter,
Circumference, and Area
l
l
l
l
l
l
Draw each of the following on graph
paper.
l A rectangle with length 5 cm and width
3 cm.
l A rectangle with base 8 cm and height
2 cm.
l A rectangle with each side 4 cm.
What are the peri
1.6 Angle Pairs
Vertical Angles: two angles whose sides
are opposite rays.
1
4
2
3
1 and 3 are vertical angles
2 and 4 are vertical angles
Linear Pair: Two adjacent angles whose
noncommon sides are opposite rays.
1.5 Segments and Angle Bisectors
n
n
n
Midpoint A point that divides a segment
into two congruent segments.
Segment Bisector A line, segment, ray, or
plane that intersects the midpoint of a given
segment.
Perpendicular Bisector A line, segment or
ray that
1.4 Angles and Their Measures
Measuring and Naming Angles
Angle () Consists of two dierent
rays (called the sides of the angle) with
the same endpoint (called the vertex).
Naming Angles: (Use the angle sy
1.2 Points, Lines and Planes
Definition using known words to
describe a new word.
Point a location (no size);
represented by a small dot.
Line an infinite amount of points
that extend in two directions without
end; represented by a straight line
with t
1.1 PATTERNS AND INDUCTIVE REASONING
Two types of patterns: visual (pictures), and
numerical.
Describing a visual pattern:
Sketch the next figure in the pattern.
Try the following.
1.
2.
3.
Describing a number pattern.
Predict the next number and describe
1.1 Hourly Pay
When you get a fixed amount of money for
each hour you work.
Hourly rate: amount paid per hour worked.
Straight-time pay: the total amount of
money you earn for a pay period at the
hourly rate.
Calculations:
Straight-Time Pay = Hourly Rate
2.1 FEDERAL INCOME TAX
FIT: (Federal Income Tax) is money withheld by
employers.
Tax Table on Pg. 790
Carlas gross pay for this week is $425.88. She
is married and claims 2 allowances- herself and
her husband. What amount will be withheld
from her pay for