Download Geometry - Kellenberg Memorial High School

2015-2016
Geometry
Kellenberg Memorial High School
Undefined Terms and
Basic Definitions
1
Click here for
Chapter 1
Student Notes
Section 1
Undefined Terms
1.1: Undefined Terms (we accept these
as true)
The word “Geometry” comes from two Ancient Greek words:
“ge”
“metron”
meaning “Earth”
meaning “measure”
1. Set: a set is a collection or group of objects with some
common characteristic.
Ex: The set of all students in your Geometry class,
the set of all odd numbers, or the set of all cars made by
Ford are examples.
2. Point: a point is basically a location in space. It can be
represented by a dot on a piece of paper, named with a
2
capital letter.
!
Ex: •P
or we can name it with a lower case letter
!
“point P”
!
Ex:
!
!
“line k”
Points have no size at all: no length, width, or thickness.
Points indicate position or location as seen when graphing
A line does not have a measurable length because it is infinitely
points on the coordinate plane.
long.
3. Line: A line is an infinite set of points. When we represent a
4. Plane: a plane is a set of points that forms a completely flat
line, arrows are placed on each end of the line to illustrate that
surface which extends infinitely in all directions. Think of it as
the line extends infinitely in both directions.
the world’s biggest, flattest, thinnest piece of paper.
(The symbol for Infinity is “∞”).
That infinite set of points, at least as far as we’re concerned,
In fact, a plane is so thin that it has no thickness at all. Think of
usually forms a straight line. (We’ll talk about curved lines much
a textbook: if you pile 800 pages, one on top of another, you’ll
later in the course.)
have a book that’s an inch or two thick. But,
To represent a line, we choose any 2 of the points on the line
because planes have no thickness at all,
and place an
you can pile 8,000 of them, one on top of
Ex:
over them. !
!
!
! !
!
“line EF”
!
the other, and the thickness won’t increase.
3
A plane is named by a single letter:
“plane N”
You try: Determine which undefined term describes
the following.
4
Section 2
Basic Definitions
1.2 BASIC DEFINITIONS
5. Line Segment: a set of points consisting of two points on a
line, called endpoints and the set of all points on the line
between the endpoints.
We can name a segment by placing a bar over the endpoints.
***A Line is named by any 2 points on the line, while a
Line Segment is always named by its endpoints.
6. Ray: the set of all points in a half line, including the dividing
point, which is called the endpoint of the ray.
5
A ray is named by placing an arrow pointing to the right over
the angle. The vertex of the angle pictured above is at A.
two capital letters
We have a couple of options when it comes to naming an
* 1st Letter- Names the endpoint of the ray
angle. We can use one letter: the vertex angle, and call the
* 2nd Letter- Names some other point on the ray
angle above Angle A.
Or we can use three letters—as long as the one in the middle is
!
EX:
the vertex. So the picture above could be called ∡ CAB or
∡BAC. Or, if we choose, we can name our angle using numbers
or lower case letters.
Examples:
7. Angle: the union of two rays having the same endpoint. Its
symbol is either
or
!
!
!
!
!
!
!
!
Vertex (of the angle): the endpoints of each ray, or the corner of
6
We measure angles by determining the number of DEGREES
c) Obtuse:
90° < θ < 180°
contained in each one.
An obtuse angle is one measuring greater than 90 and less
than 180 degrees. It can look kind of like this:
What are the different types of angles?
As you probably remember from elementary school, there are a
number of different types of angles, classified by the number of
d) Straight: θ = 180°
degrees it contains:
A straight angle is an angle of exactly 180 degrees.
a) Acute:
0° < θ < 90°
An acute angle is one measuring greater than 0 and less than
90 degrees. It can look kind of like this:
e) Reflex:
180° < θ < 360°
A reflex angle is the one of these angle types you’ve probably
b) Right:
θ = 90°
never heard of. It’s an angle whose measure is more than 180
A right angle is one measuring exactly 90 degrees.
and less than 360 degrees
It’s the kind of angle found in a square or a rectangle. A right
The problem, of course, is that reflex angles look just like acute
angle is symbolized by a little box at the vertex, like
this:
angles:
7
The reflex angle in the picture above isn’t what catches your
60 smaller units called, predictably enough, SECONDS.
eye; it’s the acute angle next to it that you tend to see. As a
result, on those rare occasions this year when we want to talk
about the reflex angle, we’ll be sure to specify it. We can also
Here are the symbols used for each of the units of measurement:
Degrees ° Minutes’ Seconds”
mark the above diagram showing the reflex angle.
Remember:
θ is just another symbol used like the variables x or y, but is
usually used with angles. It comes from the Greek alphabet:
!
!
!
1 Degree = 60 minutes (60’)
1 Minute = 60 seconds (60”)
θ (pronounced “theta”)
So that means, for example, that
Measuring Angles
As you’re already aware, angles can be measured in degrees.
But sometimes, a single degree is too wide a measurement for
a particular situation. Sometimes, we need a part of an angle in
order to provide greater precision.
Each angle can be broken down into 60 smaller units called
MINUTES. And each minute, in turn, can be broken down into
!
!
¼° = 15’
(since ¼ of 60 is 15)
!
!
½° = 30‘!
(since ½ of 60 is 30)
!
!
¾° = 45’
(since ¾ of 60 is 45)
30 minutes or 30 seconds act like the .5 in a decimal, for
rounding purposes. That means that 30 minutes or 30 seconds
is your “Round UP” number… any number smaller will
round DOWN.
8
Example 1:
4. 15°03’38”
_____________
5. 177°39’56”
_____________
!Round to the nearest degree
!
52°29’ !
!
! 52°
!Round to the nearest degree
!
52°30’ !
!
! 53°
You try: Round the angle measure to the nearest minute.
Example 2:
!Round to the nearest minute
!
27°43’29”
! !
27°43’30”
! !
2. 150°13’
2. 163°27’15”
____________
27°44’
You try: Round the angle measure to the nearest degree.
1. 46°37’
____________
27°43’
!Round to the nearest minute
!
1. 24°37’41”
3. 48°34’46”
____________
4. 115°52’08”
_____________
____________
____________
5. 22°59’31”
3. 62°51’26”
_____________
____________
9
8. Congruence: means “having same length or measure”
(think: same size & shape)
The symbol for congruence is:
How would you mark line segments to show they are not
It combines the equal sign: “ = ” (same size ) with the symbol
congruent?!
!
!
!
for similarity “~” (same shape). We will learn more about
similarity later in the year.
b) Congruent Angles are angles which have the same measure.
a) Congruent Segments are segments that have the same
length.
10
9. Collinear Points are points that lie on the same straight
13. Parallel lines: straight lines that never intersect. The
line.
symbol for parallel is ll. For example we can say AB CD.
10. Non-Collinear points, on the other hand, are points that
DO NOT lie on the same straight line.
Note that, unlike many of the definitions we’ve seen thus far,
11. Midpoint is the point on a line segment that divides the
Parallel addresses the DIRECTION a line goes, and not its
segment into 2 ≅ segments.
length. Two segments can certainly be parallel without being
congruent.
14. Perpendicular lines: straight lines that intersect and form
12. Bisection of a Line Segment: a segment is bisected at a
right angles (90°).
point if the point is the midpoint of the line segment.
The symbol for perpendicular lines is: ⊥. So we can write
AB ⊥ BC if they intersect and form a right angle at B: !
!
!
!
!
!!
11
15. Perpendicular Bisector is, as you might think, a line or
and you get degrees, not inches.
segment which does two things: it cuts the line segment in half
& forms right angles.
17. Complementary Angles are two angles whose sum is 90°.
For example, an angle of 40° and one of 50° are complements.
!
Likewise,
1 and
2 in the diagram below are
complements:
16. Angle Bisector: divides an angle into 2 congruent angles.
18. Supplementary Angles are two angles whose sum is 180°.
An angle of 116°, then, would be the supplement of an angle
of 64°, since their sum is 180°.
Please note: when an angle is bisected, it forms two congruent
ANGLES. It does NOT mean that the sides of the angles are
congruent…. Think about it for a second… cut degrees in half
12
You try:
19. Adjacent Angles are angles that share a vertex and a side,
1. Find the complement of 32°
but have no interior points in common. (The word “adjacent”
means “next to.”)
2. Find the complement of 46°17’
3. Find the supplement of 58°
Which angles x & y do represent adjacent angles?
4. Find the supplement of 38°44’
(we will place x & y in the diagrams)
5. Find the supplement of 161°
6. Find the supplement of 118°35’
13
20. A Linear Pair are two angles that are both supplementary
You try: If
and adjacent.
!
4 = 60°, find the other 3 angles.
21. Vertical Angles are formed by intersecting lines.
In the diagram below,
as are
2 and
1 and
3 are vertical angles,
4.
Which other pairs of angles can be found in this diagram?
It’s important to remember that VERTICAL ANGLES ARE
CONGRUENT.
Which pairs add to 180°?
(How would you mark the angles in the following diagram?)
The four angles add up to ____________degrees.
14
You try: In the following diagram
!
!
!
! !
!
!
! !
!
!
!
!
a) Name the pairs of vertical angles.
b) Name the adjacent angles at A.
c) Name a linear pair.
15
Section 3
More Practice
1.3 More Practice
Use your definitions to answers these questions:
1. Describe the type of angles.
a) 75°22‘
b) 90°!
!
!
c) 120° 59’ 22’‘
!
d) 180°!
e) 216°
f) 19° 14’ 34”!
g) 25° 30‘!
!
h) 300° 01’ 01”!
i) 163° ! !
j)
89° 59’ 59”
16
2. Round each angle to the nearest degree.
a) 40° 13‘
4. Points A, B & C are collinear and in that order
!
Use the diagram below to answer the following:
b) 58 48’!!
c) 120° 09‘
!
d) 175° 52‘!
a) AB = 10, BC = 4 THEN AC = ?
e) 109° 22’
b) AB = 23, AC = 72 THEN BC = ?
3. Round each angle to the nearest minute.
a) 14° 26’ 18”
!
b) 75° 19’ 38”!
c) 125° 45’ 46”
!
!
!
!
!
!
!
c) AC = 156, BC = 91 THEN AB = ?
!
d) 22° 09’ 20”
e) 158° 11’ 49”
17
5. Find the complement of the following angles:
a) 73°
7. Add the angles, use the following diagram.
!
b) 38°!
!
c) 44°
!
d) 63° 23‘!
e) 10° 52’
8. Subtract the angles, use the following diagram.
6. Find the supplement of the following angles:
a) 133°
!
b) 45°!
c) 99
!
For # 9 – 12 All points are collinear.
!
d) 161° 40‘!
e) 20° 36’
9. Add the line segments, using the following diagram.
!
18
10. Subtract the line segments, using the following diagram.
!
!
!
13. Find the complement and supplement of each of the following algebraic expressions:
a)
m°
b)
(3y)°
c)
(y + 20)°
d)
(q – 35)°
e) (3p + 56)°
19
Section 4
Distance and
Absolute Value
1.4 Distance & Review of Absolute
Value
Distance:
Absolute Value:
1.
| 98 |
= _______________
2.
| -34 |
= __________________
20
3.
- | -45 |
4.
| 5 – 18 |
= _______________
5. | -6 – (-2) |
= ___________________
= _______________
Using the above number line find the distance between the
following points.
6. | -2 + 2 – 10 | = __________________
7. - | 15 + 9 |
8. | -3 – (-12) |
9.
| -1 + -5 – (-2) |
10. -| -36 |
= ________________
= __________________
= _______________
= ___________________
11.
C&F
_______________
12.
D&F
_______________
13.
G&H
_______________
14.
H&F
_______________
15.
C&E
_______________
16.
E&G
_______________
17.
C&H
_______________
18.
H&E
_______________
19.
C&D
_______________
20.
D&H
_______________
21
Sets and Venn
Diagrams
2
Click here for
Chapter 2
Student Notes
Section 1
Definitions
Involving Sets
2.1 Definitions Involving Sets
A Set is a well-defined collection of objects or numbers.
!
Ex: Even integers greater than 0 and less than 10
!!
!
!
A = { 2, 4, 6, 8 }
We can name a set by assigning a capital letter to it, as we did
in the above example. Note, too, that the members, or
ELEMENTS, of a set are listed (or tabulated) inside the braces.
Elements are the members that are contained in a given set.
They can be numbers, letters, symbols or any other type of
object.
If we want to say that a particular number or object is an
element of a set, we can use the symbol
23
For example:
Set B is the set of common household pets
B = {cat, dog, bird, hamster, fish, snake}
Cat is an element of the set of common pets
Cat
B
“cat is an element in set B”
* When an element is not part of the set:
Elephant
B
“elephant is not an element in set B”
24
Section 2
Kinds of Sets
2.2 Kinds of Sets
A Finite Set is a set whose elements can be counted. In other
words, there is a definite number of elements in the given set.
Finite sets don’t have to be EASY to count—they can have
millions of elements—but there has to be a finite, or limited,
number of elements in the set.
!
Ex: the set of letters in the alphabet
!
Ex: {1,2,3, … , 200}
An Infinite Set is a set whose elements cannot be counted.
!
Ex: the set of real numbers
!
Ex: {1,2,3,…}
25
***Please Note:
{1, 2, 3, 4, . . ., 200}!
Examples of sets that are empty:
- This means that the set includes
“The set of days that begin with the letter Z.”
the #’s 1 to 200, consecutively. We use the three dots to
Or
represent the rest of the #’s between 4 and 200.
“The months of the year that have 53 days.”
(it would take too long to write them all)
Please note that we do NOT put the ∅ symbol into braces.
!
{1, 2, 3, . . .} - This means that the set is an infinite set.
The list of the elements in the set keeps going on forever.
Doing so would mean that they were no longer empty. Nor
do we put the “number zero” in braces. That would mean
the set is a finite set containing the element 0.
There’s one more type of set we need to discuss:
the Empty Set (or Null Set). It’s empty because it’s a set that
has no elements.
We represent the empty set one of two ways:
The empty set is a subset* of every set.
either the symbol ∅ or a pair of empty braces { }.
26
Section 3
Relationships
Between Sets
2.3 Relationships Between Sets
Equal Sets are sets that contain exactly the same elements
though the order of elements does not matter.
The symbol for equal sets is =
!
Ex 1:
!
!
A = {2, 4, 6, 8}
B = {even counting numbers < 10}
!
!
A=B
Ex 2: C = { &, @, $, # }
D = { $, &, #, @ }
!
!
!
C=D
27
Ex 3: E = { x, y, z }
The Universal Set is the entire set of elements under
F = { a, b, c }
!
!
!
!
consideration in a given situation.
E≠F
The Universal Set is represented by the symbol
Sets E and F are not EQUAL sets, but they are an
For example, if we wanted to examine the English Alphabet we
example of EQUIVALENT sets.
would state:
!
Equivalent Sets are sets that have the same number of
!
!
= {the letters of the alphabet}
We can then look at the subsets* that are found in
elements. Each element in one set can be matched to an
V = {vowels} and C = {consonants}
element in another. This is another way of saying that “there is
one-to-one correspondence between the two sets” — an idea
A Subset* is a set that contains the elements of another set.
that will become important as you go on in math.
Logically, then, every set is a subset of itself.
The symbol for subsets is
The symbol for equivalent sets is
!
Ex:
~
G = {Beth, Mary, Kelly, Sue}
H = {William, Joe, Mark, George}
!
!
!
!
G~H
Ex: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B = {3, 4, 5, 6}
We can say that “set B is a subset of set A”
B
A
Could “set A be a subset of set B”?
28
You try:
Disjoint Sets are when sets do not have any common
Rewrite these true statements using the symbol:
elements -- when their intersection yields the empty set.
1. Set A is a subset of itself. ___________________
!
Ex: C = {2, 4, 6, 8}
!
D = {1, 3, 5, 7}
2. Set B is a subset of itself. ___________________
!
C
3. The empty set is a subset of A. ________________
Union of Sets is the combination of the elements of 2 or more
D = { } or ∅
sets.
4. The empty set is a subset of B. ________________
The symbol for Union is
. It will be easy to remember,
since it looks like the capital letter U.
Intersection of Sets: Sets intersect when they share at least
!
Ex 1: A = {1, 2, 3}
one common element. That intersection contains any overlap
B = {4, 5, 6}
between the sets.
A
B = {1, 2, 3, 4, 5, 6}
The symbol for intersection is
Ex:! A= {1, 2, 3, 4, 5}!
!
!
!
!
A
Ex2: C = {5, 6, 7, 8}
B= {2, 4, 6, 8, 10}
D = {4, 5, 6, 7, 9}
B = {2, 4}
!
!
C
D = {4, 5, 6, 7, 8, 9}
29
Remember to copy each element only once.
The Complement of a Set is the set of elements in the
Universal Set that are NOT in the given set. Obviously, in order
A good tip: Putting the elements into alphabetical/
to find the Complement of a set, you need to also know which
numerical order is a helpful way to check that you have
elements are in the Universal Set.
listed all the elements needed.
The symbol for the Complement is an apostrophe placed after
the name of the set.
Using the Empty Set with UNION & INTERSECTION
Ex:
ANY SET
Ø = ANY SET
ANY SET
Ø = Ø
!
= {1, 2, 3, 4, 5}
!
For example… if set A = {1, 2, 3} then
A
Ø = A or {1, 2, 3}
A
Ø = Ø
30
Section 4
The Venn Diagram
2.4 The Venn Diagram
A Venn Diagram is a pictorial representation of a set.
Venn Diagrams sometimes make it easier to find the answers to
problems involving sets.
In using Venn Diagrams, a rectangle will represent the
Universal Set. It surrounds the rest of the diagram. Circles
represent subsets of the Universal Set and the elements of a
set are placed in the circle.
31
f) List A’
g) List B’
h) List (A
B)’
i) List (A
B)’
a) List
j) List A
b) List A! !
B’
!
!
c) List B!
!
d) List A
B!
e) List A
B
!
!
32
d) List C
!
!
!
e) List A
!
a)!
List
!!
!
!
!
!
!
!
!
!
!
!
B
!
!
f) List B
C
g) List A
C
!
!
!
!
!
b)!
List A!
!
!
!
!
!
c) List B!!
!!
!
h) List A
!
!
!
!
!
B
!
i) List A’
33
j) List C’
p) List [ A
B
C ]’
B
C
k) List (A
B)’
q) List A
l) List (A
C)’
r) List [ A
m) List A
C’
n) List A
B’
o) List A
B
B
C ]’
C
34
Introduction to
Triangles
3
Click here for
Chapter 3
Student Notes
Section 1
Definition of a Triangle
and its Classifications
3.1 Definition of a Triangle and its
Classifications
Definition: A triangle is a 3 sided polygon. A polygon is a
closed figure which is the union of line segments. (We will study
more about polygons in chapter 5) Because a triangle has 3
sides it also has 3 interior angles. These three angles always
add to 180°.
Labeling a triangle: Capital letters are used for the vertices.
The same letters in lower case are used to represent the sides
opposite those vertices.
36
Angles of the triangle are written using the single vertex letter
3. A RIGHT TRIANGLE has one right angle.
or with three letters.
(Again, why only one?)
Sides can also be written by their line segment name.
A triangle can be classified by its angles:
1. An ACUTE TRIANGLE has 3 acute
We can also classify a triangle based on the number of
angles.
congruent sides it has.
2. An OBTUSE TRIANGLE has
one obtuse angle.
(Why only one?)
37
Classifying a triangle by its sides:
3. An EQUILATERAL TRIANGLE has three congruent sides
and three congruent angles.
1. A SCALENE TRIANGLE has no congruent sides & therefore
no congruent angles.
Draw an example. Mark the sides and angles accordingly.
Draw examples. Mark the sides and angles accordingly.
What is the measure of each angle of the equilateral triangle?
Why?
2. An ISOSCELES TRIANGLE has two congruent sides and
two congruent angles.
This triangle can also be called an EQUIANGULAR
triangle, since the 3 angles are congruent.
Draw examples. Mark the sides and angles accordingly.
38
You try:
Some triangles have names for particular parts. Let’s first
EX 1: Classify a triangle with angles of 40°, 60° and 80°.
discuss the right triangle:
!
The easiest part of a right triangle to spot is its right angle.
EX2: Classify a triangle with angles of 120°, 30°and 30°
It is symbolized with the small box in the right angle.
The side across from the right angle is known as the
EX 3: Classify a triangle with angles of 25°, 90° and 65°.
HYPOTENUSE. It is always the longest side of the right
triangle.
EX 4: Classify a triangle with angles of 100°, 60° and 20°.
The remaining two sides are the LEGS. They are always
perpendicular to each other forming the right angle.
EX 5: Classify a triangle with angles of 51°13’ and 70°25’.
!
Label the parts of the right triangle below if AC ⊥ CB.
EX 6: Classify a triangle with angles of 90° and 45°.
EX 7: Classify a triangle with angles of 60° and 59°60’.
Since
C is the right angle, the other two angles of a right
triangle must be acute.
Why?
What angle pair name can you give these two angles?
39
Now let’s look at the Isosceles Triangle. As was stated earlier,
You try:
it has 2 congruent sides. Those congruent sides are called the
1. If the vertex angle is 106°, what is the vertex angle?
LEGS; the non-congruent side is the BASE. The two angles
that share the base are called the BASE ANGLES. These
angles are congruent. The angle formed by the legs is the
VERTEX ANGLE.
2. If a base angle is 68°, what is each base angle?
Label the parts of the Isosceles Triangle below, if
AB ≅ AC.
40
Section 2
Interior Angles of a
Triangle
3.2 Interior Angles of a Triangle
•
The sum of the 3 interior angles of any triangle is
180 degrees
•!
A triangle can have only 1 right angle
•!
A triangle can have only 1 obtuse angle
•
If the triangle is a right triangle, then the remaining two
angles must add up to the remaining 90 degrees. In other
words, the acute angles of a right triangle are complements.
•
If the triangle is equilateral, then it’s also equiangular.
(Remember, as stated earlier, that a triangle always has as
many congruent angles as sides.) As a result, each angle of an
equilateral triangle measures 60 degrees. (180°/3 = 60°)
•
If the triangle is an isosceles right triangle then each acute
base angle, measures 45°.
41
•
The sum of the angles of a quadrilateral is 360 degrees.
You try:
The logic is simple: take any quadrilateral and draw a diagonal.
The quadrilateral is now a pair of triangles, each having
1. Two angles of a triangle are 78° and 45°. What is the
180 degrees.
measure of the third angle?
Then classify the triangle.
2. Two angles of a triangle are 24°13’ and 36°24’. What is the
measure of the third angle?
Then classify the triangle.
3. The angles of a triangle are represented by (3x + 1)°,
(4x - 12)° and (7x + 9)°.
Solve for x, find the measure of each angle and then classify
the triangle.
42
Section 3
Exterior Angles of a
Triangle
3.3 Exterior Angles of a Triangle
When one side of a triangle is extended, the angle between that
extension and the adjacent side is known as an
Exterior Angle.
(Remember, “exterior” means “outside.”)
In the diagram above,
1 is an exterior angle.
The measure of an exterior angle of a triangle equals the sum
of the two angles inside the triangle that are NOT adjacent to
it… the two interior angles that don’t share a side with the
exterior angle.
43
In the diagram above, that makes m 1 = m 2 + m 3.
Below, draw examples of an obtuse triangle and a right triangle
with their interior and exterior angles.
What is the measure of exterior
1 if
2 = 46° and
3 = 77°?
What can be concluded about the sum of the exterior angles of
any triangle?
How many exterior angles does a triangle have?
What are the degree measures of the exterior angles at A and
C in the above diagram? Draw and label the angles.
44
You try:
3. Find the measure of a base of an isosceles ∆ if the exterior
angle at the vertex measures 132°.
1. Solve for x and find the measure of the angles Q & M.
4. In ∆RST, angle S is a right angle and the m T = 38°. Find
the measure of the exterior angle at R.
2. Find the measure of the vertex of an isosceles ∆ if either of
the exterior angles formed by extending the base measures
144°.
45
Section 4
Line Segments
Associated with the
Triangle
3.4 Line Segments Associated with the Triangle
There are three types of line segments that exist in the triangle.
1. A MEDIAN is a line segment that is drawn from a vertex to
the midpoint of the opposite side.
!
In ∆ABC, above, if M is the midpoint of BC, then AM is a
median. We can also draw medians to sides AB and AC, once
we locate their midpoints.
46
2. An ALTITUDE is a line segment drawn perpendicularly from
How do you know which you’re dealing with? The problem has
a vertex to the opposite side, forming right angles.
to tell you, either directly (using the words “altitude” or “median”
or “angle bisector”) or indirectly, by giving you the information
that permits you to draw the correct conclusion.
You try: For #’s 1- 6 Describe line segment DF in each of the
following triangles.
How many altitudes does a triangle have?
3. An ANGLE BISECTOR in a triangle does the same job it
does when it’s not in a triangle: it cuts an angle of the triangle
into two congruent angles.
47
There are times when a single line segment can perform two or
even all three of these jobs.
In which triangle(s) can this occur?
Illustrate below:
48
Section 5
Triangle
Inequalities
Triangle Inequalities
Let’s say your older brother has his driver’s license and drives
you to school each day.
!
!
In the diagram above, let’s let H symbolize your home. C can
be your brother’s favorite source of coffee, and K, of course, is
Kellenberg.
No matter how many shortcuts he knows, how fast a driver he
is, how early he gets started, it’s a basic fact of geometry that a
detour for coffee on the way to school will add more mileage to
the car than going straight to school… the shortest distance
between two points is a straight line.
49
As a result, we can make the following statement:
the sum of two sides of a triangle is greater than the third
side.
You try:
1. Can these sets of numbers be the sides of a triangle?
In particular, the sum of the lengths of two shortest sides must
be greater than the longest side.
a)
{3, 4, 5}!
b) {11, 6, 9}!
!
c) {2, 8, 10}
d)
{.5, 12, 12}!
e) {7, 7, 7}!
!
f) {13, 30, 13}
g)
{6¼, 4½, 11}!
h) {1, 1, 3}!
!
i) {15, 8, 17}
As far as angles in a triangle go, recall that the exterior angle of
a triangle is equal to the sum of the non-adjacent interior
angles. As a result, that exterior angle must be greater than
either non-adjacent interior angle. (Think about it for a
second… if you have to add two interior angles to get the
exterior, then that exterior MUST be greater than either of the
two angles you added.)
50
Section 6
Lengths of Line Segments
within Triangles
3.6 Lengths of Line Segments within Triangles
Some curious things happen when we draw lines within
triangles.
The first happens when we connect midpoints. Here’s the rule:
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
!
In the diagram above, D is the midpoint of AB and E is the
midpoint of AC. As a result, DE is parallel to BC, and
half its length.
51
The second rule is this: The median to the hypotenuse of a
right triangle is half the length of the hypotenuse.
You try:
For # 1 & 2 use this information:
!
!
!
In triangle RST, QP joins the midpoints of sides RS and TS,
respectively.
1. Find the length of QP if RT is:
In the diagram above, BD is the median to hypotenuse AC.
BD is half the length of AC.
a) 14!
!
!
b)
d)
!
!
e) 31½ !
6x!
17!!
!
!
c)
26.5
!
!
f) 8¾
!
!
c)
10.36
!
!
f)
(x+3)
2. Find the length of RT if QP is:
a) 9!
!
!
b)
13!
d) 6¾ ! !
!
e) 29½ !
52
Section 7
Angle - Side Relationship
in a Triangle
3.7 Angle - Side Relationship in a
Triangle
Here are more interesting facts about the triangle:
1. The longest side of a triangle is opposite the triangle’s
largest angle. Likewise, the largest angle will be
opposite the triangle’s longest side.
This is clearly demonstrated in the Right Triangle:
! The hypotenuse of the right
triangle is the longest side of this
triangle & it is opposite the
90° angle, the largest angle.
53
2. The shortest side of a triangle is opposite the triangle’s
smallest angle. Likewise, the smallest angle will be
opposite the triangle’s shortest side.
You try:
1. In ABC,
A = 50° &
shortest sides of
B = 60°. Name the longest and
ABC.
2. In ABC, AB = 11, BC = 10, and AC = 15. Name the largest
and smallest angles of
ABC.
54
Section 8
The Isosceles and
Equilateral Triangles
3.8 The Isosceles and Equilateral
Triangles
The Isosceles and Equilateral triangles have some very useful
properties.
Let’s go back to that basic rule: a triangle always has as many
congruent angles as it has congruent sides. As a result, we
have the rule:
Base angles of an isosceles triangle are congruent. As you’ll
recall from section 3.1, the base angles are the angles touching
the base. (The other angle is referred to as the vertex angle.)
55
The Converse (that’s the reverse) of that rule is true as well:
Another special property of the isosceles triangle is that the
If two angles of a triangle are congruent, the sides opposite
altitude is also the median is also the bisector of the vertex
them are as well.
angle; all three segments fall in the same place.
(Note: very often, the converse of a true statement is NOT true;
So, for example, in the diagram below, if we know that AD is an
altitude, we know the following:
this is one of the rare occasions when both are true.)
-
AD is perpendicular to BC. ( ADB and
ADC are right
angles.)
-!
AD bisects BC (so BD ≅ DC)
-!
AD bisects A (so
BAD ≅
CAD)
!
56
The equilateral triangle also has some unique properties. Again
going back to that basic rule, an equilateral triangle has 3
congruent angles. In other words, every equilateral triangle is
also equiangular.
(That means exactly what it sounds like: equal angles.)
And since we know that a triangle has 180 degrees, we know
that each angle of an equilateral triangle measures 60 degrees.
Also, because an equilateral triangle is, by definition, also
isosceles, all the properties we just discussed for the isosceles
triangle also apply to the equilateral.
57
Section 9
The Pythagorean
Theorem
3.9 The Pythagorean Theorem
Pythagoras was a Greek philosopher and mathematician who
lived in the 6th century BC. Among his many contributions to
both fields is the theorem that bears his name:
the Pythagorean Theorem.
In a right triangle, the sum of the squares of the lengths of the
legs is equal to the square of the length of the hypotenuse.
Or, to put it simply:
In a right triangle,
, where a and b are the legs
and c is the hypotenuse.
That last bit is very important! The a and b values are
interchangeable, but c MUST be the hypotenuse.
(Remember that the hypotenuse of a right triangle is the side
across from the right angle.)
58
The Pythagorean Theorem can ONLY be used in a right
For example, the world’s most common Pythagorean Triple is
triangle—no other triangles have a hypotenuse. And it can only
the 3-4-5 triple. Do the math, and you’ll see that
be used to find the SIDES of a right triangle, never the angles.
So if you’re given a right triangle with legs of 3 and 4, you can
simply state that the hypotenuse is 5, because it’s a 3-4-5 triple.
Much of the time, in using Pythagorean Theorem, your answer
(Note, it’s not any 3 consecutive integers that will work; it’s
will be an irrational number—the square root of a number that’s
these particular three.)
not a perfect square. In such cases, please remember to leave
your answer in simplest radical form.
If you take a triple, and multiply each side by the same amount,
you get another triple. So, for example, if you take that same
3-4-5 triple, and multiply each side by 2, you get a 6-8-10 triple.
Pythagorean Triples—three whole numbers that work in the
Check and you’ll see that it, too, works for the Pythagorean
Pythagorean Theorem.
Theorem.
The most popular triples are, in order:
Pythagorean Triples are common right triangles. If you have
!
two of the three numbers in a triple, and they’re in the correct
3-4-5
! 5-12-13
positions, you can know the third number without doing the
!
8-5-17
math.
!
7-24-25
59
But they’re not, by any stretch of the imagination, the only
You try:
triples that exist. Here’s a list of a few more:
http://www.tsm-resources.com/alists/trip.html
1. Find the hypotenuse of a right triangle, if the legs are:
a) 9 & 12
b)
2&3
c) 5 & 6
d) 1.5 & 2
For a great take on the Pythagorean Theorem and what it
DOESN’T say, take a look at what happened when the
Scarecrow from the Wizard of Oz was granted a brain:
2. Find the other leg when the hypotenuse and one leg is
!
http://www.teachertube.com/video/wizard-of-oz-and-the-pythag
orean-theorem-145155
given:
a)
26 & 10
b) 8 & 4
c) 17 & 3
d) 50 & 30
3. In an isosceles right triangle, what are the measures of the
legs if the hypotenuse is 10?
60
Section 10
Special Right
Triangles
3.10 Special Right Triangles
The first is derived from the Equilateral Triangle.
It is the 30-60-90 degree triangle.
Here are some very special rules:
-!
The shorter leg is half the hypotenuse.
-!
The longer leg equals the shorter leg times √3.
(Remember, of course, that the short leg is opposite the
30 degree angle, the long leg is opposite the 60, and the
hypotenuse is opposite the right angle.)
61
Sometimes you will see the rule shown this way:
The second special triangle is the 45-45-90 degree triangle, or
the Isosceles Right Triangle. In that triangle, the following rules
apply:
-!
The legs are congruent.
-!
The hypotenuse equals the leg times √2.
Examples:
***We will also investigate what happens when the side
opposite the 60 degree angle is whole number.***
62
When the hypotenuse of the Isosceles Right Triangle is a
You try: (remember to draw pictures for each when solving)
whole number then this rule applies:
30-60-90 triangle
-!
1. Find the remaining two sides when the hypotenuse is 12.
A leg equals the hypotenuse times √2/2.
2. Find the remaining two sides when the side opposite the
30 degree angle is 7.
3. Find the remaining two sides when the side opposite the
60 degree angle is
.
***However, if you forget these rules for the Isosceles Right
Triangle, you can always use the Pythagorean Theorem to find
4. Find the remaining two sides when the hypotenuse
the lengths of the legs or hypotenuse.
is
.*
63
You try: (remember to draw pictures for each when solving)
45-45-90 triangle
1. Find the remaining two sides when the hypotenuse
is
.
2. Find the remaining two sides when one leg is 6.
3. Find the remaining two sides when the hypotenuse is 14.
4. Find the remaining two sides when one leg is
.
64
Parallel Lines
4
Click here for
Chapter 4
Student Notes
Section 1
Properties of
Parallel Lines
4.1 Properties of Parallel Lines
Parallel lines are two or more straight lines that do NOT
intersect. You’re familiar with the old example of railroad tracks
as being parallel; if they weren’t, the wheels of the train
wouldn’t be able to stay on the tracks.
The symbol for parallel is ||. (That’s convenient both because it
shows you what it’s symbolizing, and because it’s contained
within the word parallel. )
For example, we might write p || q to describe the lines below:
*Mark the above diagram to show lines p & q are ||
66
A Transversal is any line that intersects (cuts) 2 lines at
-
2 different points.
opposite sides of the transversal.
-
Alternate Interior Angles: The pairs of interior angles on
Alternate Exterior Angles: The pairs of exterior angles
on opposite sides of the transversal.
Line c is the transversal.
same side of the transversal. They are supplementary.
-
When a transversal intersects any 2 lines, it creates 8 types of
Same Side Interior Angles: The interior angles on the
Same Side Exterior Angles: The exterior angles on the
same side of the transversal. They are supplementary.
angles (not including the linear pairs that exist).
We will discuss these types of angles when a transversal
intersects any 2 || lines.
-!
Interior Angles: The angles between the || lines
-!
Exterior Angles: The angles NOT between the || lines
-!
Vertical Angles: Angles defined earlier in Chapter 1
-
Corresponding Angles: The pairs of angles that are
In the diagram below, we can identify the following angles.
a || b with transversal c
“in the same matching position”
67
Interior Angles:
Exterior Angles:
Vertical Angles:
Corresponding Angles:
Alternate Interior Angles:
Alternate Exterior Angles:
Same Side Interior Angles:
Same Side Exterior Angles:
68
Section 2
Exercises with
Parallel Lines
4.2 Exercises with Parallel Lines
You Try: (not all diagrams are drawn to scale)
1. Given the diagram m || n with transversal k
If m
1 = 127°, fill in the remaining angles.
69
2. Given the diagram, e || f with transversal g.
If m
7 = 32°24’, fill in the remaining angles.
4. What happens when there are 2 transversals?
Given: m || n with transversals p & q,
m
1 = 54° and m
13 = 126°.
Find all of the missing angles in the diagram.
3. Given the diagram, p || q with transversal r.
If m
6 = 105°41’, fill in the remaining angles.
70
Applying Algebra to Parallel Lines
You try:
For #1 & #2 AB // CD with transversal EF
If the m
1 = 3x + 30 and the m
measure of
1) m
EGA = 2x and m
GHC = 5x – 54.
Find:
a) x = ___________
8 = x + 60, find the
3.
b) m
EGA =________
c) m
EGB =________
The steps:
1.!
What type of angles?
! !
2.!
Relationship of angles!
! !
3.!
Set up Equation!
4.!
Solve equation!
5.!
Substitute!!
!
!
!
!
!
!
!
!
!
!
71
2) m
Find:
AGH = 3x - 40 and m
a) x = ____________
CHG = x + 20.
3. Given: g || h with transversals w & z, r
g,
m
7 = (5x-35)°, m
16 = (x+14)° and
m
17= (4x+1)°. Solve for x & then find all of the missing
angles in the diagram.
b) m
AGH = ________
c) m
CHG = _______
d) m
BGH = _______
72
You try: (how well do you know the angles?)
Use the following diagram: a // b cut by transversal c.
Write the types of angles in the spaces provided.
a. 2 and 7 are called __________________________ angles.
b. 1 and 5 are called __________________________ angles.
c. 4 and 1 are called __________________________ angles.
d. 3 and 5 are called __________________________ angles.
e. 2 and 8 are called __________________________ angles.
f. 4 and 5 are called ___________________________ angles.
g. 6 and 8 is an example of a(n) _______________________ .
!!
!
73
Polygons
5
Click here for
Chapter 5
Student Notes
Section 1
Polygons
5.1 Polygons
A POLYGON is a closed figure which is the union of line
segments. Polygons have sides and corners. Those corners
are called VERTICES. (A single one is called a VERTEX.)
We can classify polygons according to the number of sides they
have:
75
When we’re labeling a polygon, we choose one vertex as a
starting place. Then we go from one vertex to the next, either in
clockwise or counter-clockwise order—it doesn’t matter which,
but it is important that the vertices be labeled in order.
CONSECUTIVE VERTICES of a polygon are vertices that
share a side. So, for example, in the diagram below of
quadrilateral ABCD, A & B, B & C, C & D and D & A are all
consecutive vertices.
CONSECUTIVE SIDES are, predictably enough, sides that
share a common vertex. Using the same quadrilateral,
AB & BC, BC & CD, CD & DA and DA &AB are all consecutive
sides.
76
Section 2
The Interior and
Exterior Angle of
Polygons
5.2 The Interior and Exterior Angles of
Polygons
We will explore the angles of various polygons and develop the
formulas needed.
Summary of Formulas:
77
Section 3
The Regular
Polygon Chart
5.3 The Regular Polygon Chart
78
Coordinate
Geometry
6
Click here for
Chapter 6
Student Notes
Section 1
Plotting Points & the
Coordinate Plane
6.1 Plotting Points & the Coordinate
Plane
If you’ve ever used MapQuest or Google Earth, you know that
every location has its own unique address. Those addresses
are actually based on the idea of graphing points in the
coordinate plane.
The Coordinate Plane, also known as the Cartesian or
Rectangular Plane, is made up of two perpendicular lines called
axes. The x-axis is a horizontal number line and the y-axis is a
vertical number line. Every point on the plane can be located by
its coordinates: the ordered pair made up of its x coordinate or
“abscissa” followed by its y coordinate or “ordinate”. A point is
written as (x, y).
80
The x and y axes divide the plane into 4 Quadrants (or
quarters) numbered as shown:
!
!
Quadrants are numbered counter clockwise. Each quadrant has specific x
and y values.
When plotting a point, the abscissa tells you to move left or
right along the x-axis and the ordinate tells you to move up or
down along the y-axis.
81
You try:
1. If a point is on the x-axis, what is the value of its ordinate?
2. What is the value of the abscissa of every point which is
on the y-axis?
3. What are the coordinates of the origin?
4. Tell the sign of the abscissa and the ordinate of a
coordinate point if the point lies in quadrant:
a) I!!
!
b) II!
!
c) III!
!
d) IV
5. Tell which quadrant or axis the following points lie:
!
a) (-9, -10)!
!
c) (0, -15)
e) (14, 0)!
g) (16, 24)
!
!
b) (3, -11)
d) (20, -18)
!
f)
(-17, 13)
h) (-14, -29)
82
Section 2
Areas in Coordinate
Geometry
6.2 Areas in Coordinate Geometry
Once we plot a polygon on the coordinate plane, it’s a fairly
simple matter to find its area.
If the polygon has horizontal and vertical line segments that
represent sides and/or altitudes, it’s really just a matter of
counting boxes and using basic formulas.
But if the polygon has slanted sides, the process is just a little
more extensive. Here is an example and the steps
to do so:
83
You try:
1. Plot the points and then find the area of each triangle.
a) A(4,-3) B(1, 3) C(-2, -1)! !
b) D(-3, 3) E(4, 5) F(2, -4)
c) T(3, -3) R(0, 5) I(-4, 1)! !
!
d) S(-4, -3) K(1, 5) Y(6,2)
e) P(-5, 4) Q(5, 6) R(-2, -5)!!
f) K(-3, -2) L(1, 8) M(3, -4)
2. Plot the points and then find the area of each quadrilateral.
a) A(6, 4) B(-3, 2) C(-2, -3) D(9, 0)!
b) Q(0, 3) R(6, 1) S(2, -3)
T(-3, 1)
c) E(-3, 6) F(6, 2) G(-2, -6) H(-7, -1)
d) J(-5, -2) K(-3, 5) L(2, 3)
M(-1, -1)
e) P(1, 2) Q(-2, 7) R(6, 10) S(9, -1)
f) W(-4,3) X(1,7) Y(6,6) Z(10,4)
g) G(-6, 7) H(2, 3) I(4, -4) J(-4, -3)
84
3. Plot the points and find the area of the pentagon if the
coordinates are A(0, 7) B(2, 8) C(6, 4) D(0, 0) and E(2, 3).!
85
Section 3
Distance Between
Two Points
6.3 Distance between Two Points
When we want to find the LENGTH of a line segment, it’s
another way of saying we want to find the DISTANCE between
the two endpoints.
Points A (2, 5) and B (2, 1) form a vertical line segment. Since
the abscissas (x-values) are the same, the length of line
segment AB can be found by taking the absolute value of the
86
difference of the ordinates (y-values).
Of course, not all line segments are horizontal or vertical.
Let’s look at points E (3, 4) and F (-2,-3).
Points C (-3, -3) and D (5,-3) form a horizontal line segment.
Since the ordinates (y-values) are the same, the length of line
segment CD can be found by taking the absolute value of the
difference of the abscissas (x-values).
What is the length of line segment EF?
87
One way to find the length of line segment EF is to create
However, there is another method to find the Distance between
a right triangle and use the Pythagorean Theorem, since
any 2 points or the Length of any line segment.
EF will become the hypotenuse of the right triangle.
Find the horizontal and vertical line segments’ lengths and then
substitute the values into
88
You try:
1. Find the distance between points G (-2, 5) and H (4, -3)
2. Points R (4, 4) and S (-2, 3) form line segment RS. Find its
length as a radical answer and as a decimal answer to the
nearest tenth.
3. Find the distance between points (m, 0) and (0, p).
89
Section 4
Midpoint of a Line
Segment
6.4 Midpoint of a Line Segment
The midpoint of a line segment is exactly what it sounds like:
the point in the middle of the two endpoints.
To easily remember the formula: Take the average of the
x-values and then take the average of the y-values.
90
Find the midpoint of the line of the line segment which joins the
point R (2, -5) & the point S (4, 1)
The midpoint is (3, -2)
*Always remember that you want to express your final answer
as a coordinate point, an ordered pair
You try:
1. Find the midpoint of the following sets of point:
a) !(3, 8) & (5, 6)! !
!
b) (-2, 7) & (-8, -10)
Now use the distance formula to show that (3,-2) is indeed
c)! (16, -9) & (0, 5)!
!
!
d) (-17, 1) & (-5, -6)
the midpoint of RS. Label (3,-2) as point M and show that
e) !(3, 12) & (-3, 10)!
!
!
f) (k, 0) & (0, m)
RM is congruent to SM.
2. In a circle, the diameter’s endpoints are (4, 3) and (-2, -9).
Find the center of the circle.
91
3. Line segment RS has endpoint R (-14, 6) and midpoint M
(-3, 2). Find the coordinates of endpoint S.
4. The midpoint of segment QR is (-5, 2½). Find endpoint Q if
endpoint R has coordinates (-20, 7).
92
Section 5
Slope of a Line
6.5 Slope of a Line
The slope of a line tells you something about the direction in
which the line slants. (Slope indicates Direction.)
The formula for slope is:
In other words, it’s the change in the y values (vertical
movement) divided by the change in the x values (horizontal
movement).
It’s important to note that the order in which you use the points
can’t change; if you use Point A first on top, then Point A must
also be used first on the bottom.
93
Find the slope of line AB if points A (-2, 5) and B (4, 5)
Find the slope of line CD if points C (2, 4) and D (2, -3)
are on the line.
are on the line.
Vertical lines will always have slopes that are undefined.
Horizontal lines will always have slopes of ZERO.
They are said to have NO SLOPE.
94
Find the slope of line EF if points E (-2, -2) and F (-5, -4)
Find the slope of line GH if points G (5, -2) and H (0, 5)
are on the line.
are on the line.
This is a line that has a POSITIVE SLOPE.
This is a line that has a NEGATIVE SLOPE.
It is a line that “leans to the right”.
It is a line that “leans to the left”.
95
You try:
1. Determine the slope of a line formed by each set of points:
a) (4, 7) & (8, 3) !
!
b) (-3, 6) & (-3, -2)
c) (-5, 4) & (2, 1) !
!
d) (2, 3) & (0, -3)
e) (-6, -2) & (2, 2)
f) (b, a) & (0, a)
2. Find the missing value of y so that the line passing through
the points (5, 3) and (-5, y) has a slope of 1/2.
3. Find the missing value of x so that the line passing through
the points (4, 1) and (x, 3) has a slope of -2/3.
4. Determine if the following points are collinear.
a) (-6, 8) (0, 5) (4, 4)
b) (-1, -8) (1, -2) (4, 8)
96
Section 6
Parallel and
Perpendicualar Lines
6.6 Parallel and Perpendicular Lines
As we’ve been discussing, slope tells you something about the
direction of a line. So it stands to reason that if two lines go in
the same direction—or are parallel—they would have the same
slope.
1. Plot line AB: A (3, 5) and B (-1, 2) and then find its slope.
2. Plot line CD: C (2, -2) and D (-2, -5) and then find its slope.
97
Parallel lines have equal slopes.
Perpendicular lines have slopes that are negative
reciprocals.
However, perpendicular lines intersect to form right angles.
Their slopes have a special relationship than just any pair of
intersecting lines.
You try:
1. Plot line EF: E (-3, 4) and F (1, -1) and then find its slope.
2. Plot line GH: G (2, 3) and H (-3, -1) and then find its slope.
1. Determine if line EF is parallel or perpendicular to line GH?
a) E (1, 6) F (5, 4) & G (-1, 2) H (3, 0)
b) E (2, 2) F (-1, 5) & G (-6, -8) H (0, -2)
c) E (4, -1) F(4, 7) & G (-3, 1) H (7, 1)
2. Plot triangle ABC with vertices A (6, 8), B (10, -2) and
C (-4, 4). Show that if you join the midpoints of AC and AB, that
the line segment formed is both parallel to CB and is half its
length.
98
Section 7
Coordinate Proofs
6.7 Coordinate Proofs
Now that you’re comfortable with slope, distance and midpoint,
we can use them to prove figures in coordinate geometry.
!
Remember: Slope means Direction and
! !
Distance means Length.
Here’s how a proof works:
First, you plot the coordinates and label the lines or vertices.
Second, you determine which formula(s) you’ll need, depending
on what you’re asked to prove. Write the formula(s) out, at least
the first time you use it.
99
Third, use the formula(s) as many times as necessary to prove
what you’re asked.
Last, write a sentence or two, starting with the word “Since…”
to explain how the work you’ve done proves what you’ve been
asked to prove.
Triangles will be the first type of polygon to be used with
coordinate proofs. (section 6.8)
Quadrilaterals will be used after first learning about their
properties. (section 7.9)
100
Section 8
Proving Triangles Using
Coordinate Geometry
6.8 Proving Triangles Using Coordinate
Geometry
To Prove a Triangle is Isosceles:
-!
Show that 2 sides are congruent (distance formula)
To Prove a Triangle is a Right Triangle:
- Show that there is 1 right angle sides by showing that 2
consecutive sides are perpendicular
(slope formula)
- Show that the Pythagorean Theorem works after finding the
length of each side
(distance formula &
101
To Prove a Triangle is a Right Isosceles Triangle
-
Show that 2 sides are congruent & that there is 1 right
angle by showing that 2 consecutive sides are perpendicular
(distance & slope formulas)
-
Show that 2 sides are congruent & then use the
Pythagorean Theorem to show it is a right triangle
(distance formula &
102
Section 9
Equation of a Line
6.9 Equation of a Line
Any straight line can be expressed in the form of an equation.
That equation is typically written in standard form:
y = mx + b
Where
m = slope
b = the y-intercept
where the line intersects the y-axis at the point (0,b)
x & y = the coordinates of any point that is on the line
Example: Write in standard form: 3x – 2y + 8 = 0
Find the slope and y-intercept.
103
Section 10
Writing the
Equation of a Line
6.10 Writing the Equation of a Line
Just as every equation can be graphed, we can also write the
equation of any line we graph.
We will be writing the linear equation in standard
(slope-intercept) form.
Here are the steps:
First determine the slope if it is not given. If necessary
use the formula for slope:
1. Write the standard form: y = mx + b
2. Substitute that value of “m” in y = mx + b
3. Substitute the “x” and the “y”, the values of the coordinates
of a point on the line, in y = mx + b. Be careful to put them into
104
the right places.
Ex3: Write a linear equation that passes through the points
4. Solve for b.
(6, 4) and (8, 5).
5. Rewrite y = mx + b, replacing the values of m and b into
the equation: y = mx + b
Ex1: Write a linear equation with a slope of -3 and y-intercept
of 12.
m = -3 b = 12
y = mx + b
y = -3x + 12 substitute in m & b
Ex2: Write a linear equation with a slope of 3 and passes
through the point (2, -7).
105
Equation of a Horizontal Line:
Since a horizontal line has a slope of Zero then
y = mx + b
y = 0x + b
Equation of a Vertical Line:
Since a vertical line has an undefined slope or No Slope then
* # is the number on the x-axis where the
vertical line intersects the x-axis.
106
Section 11
Linear Equation
Practice Problems
6.11 Linear Equation Practice Problems
1. Write each equation in standard form. Find the slope and
y-intercept of each.
a) 14 + 2y = -8! !
b) x – 3 + y = 2!!
c) 12 x – 3y = 9
d) ½x + y = 2!
e) 4y + 2x – 6 = 0! f) -2x + y + 7 = 0
2. Write a linear equation in standard form given the slope and
y-intercept:
a) m = 1/3
b = -6!
!
b) m = 2 b = 4/5!
c) m = -1
b = -9!
!
d) m = -1/2 b = -8!
e) m = 6 b = -3!!
f) m = 2/3 b = 5
107
3. Write a linear equation in standard form given the slope and
a point:
a) m = 3 & (2, -7)!!
!
b) m = 5 & (-1, -3)
c) m = ½ & (8, -3)!
!
d) m = 2 & (1, -4) ! !
e) m = 2/3 & (-6, -5)
g) m = -3 & (4, -2) !
!
!
6. Write the equation of line that has a slope of -1/4 and
passes through the point (0, -3).
7. Write the equation of a horizontal line that passes through
f) m = 4/5 & (10, 1)
the point (8, -3).
h) m = -1/3 & (3, -9)
i) m = -1 & (-11, -3)
8. Write the equation of a vertical line that passes through the
point (-5, 9).
4. Write a linear equation given two points:
a) (2, -3) (1, -1)
b) (3, -6) (6, -8)
c) (0, -3) (-6, 0)! !
d) (3, 5) (8, 5)
e) (-4, 7) (-4, 9)
f) (5, -2) (7, -8)
9. Given line: y = 3x – 5
Write the equation of a line that is parallel to this line and
has a y–intercept of 7.
New line’s equation: ___________________
5. Which point(s) satisfy the given equation?
a) y = 3x + 1
(4, 1)
(3, 10)
(-2, -5)
(0,1)
b) y – x = -3
(8,5)
(-2, -5)
(-3, 0)
(0, -3)
108
10. Given line: y - 2x = 4
13. Given line: y = ½ x + 6
Write the equation of a line that is parallel to this line and
Write the equation of a line that is perpendicular to this line
has a y-intercept of -1.
and has a y–intercept of 1.
New line’s equation: ___________________
New line’s equation: ___________________
11. Given line: 2x + 3y = 6
14. Given line: y - 3x = 4
Write the equation of a line that is parallel to this line and
Write the equation of a line that is perpendicular to this line
has a y–intercept of 5.
and has a y–intercept of -2.
New line’s equation: ___________________
New line’s equation: ___________________
12. Given line: y = -2x + 5
15. Given line: 2y = -3x + 1
Write the equation of a line that is parallel to this line and
Write the equation of a line that is perpendicular to this line
has the same y–intercept as y = 3x – 9.
and has the same y-intercept as y = 2x + 5.
New line’s equation: ___________________
New line’s equation: ___________________
109
16. Given line: 2y - 6x – 5 = 0
Write the equation of a line that is perpendicular to this line
and has the same y–intercept as 2x + y + 4 = 0.
New line’s equation: ___________________
17. Determine if each pair of lines are parallel or perpendicular
to each other:
a) y – 3x = 5
and 2y = 6x – 7
b) 2x – 4y = 1 and y = -2x + 3
c) -3x = 2y
and 4y + 6x – 9 = 0
d) -5x – y = -7 and 10y = 2x + 5
110
Section 12
Graphing a Line Using the
Slope-Intercept Form
6.12 Graphing a Line using the
Slope-Intercept Form
To graph using the slope-intercept form or standard form:
y = mx + b.
- Start with the y-intercept “b”. Plot the point (0,b).
Remember this is the starting point that is always on
the y-axis.
- Now use the slope “m”.
If “m” is positive and the line will lean to the right.
If “m” is negative and the line will lean to the left.
111
At the starting point, use the numbers in the fraction
to move up and down (vertical movement/numerator)
and left and right (horizontal movement/denominator)
in the direction that the line leans.
Extend the line at both ends.
- Label the line.
112
Section 13
Graphing a Line Using
X and Y Intercepts
6.13 Graphing a Line using X and Y
intercepts
To find the x – intercept (where the line crosses the x-axis) set y
= 0 and solve.
To find the y – intercept (where the line crosses the y-axis) set x
= 0 and solve.
The x- and y- intercepts are almost always 2 separate points.
(Except when??)
To graph using the intercepts:
- Find the x-intercept. Plot it on the x-axis.
- Find the y-intercept. Plot it on the y-axis.
- Connect those two intercepts, using a straight edge.
- Label the graph with its equation
113
Section 14
Graphing a Line Using a
Table of Values
6.14 Graphing a Line Using a Table of
Values
Sometimes it’s more convenient to graph a line (or, as we’ll
soon see, a curve) using a table of values.
Steps:
-!
Place equation in standard form: y = mx + b
-!
Make a table of values . Most of the time it’s convenient to
use 0, 1, and 2 as your x values.
-
Plug each x into the equation and determine the
corresponding y value
-!
Plot the points represented in the table
-!
Draw a line through the points
-!
Label with the original equation
114
Section 15
Graphing a System
of Linear Equations
6.15 Graphing a System of Linear
Equations
As you’ll remember from Algebra last year, a system of
equations can be solved algebraically—you can use either the
Substitution or Elimination methods to find the solution. You
may also recall a third method: graphing the two lines on the
same graph and finding their point of intersection.
You can graph the two lines using whichever method you
prefer; you can even use two different methods in the same
problem. But accuracy is key; be sure to use a straightedge.
115
Section 16
Graphing a
Parabola
6.16 Graphing a Parabola
Up until now, all our graphs have been of LINEAR equations;
equations of the form y = mx + b whose graph is a straight line.
Now we’re going to see what happens when we graph a
QUADRATIC equation of the form
y = ax² + bx + c
The result will be a curved line called a PARABOLA.
In order to graph a parabola, first find the values of a, b, and c.
116
Now find the equation of the Axis of Symmetry: The line in
Now plot the coordinate points from the table to form the
which the parabola is reflected.
parabola on the set of axes.
The formula is:
!
!
Please remember to label your parabola with the equation!
Use the axis of symmetry to set up your table. It gives you the
You may even be asked to label the turning point, the axis of
x value of your turning point. Once you have that x value go at
symmetry and the roots of the quadratic equation.
least 2 units higher and at least 2 units lower than it to set up
your table. (*Leave spaces in your table in case you need to
extend your parabola.)
117
Section 17
Graphing a
Quadratic-Linear System
of Equations
6.17 Graphing a Quadratic-Linear
System of Equations
Just as you can solve a system of linear equations graphically,
you can do the same with a Quadratic-Linear system. Now,
instead of two lines intersecting, you’ll have a line intersecting a
parabola.
To solve a Quadratic-Linear system, it’s probably best to first
graph the parabola, using the steps you already know. Once it
has been graphed, use any method you choose to graph the
line. Label any points of intersection.
Sometimes there will be just one point of intersection;
sometimes there will be two. There may even be no points of
intersection!
118
Section 18
Equation of a Circle
6.18 Equation of a Circle
The equation of a circle is:
where (h, k) is the center of the circle
and “r” is the length of the circle’s radius.
Ex 1: What is the center and radius of a circle whose equation
is
Answer: Center (0, 0) r = 6
Ex 2: What is the center and radius of a circle whose equation
is
Answer: Center (3, 0) r = 2
119
Ex 3: Write the equation of a circle with a center at (-5, 1)
and with a radius of 10.
Answer:
You try:
1. Write the equation of a circle given the center and the radius.
a) (0, 9) r = 4 ! !
b) (2, -3) r = 1!
c) (-8, -1) r = 11
d) (0, 0) r = 3! !
e) (3, 6) r = 7!
f) ( 1, -5) r = 1.5
Ex 4: Write the equation of a circle with a center at (-7, -2)
and with a radius of
Answer:
2. What is the center and radius of a circle whose equation
is
3. What is the center and radius of a circle whose equation
is
4. What is the center and radius of a circle whose equation
is
5. Write the equation of a circle whose diameter has endpoints
(2, -6) and (-8, 8).
120
Section 19
Graphing a Circle
6.19 Graphing a Circle
- To graph the circle on a set of axes, first plot the center.
- Then plot 4 points “r” units from the center in the
4 directions: “up, down, left, right”
- Connect the 4 points with curved lines to form a circle.
121
Section 20
Coordinate Geometry
Summary of Facts
6.20 Coordinate Geometry Summary of
Facts
122
123
Quadrilaterals
7
Click here for
Chapter 7
Student Notes
Section 1
Quadrilateral
Family Tree
7.1 Pictorial Representation of the
Quadrilateral Family Tree
!
!
A quadrilateral is a 4 sided polygon.
125
Section 2
7.2 The Parallelogram
The Parallelogram
As you know, a quadrilateral is a 4 sided polygon. There are
many different types of quadrilaterals.
A PARALLELOGRAM is defined as a quadrilateral with
the following properties:
1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. Consecutive angles are supplementary.
5. Diagonals bisect each other.
6. One diagonal divides the parallelogram into 2 congruent
triangles.
126
**Note that there are 2 properties on sides, 2 on angles, and 2
To prove a quadrilateral is a parallelogram, use one of the
on diagonals.
following options:
symbol:
- Use slope to prove both pairs of opposite sides are
parallel.
Also, remember that when labeling the vertices of any polygon
- Use distance to prove that both pairs of opposite sides are
— including a parallelogram — choose a vertex for the first
congruent.
letter, then label in either clockwise or counter-clockwise order.
- Use both slope and distance to prove that one pair of
For example, here’s how you might choose to label
opposite sides are both parallel and congruent.
parallelogram ABCD:
- Use midpoint to prove that the diagonals bisect each other.
127
Section 3
The Rectangle
7.3 The Rectangle
A RECTANGLE is a parallelogram.
!
!
!
!
A rectangle has the following properties:
1.!
All of the properties of the parallelogram.
2.!
All angles are right angles. As a result, it is equiangular.
3.!
Diagonals are congruent.
128
To prove that a quadrilateral is a rectangle you must first
prove that it is a parallelogram using one of the methods
already discussed. Then, you must either:
- Use the distance formula to show that both diagonals
are congruent.
OR
- Use the slope formula to show that 2 consecutive sides
are perpendicular, forming a right angle.
**NOTE: Using the slope formula is easiest. Finding the slope
of all 4 sides will show that opposite sides are parallel (same
slope), making it a parallelogram. This will also show that
consecutive sides have slopes that are negative reciprocals,
meaning that those two sides are perpendicular, giving you that
one right angle you need to make it a rectangle.
129
Section 4
The Rhombus
7.4 The Rhombus
A RHOMBUS is another type of parallelogram.
A rhombus has the following properties:
1. All of the properties of a parallelogram.
2. All sides are congruent, therefore it is equilateral.
3. Diagonals are perpendicular.
4. Diagonals bisect the angles.
130
To prove a rhombus, you again must first prove that it is a
parallelogram using one of the previously discussed methods.
Then, you must:
- Use the distance formula to show that 2 consecutive
sides are congruent.
OR
- Use the slope formula to show that the diagonals are
perpendicular.
**NOTE: For a rhombus it is easiest to use the distance
formula. Finding the length (distance) of all 4 sides will
show that opposite sides are congruent, proving that it is a
parallelogram, but will also show that all 4 sides are congruent,
making it equilateral and hence a rhombus.
131
Section 5
The Square
7.5 The Square
A SQUARE is pretty much a hybrid of all the shapes we’ve
been discussing. The definition of a square is either:
-!
A rectangle with 2 congruent consecutive sides
or
-!
A rhombus with a right angle
132
As a result, it has all the properties of a parallelogram, a
Proving that a quadrilateral is a square requires proving that it
rectangle, and a rhombus combined.
has a property of all three figures: parallelogram, rectangle, and
1.!
Opposite sides are parallel
rhombus.
2.!
Opposite sides are congruent.
3.!
Opposite angles are congruent
To prove a square you must first prove that it is a parallelogram,
4.!
Consecutive angles are supplementary
then:
5.!
Diagonals bisect each other
6.!
Diagonals divide the rectangle into 2 congruent
triangles.
7.!
- Show that the diagonals are BOTH congruent and
perpendicular (use distance and slope)
All angles are right angles. As a result, it is
equiangular.
8.!
Diagonals are congruent.
9.!
All sides are congruent. As a result, it is equilateral.
10. Diagonals are perpendicular
OR
- Show that two consecutive sides are BOTH congruent
and perpendicular (use distance and slope)
11. Diagonals bisect the angles.
133
Section 6
The Trapezoid
7.6 The Trapezoid
The first thing to remember about the trapezoid is that
a trapezoid is NOT a parallelogram.
A TRAPEZOID is defined as a quadrilateral with ONLY ONE
pair of opposite parallel sides. Those parallel sides are called
the BASES; the non-parallel sides are called the LEGS.
A trapezoid has only one major property:
!
1. Only one pair of opposite sides is parallel.
134
In order to prove a quadrilateral is a trapezoid, use slope on all
4 sides. One pair of opposite sides will have equal slopes,
showing that they are parallel (these are the bases) and the
other pair of opposite sides will not have equal slopes, showing
that they are non-parallel.
135
Section 7
The Isosceles
Trapezoid
7.7 The Isosceles Trapezoid
A special type of trapezoid is the ISOSCELES TRAPEZOID.
The properties of an isosceles trapezoid are:
1.!
Only one pair of opposite sides is parallel.
2.!
The legs (non-parallel sides) are congruent.
3.!
Each pair of base angles are congruent.
4.!
Diagonals are congruent.
136
To prove that a trapezoid is isosceles, first show that it is a
trapezoid using the slope formula (two sides parallel and two
sides non-parallel). Then:
- Use the distance formula to show that the non-parallel
sides are congruent.
OR
- Use the distance formula to show that the diagonals
are congruent.
137
Section 8
Quadrilateral
Properties
7.8 Review the Properties of the
Quadrilaterals
138
Section 9
Proving
Quadrilaterals
Using
Coordinate
Geometry
7.9 Proving Quadrilaterals Using
Coordinate Geometry
To Prove a Quadrilateral is a Parallelogram, choose 1 of the
following methods:
-!
Show that both pairs of opposite sides are parallel
(slope formula)
-!
Show that both pairs of opposite sides are congruent
(distance formula)
-!
Show that only 1 pair of opposite sides are parallel and
congruent (slope & distance formulas)
-!
Show that the diagonals bisect each other, by showing
they have the same midpoint (midpoint formula)
139
To Prove a Quadrilateral is a Rectangle, *first show it is a
parallelogram & then choose 1 of the following methods:
-!
Show that both diagonals are congruent
(distance formula)
-!
Show that it has 1 right angle, by having 2 consecutive
sides perpendicular (slope formula)
To Prove a Quadrilateral is a Rhombus, *first show it is a
parallelogram & then choose 1 of the following methods:
- Show that 2 consecutive sides are congruent
(distance formula)
To Prove a Quadrilateral is a Square, *first show it is a parallelogram & then choose 1 of the following methods:
-!
Show that the 2 diagonals are congruent & perpendicular
(distance & slope formulas)
- Show that 2 consecutive sides are congruent &
perpendicular (distance & slope formulas)
To Prove a Quadrilateral is a Trapezoid:
- Show 1 pair of opposite sides are parallel and the other pair
not parallel (slope formula)
- Show that the 2 diagonals are perpendicular
(slope formula)
140
To Prove a Quadrilateral is an Isosceles Trapezoid, *first
show it is a trapezoid & choose 1 of the following methods:
-!
Show that the non-parallel sides are congruent
(distance formula)
-!
Show that the 2 diagonals are congruent
(distance formula)
141
Section 10
The Quadrilaterals
and Algebra
7.10 The Quadrilaterals and Algebra
* E is the intersection of diagonals AC & BD
* T is the intersection of diagonals PR & QS
* V is the intersection of diagonals WY & XZ
1. In parallelogram ABCD,
BCD = 2x + 30 and
CDA = 3x – 50, find the measure of each.
2. In parallelogram ABCD, AE = x + 2, EC = y + 4,
DE = 3x – 4, EB = 2y + 4. Find x and y.
3. In parallelogram ABCD, AB = x + 3y, BC = 20,
CD = 2(x + y – 1), DA = 5y. Find x and y.
4.!
Square WXYZ: WX = 1-10x, YZ = 14+3x. Find x.
142
5. In parallelogram PQRS,
P = y and
Q = 4y + 20. Find
11. In parallelogram ABCD, BC = 4x + 7 and DA = 8x – 5, find
the measure of each angle.
the value of x.
6.!
12. In parallelogram ABCD, opposite angles are 9x + 12 and
In parallelogram ABCD, BC = 8y - 6, DA = 3y + 14. Find y.
15x. Find all the angles of the parallelogram.
7.!
In parallelogram ABCD, AC = 5x - 12, AE = 14. Find x.
8. In parallelogram ABCD, BD = 3x + 56, BE = 4x + 8.
13.! In rectangle PQRS, PT = 3x – y, ST = x + y, TQ = 5. Find
PT and ST.
Find BD.
14. In rectangle PQRS, PS = y, QR = x + 7, PQ = y-2x,
9. In parallelogram ABCD,
BCD = 3x + 14,
ADC = x + 10.
SR = x +1. Find the length of the sides.
Find the measure of both angles.
15.! In rhombus PQMN, PN = 7x - 10 and PQ = 5x + 6, find
the value of x.
10. In parallelogram ABCD,
find the measure of each.
C = x + 75 and
D = 3x – 199,
16. In rhombus ABCD, AB = 5x + 24 and BC =
Find the length of each side.
143
17. In parallelogram WXYZ, WY = 4x – 14 and VY = x + 8.
23. In parallelogram ABCD, BE = 4x – 12 and DE = 2x + 8.
Find x.
Find the value of x.
18. In parallelogram ABCD, A = 5x - 10 and C = 3x + 4, find
24.! In rhombus RSTW,
RST = 108. Find
SRT.
the measure of each.
25. In parallelogram MATH,
19. In parallelogram ABCD, consecutive angles are 2x and
T exceeds
H by 50. Find the
angles.
2x – 20. Find all the angles of the parallelogram.
26. In parallelogram TRIG,
20. In rectangle ABCD, DE = 3x + 1, AC = 5x + 4. Find the
R = 2x + 19 and
G = 4x – 17.
Find the angles.
length of the diagonals.
27. In parallelogram RSTW, SA = x – 13, AW = 2x – 37,
21. In rectangle ABCD, AC = 2x + 15 and BD = 4x -5, find the
diagonals RT and SW intersect at A. Find x.
measure of AC.
22. In parallelogram ABCD,
A is 30 more than
B. Find all
the angles in the parallelogram.
144
Ratio and
Proportion
8
Click here for
Chapter 8
Student Notes
Perimeter and Area of Polygons
Circumference and Area of
Circles
9
Click here for
Chapter 9
Student Notes
Section 1
Perimeter of
Polygons
9.1 Perimeter of Polygons
Perimeter is the distance around a polygon. So, for example,
we could use perimeter if we were fencing in a garden or
putting molding into a room. In fact, if you pick up someone at
JFK, you’ll find that Perimeter Road goes around the outside of
the airport.
The perimeter is considered the distance around a
two-dimensional figure, namely a polygon.
The perimeter of any polygon can be found by adding the
lengths of all its sides. In polygons where several sides have
the same length, as in a square or a rectangle, those congruent
sides can be combined. As a result, we can come up with
formulas such as:
147
!
Perimeter of a square = 4s, where s is the length of a side
You try:
!
!
1. Find the perimeter of each of the following:
(since s + s + s + s = 4s)
!
!
or
Perimeter of a rectangle = 2l + 2w,
where l = length and w = width
a) a square with a side of 17 inches
b) a rectangle that has a length of 12.3cm and a width of !
7.8cm
c) a rhombus with a side of 2½ feet
What formula for perimeter could you write for a regular
d) a triangle with sides 7.09, 3.46 and 6.124 millimeters
heptagon or regular decagon?
e) an isosceles trapezoid with bases of 32 and 45 and
congruent legs of 27
f) a right triangle with legs 8 and 15 centimeters
Since perimeter is the sum of the lengths of sides, it is
g) a regular nonagon with sides of 8.23 feet
measured the same way the sides are: in linear units. So the
h) a rectangle with consecutive sides of 17 inches and 2 feet
perimeter of a polygon might be 14 inches or 20 miles.
i) a square with a diagonal equal to 42
j) a rhombus with diagonals of 6 and 8 inches
148
2. Find the perimeter of each of the following in terms of an
4. The perimeter of a regular polygon measures 84 cm. If one
algebraic expression:
of the sides measures 14 cm, what is the name of the polygon?
a) an equilateral triangle with a side of 4x
b) a rectangle that has a length of 2x+7 and a width of x+3
c) a regular pentagon with a side 3x-2
d) a parallelogram with consecutive sides 3x and x+1
e) an isosceles triangle with congruent legs 5x-4 and base
of x+5
3. Find the length of a side of each of the following given its
perimeter. (some answers may be left as an algebraic
expression)
a) an equilateral triangle with a perimeter of 324.75
centimeters
b) a regular hexagon with a perimeter of 24x+66
c) a square with a perimeter of 183½ feet
d) find the width if the perimeter of a rectangle is 44
149
Section 2
Understanding Area of the
Polygon
9.2 Understanding Area of the Polygon
Area is very different from perimeter. While perimeter tells the
distance around a polygon, area tells you the amount of space
within the polygon.
(In fact, don’t laugh, there’s an episode of Cyberchase for Real
that does a great job explaining the relationship between Area
and Perimeter. It’s called the Dumas Diamond and can be
found on the PBS site:
http://pbskids.org/cyberchase/videos/cyberchase-the-dumas-di
amond/ )
Area can be used to determine how much carpet you need for a
floor, or how much paint you need for a wall. Area tells you the
number of square units you need to fill up a particular polygon.
150
For example, the rectangle below can be divided, as shown,
into 2 rows of 3 small squares. Hence, the area is 6 square
units.
Area is ALWAYS measured in square units, square feet, square
inches, square centimeters, etc.
151
Section 3
Area of a Rectangle
9.3 Area of a Rectangle
We’ll start our discussion of area with an old favorite formula:
the Area of a Rectangle. Remember, our objective is to
determine the number of small square units that can fit within a
rectangle.
In the pictured rectangle below, we say the base has a length of
3 units and the height has a length of 2 units
152
While we could choose to simply count the number of squares,
You try:
you can imagine a much larger rectangle where that wouldn’t
1. Find the area of a rectangle that has a length of 12 inches
be practical. Instead, we can go back to our basic arithmetic
and a width of 7 inches.
and realize that the number of squares is 3 x 2, or 6 square
units.
2. Find the area of a rectangle that has a length of 23.8 cm and
a width of 14.65 cm
As a result, we can generalize that
3. A rectangular plot of land has an area of 3130 square feet.
!
Area of a Rectangle = base x height
!
!
!
Area of a Rectangle = bh
!
!
If the width is 40 feet, find the length of the rectangular plot.
or
4. Find the area of throw rug that has the dimensions
3½’ by 5½’.
but most commonly known as
5. The length of a rectangle is 8 more than its width. Represent
!
Area of a Rectangle = length x width
!
!
!
Area of a Rectangle = lw
!
!
the area of the rectangle as an algebraic expression.
or
153
6. Represent the area of a rectangle as an algebraic
expression if the length and width are represented by x+10 and
x-3, respectively.
7. The area of a rectangle is represented by
What are the dimensions of the rectangle if the area is
112 sq. units.
154
Section 4
Area of a Square
9.4 Area of a Square
I’m sure you remember from our study of parallelograms, a
square is a kind of rectangle. As a result, if we want to find the
area of a square, we can use the same formula that we used to
find the area of any rectangle: length times width.
But in a square, the length and the width are the same number.
As a result, we can also say that
!
But wait, we’re not done. Think back to all you know about
special triangles. You know that there is a definite relationship
between the diagonal of a square and the length of its leg.
155
As a result, we can also say that
So to find the area of a square with a diagonal of 4,
it’s
or ½(16) or 8 square units. Remember to follow
PEMDAS and to square “d” before you multiply it by ½.
156
Section 5
Area of a
Parallelogram
9.5 Area of a Parallelogram
Let’s go back to that rectangle we spoke about a few sections
ago. We said that the area of a rectangle was its length times
its width, since that formula would give us the number of square
units within the rectangle.
Let’s take a rectangle, and play with it a bit:
See the diagonal line inserted into the rectangle? It cuts off a
right triangle.
157
If we take that right triangle, and move it to the left of the
Remember, of course, that the base and height must be
rectangle, look at what we get:
perpendicular. That means, unlike in a rectangle, you’ll need to
have an altitude in order to find the area of a typical
parallelogram.
But that’s not the only formula for area of a parallelogram.
The area—the amount inside the shape—hasn’t changed. As a
Another formula is based on the idea that the sin(90°) = 1.
result, we can say that the area of the resulting parallelogram is
That formula is:
the same as the area of the original rectangle:
Area of a Parallelogram = absinC,
!
Area of a Parallelogram = base x height.
!
!
!
Area of a Parallelogram = bh
!
!
or
where a and b are consecutive sides and C is the included
angle between them.
158
To find the area of a parallelogram with sides of 12 and 14, and
the included angle of 24°, we evaluate (12)(14)sin24° and get
168(0.406736643) or approximately 68.33 square units.
If you use the parallelogram’s obtuse angle, what happens to its
area?
159
Section 6
Area of a Triangle
9.6 Area of a Triangle
You may remember from elementary school that the standard
version of the formula is
!
Area of a Triangle = ½base x height
!
!
!
Area of a Triangle = ½bh
!
!
or
(Do you remember why? It’s because you can insert a diagonal
into any parallelogram, forming two congruent triangles. Each
has half the area of the original parallelogram.
160
Remember that there were two formulas for the parallelogram.
where a, b, and c are the sides and s is the semi-perimeter.
Since a triangle is half a parallelogram, we can also cut the
(Semi-perimeter is half the perimeter.)
other area formula in half and get
!
Area of a Triangle = ½absinC,
where a and b are consecutive sides, and C is the included
For example: A triangle has sides of 8, 12, and 16. The
perimeter is 36, and semi-perimeter is 18.
angle between them.
To find the area of a triangle with sides of 6 and 10, connected
by a 30° angle, it’s ½(6)(10)sin30°, or 15 square units.
There is a third formula to find the area of a triangle called
Heron’s Formula. Heron was a Greek mathematician and
philosopher in the first century BC. His formula will allow you to
find the area of any triangle in which you’ve been given the
three sides:
161
Just when you thought you knew all there was to know about
You try:
the area of a triangle, along comes yet another formula. This
1. Find the area of a triangle when the base and height are
formula is the result of using the formula for area of a triangle
given, respectively.
“ ½absinC” and the fact that the “sin60°= √3/2”.
You know that the angles of an equilateral triangle each
measure 60°. Put all that information together, and you get the
a) 28 & 13! !
!
b) 17.8 & 8!
!
c) 15.3 & 9.8
2. Find the altitude of a triangle when the area is 190
and the base is 16 cm.
formula:
3. Two consecutive sides of a triangle are 22 and 18. Find the
area, to the nearest tenth, when the included angle is 40°.
4. Two consecutive sides of a triangle are 12 and 14. Find the
measure of the included angle, to the nearest degree, if
So to find the area of an equilateral triangle with sides of 6,
the area is 67
we substitute in:
162
5. Find the area of a triangle if the sides are 8, 11 and 13.
a) Simplify the radical answer.
b) Round the decimal answer to the nearest thousandths
place.
6. a) In simplest radical terms, find the area of an equilateral
triangle with a side of 14 inches.
b) Convert your radical answer in part a to a decimal
answer rounded to the nearest hundredths place.
7. Find the length of a side of an equilateral triangle when its
area is
163
Section 7
Area of a Trapezoid
9.7 Area of a Trapezoid
As I’m sure you remember from the work we did on
quadrilaterals, a trapezoid is a quadrilateral with one pair of
parallel sides, called the bases.
As much as we would love to just say that its area is the base
times the height, that’s not possible, since the trapezoid has
one long base and one short base. So, instead, we average the
bases and multiply that average by the height
!
If our trapezoid had bases of 12 and 18, with an altitude of 10,
its area would be ½(10)(12+18) or 5(30) or 150
164
Remember that the “altitude” or “height” is a line perpendicular
You try:
to the bases. In some, but not all, cases it may be one of the
1. The bases of a trapezoid are 16 and 28. Find the area when
the altitude is
legs.
a) 12!!
b) 7!
c)
9.3! !
d) 6½ ! !
e) 11¾
There’s another formula for area of a trapezoid. Remember that
we defined the median of a trapezoid as a line parallel to the
bases, midway between them. The length of the median is the
2. Find the area of a trapezoid when the height and median are
a) 3 & 9!
b) 12.6 & 3.5
c) 7.03 & 15
average of the lengths of the bases. As a result, we can also
3. The area of a trapezoid is 156.24
when
say
!
Area of a Trapezoid = (height)(median)
!
!
!
Area of a Trapezoid = hm
!
!
Find its altitude
the median is 16.8 inches.
or
4. The area of a trapezoid 360
Find the length of the
shorter base when the trapezoid’s height is 16 cm and the
longer base is 25 cm.
165
Section 8
Area of a Rhombus
9.8 Area of a Rhombus
A rhombus, as you’ll recall, is a parallelogram with congruent
sides. As a result, we can use any of the parallelogram
formulas to find the area of a rhombus.
But the fact that a rhombus has perpendicular diagonals gives
us another formula:
A rhombus with diagonals of 20 and 26 would have an area of
½(20)(26) or 260
166
You try:
1. Find the area of a rhombus given the following diagonal
lengths.
a) 13 & 15! !
b) 11.2 & 16.4!
!
c) 26 & 28!
2. Find the length of the longer diagonal if the shorter diagonal
is 10 feet and the area of the rhombus is 70 square feet.
3. The length of a side of a rhombus is 17 cm. One diagonal
is 16 cm. Find the area of the rhombus.
4. The length of one side of a rhombus is 32 mm. If an angle
of the rhombus is 24°, what is the area to the nearest
hundredth millimeter?
167
Section 9
Area Formulas
Reference Guide
9.9 Area Formulas Reference Guide
168
169
Section 10
Area Practice
Problems
9.10 Area Practice Problems
170
171
Section 11
Circumference of a
Circle
9.11 Circumference of a Circle
As you know, we can find the perimeter of a polygon by adding
the lengths of each of its sides. But when it comes to circles,
that method gets tricky; we can’t take a ruler and measure the
length of a curved shape.
Instead of talking about perimeter, we find the Circumference of
a circle (“Circum” means “around” in Latin).
The formula for circumference is
!
C = 2πr,
where r = the length of a radius.
172
Or, since one diameter is equal to two radii, we can say
!
C = πd,
where d = the length of the diameter.
173
Section 12
Area of a Circle
9.12 Area of a Circle
The formula for Area of a Circle is another one you’ve probably
known since elementary school.
where r = the length of the radius.
There are a lot of people who confuse the formulas for Area
and Circumference, and a variety of mnemonic devices to help
you remember which is which. Probably the easiest way to
remember that Area is always measured in SQUARE units, and
it’s the one that includes
“r squared.”
174
In the Circle Reference Guide (9.14) you will also see two
lesser known area formulas for the circle.
175
Section 13
Circle Reference
Guide
9.13 Circle Reference Guide
The diameter of the circle is double its radius.
d = 2r
The radius of a circle is half its diameter.
r = ½d
Circumference is the distance around a circle.
176
Area is the space inside the circle.
177
Section 14
Area Ratios
9.14 Area Ratios
When we spoke about similar polygons, we said that the ratio of
their corresponding sides or the ratio of similitude, was the
same as the ratio of their corresponding altitudes, angle
bisectors, medians and perimeters.
What we did NOT include was the ratio of their areas. It turns
out that, because area is the product of two factors, the rule
changes with area.
The ratio of the areas of two similar polygons is the SQUARE of
the ratio of their corresponding sides.
So if two similar polygons have sides in the ratio 2:3, the ratio of
their areas is 4:9.
178
You try:
1. Find the area ratios of 2 similar polygons when the ratio of
their corresponding sides are given.
a) 1:3!
!
b) 4:7!
d) 8:1!
!
e) x:y
!
c) 3:5!
!
!
2. Two corresponding sides of 2 similar polygons are 8 & 12,
find the ratio of the areas in simplest fraction form.
These ratios also apply to circles!
3. Find the ratio of similitude of 2 similar polygons when the
* the corresponding radii, diameters and Circumferences will
ratio of their areas are given.
have the same “ratio of similitude”.
a) 81:4! !
!
!
!
b) 25:36!
c) 16:81!
!
d) 1:121!
e)
4. The area ratios of two rectangles is 4:25. What is the ratio of
their perimeters?
179
5. The diameters of two circles are 9 and 16.
a) What is the ratio of their radii?
b) What is the ratio of their circumferences?
c) What is the ratio of their areas?
180
Absolute Value
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Chapter 1 - Distance and Absolute Value
Acute
- measuring greater than 0 and less than 90 degrees
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ACUTE TRIANGLE
- has 3 acute angles
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Chapter 3 - Definition of a Triangle and its Classifications
Adjacent Angles
- angles that share a vertex and a side, but have no interior points in common.
(The word “adjacent” means “next to.”)
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Chapter 1 - Basic Definitions
Alternate Exterior Angles
- pairs of exterior angles on opposite sides of the transversal
- they are CONGRUENT
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Chapter 4 - Properties of Parallel Lines
Alternate Interior Angles
- pairs of interior angles on opposite sides of the transversal
- they are CONGRUENT
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Chapter 4 - Properties of Parallel Lines
ALTITUDE
- in a triangle, a line segment drawn perpendicularly from a vertex to the opposite side,
forming right angles
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Angle
- the union of two rays having the same endpoint
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Angle Bisector
- divides an angle into 2 congruent angles
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ANGLE BISECTOR
- In a triangle, it cuts an angle of the triangle into two congruent angles
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Chapter 3 - Line Segments Associated with the Triangle
BASE
- IN AN ISOSCELES TRIANGLE, it is the non-congruent side
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BASE ANGLES
- IN AN ISOSCELES TRIANGLE, the two angles that share the base and are
CONGRUENT
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Chapter 3 - Definition of a Triangle and its Classifications
Bisection of a Line Segment
- segment is bisected at a point if the point is the midpoint of the line segment
- bisector cuts a line segment into two congruent segments
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Chapter 1 - Basic Definitions
Collinear Points
- points that lie on the same straight line
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Complement of a Set
- the set of elements in the Universal Set that are NOT in the given set
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Complementary Angles
- two angles whose sum is 90°
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Congruence
- “having same length or measure” (think: same size & shape)
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CONSECUTIVE SIDES
- (in a polygon) sides that share a common vertex
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Chapter 5 - Polygons
CONSECUTIVE VERTICES
- (of a polygon) are vertices that share a side
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Corresponding Angles
- pairs of angles that are “in the same matching position”
- they are CONGRUENT
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Chapter 4 - Properties of Parallel Lines
Disjoint Sets
- sets that do not have any common elements -- their intersection yields the empty set
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Chapter 2 - Relationships Between Sets
Distance
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Chapter 1 - Distance and Absolute Value
Elements
- members that are contained in a given set.
- They can be numbers, letters, symbols or any other type of object.
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Chapter 2 - Definitions Involving Sets
Empty Set (or Null Set)
- a set that has no elements
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Equal Sets
- sets that contain exactly the same elements
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EQUIANGULAR
- 3 angles are congruent
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EQUILATERAL TRIANGLE
- has three congruent sides and three congruent angles
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Equivalent Sets
- sets that have the same number of elements
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Chapter 2 - Relationships Between Sets
Exterior Angle
- When one side of a triangle is extended, the angle between that extension and the
adjacent side (“outside” angle)
- the measure of an exterior angle of a triangle equals the sum of the two nonadjacent
interior angles
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Chapter 3 - Exterior Angles of a Triangle
Exterior Angles
- angles NOT between the || lines
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Finite Set
- a set whose elements can be counted.
- In other words, there is a definite number of elements in the given set
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Chapter 2 - Kinds of Sets
Geometry
comes from two Ancient Greek words:
“ge”
“metron”
meaning “Earth”
meaning “measure”
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HYPOTENUSE
- in a right triangle, the side across from the right angle
- the longest side of a right triangle
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Infinite Set
- a set whose elements cannot be counted
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Interior Angles
- angles between the || lines
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Intersection of Sets
- Sets intersect when they share at least one common element
- i.e. the set of common elements
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Chapter 2 - Relationships Between Sets
ISOSCELES TRAPEZOID
- a special type of trapezoid
has the following properties:
1.!
Only one pair of opposite sides is parallel.
2.!
The legs (non-parallel sides) are congruent.
3.!
Each pair of base angles are congruent.
4.!
Diagonals are congruent.
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Chapter 7 - The Isosceles Trapezoid
ISOSCELES TRIANGLE
- has two congruent sides and two congruent angles
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LEGS
- IN A RIGHT TRIANGLE, the two sides forming the right angle(perpendicular)
- IN AN ISOSCELES TRIANGLE, the two congruent sides
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Chapter 3 - Definition of a Triangle and its Classifications
Line
- an infinite set of points
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Line Segment
- a set of points consisting of two points on a line, called endpoints and the set of all
points on the line between the endpoints
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Linear Pair
- two angles that are both supplementary and adjacent
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MEDIAN
- in a triangle, a line segment that is drawn from a vertex to the midpoint of the
opposite side
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Midpoint
- the point on a line segment that divides the segment into 2 ≅ segments
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Non-Collinear points
- points that DO NOT lie on the same straight line
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Obtuse
- measuring greater than 90 and less than 180 degrees
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OBTUSE TRIANGLE
- has one obtuse angle
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Parallel lines
- lines that never intersect
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PARALLELOGRAM
- a quadrilateral with the following properties:
1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. Consecutive angles are supplementary.
5. Diagonals bisect each other.
6. One diagonal divides the parallelogram into 2 congruent
triangles.
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Chapter 7 - The Parallelogram
Perpendicular Bisector
- cuts the line segment in half & forms right angles
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Perpendicular lines
- lines that intersect and form right angles (90°)
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Plane
- a set of points that forms a completely flat surface which extends infinitely in all
directions
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Point
- a location in space, indicates position
- represented by a dot on a piece of paper, named with a capital letter
!
Ex: •P
!
“point P”
- no length, width, or thickness
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POLYGON
- a closed figure which is the union of line segments
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Chapter 5 - Polygons
Pythagorean Theorem
- IN A RIGHT TRIANGLE, the sum of the squares of the lengths of the legs is equal to
the square of the length of the hypotenuse.
i.e.:
- where a and b are the legs and c is the hypotenuse.
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Chapter 3 - The Pythagorean Theorem
Pythagorean Triples
- three whole numbers that work in the Pythagorean Theorem
- common right triangles. If you have two of the three numbers in a triple, and they’re
in the correct positions, you can know the third number without doing the math
!
i.e.: 3-4-5
!
!
5-12-13
!
!
8-15-17
!
!
7-24-25
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Chapter 3 - The Pythagorean Theorem
Quadrilateral
- a 4 sided polygon
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Chapter 7 - Quadrilateral Family Tree
Ray
- the set of all points in a half line, including the dividing point, which is called the
endpoint of the ray
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Chapter 1 - Basic Definitions
RECTANGLE
has the following properties:
1.!
All of the properties of the parallelogram.
2.!
All angles are right angles. As a result, it is equiangular.
3.!
Diagonals are congruent.
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Chapter 7 - The Rectangle
Reflex
- measure is more than 180 and less than 360 degrees
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Chapter 1 - Basic Definitions
RHOMBUS
has the following properties:
1. All of the properties of a parallelogram.
2. All sides are congruent, therefore it is equilateral.
3. Diagonals are perpendicular.
4. Diagonals bisect the angles.
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Chapter 7 - The Rhombus
Right
- measuring exactly 90 degrees
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Chapter 1 - Basic Definitions
RIGHT TRIANGLE
- has one right angle
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Chapter 3 - Definition of a Triangle and its Classifications
Same Side Exterior Angles
- exterior angles on the same side of the transversal.
- They are SUPPLEMENTARY.
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Chapter 4 - Properties of Parallel Lines
Same Side Interior Angles
- interior angles on the same side of the transversal.
- They are SUPPLEMENTARY.
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Chapter 4 - Properties of Parallel Lines
SCALENE TRIANGLE
- has no congruent sides & therefore no congruent angles
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Chapter 3 - Definition of a Triangle and its Classifications
Set
a collection or group of objects with some common characteristic
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Chapter 1 - Undefined Terms
SQUARE
a hybrid of parallelogram, rectangle, and rhombus.
The definition of a square is either:
-!
A rectangle with 2 congruent consecutive sides
or
-!
A rhombus with a right angle
i.e.: has all the properties of the parallelogram, rhombus, and rectangle
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Chapter 7 - The Square
Straight
- an angle of exactly 180 degrees
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Chapter 1 - Basic Definitions
Subset
- a set that contains the elements of another set
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Chapter 2 - Relationships Between Sets
Sum of the 3 interior angles of any
triangle
- 180 degrees
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Chapter 3 - Interior Angles of a Triangle
Sum of the angles of a quadrilateral
- 360 degrees
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Chapter 3 - Interior Angles of a Triangle
Supplementary Angles
- two angles whose sum is 180°
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Chapter 1 - Basic Definitions
Transversal
- any line that intersects (cuts) 2 lines at 2 different points
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Chapter 4 - Properties of Parallel Lines
TRAPEZOID
- a quadrilateral with ONLY ONE pair of opposite parallel sides.
- Those parallel sides are called the BASES; the non-parallel sides are called the
LEGS
- it is NOT a parallelogram
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Chapter 7 - The Trapezoid
Union of Sets
- the combination of the elements of 2 or more sets
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Chapter 2 - Relationships Between Sets
Universal Set
- entire set of elements under consideration in a given situation
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Chapter 2 - Relationships Between Sets
Venn Diagram
- pictorial representation of a set
!
- a rectangle will represent the Universal Set.
!
- Circles represent subsets of the Universal Set and the elements of a
!
set are placed in the circle.
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Chapter 2 - The Venn Diagram
Vertex
- the endpoints of each ray, or the corner of the angle
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Chapter 1 - Basic Definitions
VERTEX ANGLE
- IN AN ISOSCELES TRIANGLE, the angle formed by the legs (the noncongruent
angle)
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Chapter 3 - Definition of a Triangle and its Classifications
Vertical Angles
- formed by intersecting lines
- opposite angle pairs which are congruent
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Chapter 1 - Basic Definitions
VERTICES
- the “corners” or angles of a polygon
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Chapter 5 - Polygons