Read each
question
carefully.
Read the
directions for
the test
carefully.
For Multiple Choice Tests
•Check each answer – if impossible or silly cross it out.
•Back plug (substitute) –
one of them has to be the answer
•For factoring – Work the problem backwards
•Sketch a picture
•Graph the points
•Use the y= function on calculator to match graphs
Do the Easy Ones First
Then go Back and
do the Hard Ones!
Beware of the Sucker Answer
Make sure you answer
the question that is asked!
Double check the question
before you fill in the bubble!!
X
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Factors
(6 ) (4 )
1 2 3
0 0 0
1 2 3
2 4 6
3 6 9
4 8 12
5 10 15
6 12 18
7 14 21
8 16 24
9 18 27
10 20 30
11 22 33
12 24 36
13 26 39
14 28 42
15 30 45
=
4 5
0 0
4 5
8 10
12 15
16 20
20 25
24 30
28 35
32 40
36 45
40 50
44 55
48 60
52 65
56 70
60 75
Multiples
24
6 7 8 9 10 11 12 13 14 15
0 0 0 0 0 0 0 0 0 0
6 7 8 9 10 11 12 13 14 15
12 14 16 18 20 22 24 26 28 30
18 21 24 27 30 33 36 39 42 45
24 28 32 36 40 44 48 52 56 60
30 35 40 45 50 55 60 65 70 75
36 42 48 54 60 66 72 78 84 90
42 49 56 63 70 77 84 91 98 105
48 56 64 72 80 88 96 104 112120
54 63 72 81 90 99 108117 126135
60 70 80 90 100 110 120130 140150
66 77 88 99 110121 132143 154 165
72 84 96 108120 132144 156168 180
78 91 104 117130 143156 169182 195
84 98 112 126140 154168 182196 210
90 105 120 135150 165180 195 210225
Perfect Squares
Geometry Basics
A
Point (Name with 1 capital letter)
Line
(Name with 2 capital letters, AB
)
•
A
Ray
(Name with 2 capital letter,
)
•
A
AB
•
B
•
B
C
Angle (Name with 3 letters.
Middle letter is vertex
ABC
)
Line Segment (Name with two letters,
B
A
AB)
•
B
A
Plane (Name with 3 non-collinear points, ABC)
C
A
B
90
Complementary Angles
Right Angles
Symbol (┌ or ┐)
Perpendicular ┴
A corner
180
Straight Angle (line)
Supplementary Angles
Half Circle
Sum of angles in a triangle
360
Also called linear pair
Circle
Sum of angles in a 4 sided figure (quadrilateral)
90
Complementary Angles
Right Angles
Symbol (┌ or ┐)
Perpendicular ┴
A corner
180
Straight Angle (line)
Supplementary Angles
Linear Pair
Half Circle
Sum of angles in a
triangle
Supplementary Angles
Two Angles are Supplementary if they add up to 180 degrees.
These two angles (140° and 40°) are
Supplementary Angles, because they add up
to 180°.
Notice that they are also a linear pair.
But the angles don't have to be together.
These two are supplementary because 60° +
120° = 180°
HINT: S
Straight
or S
Splits
Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html
Vertical Angles
Angles opposite each other when two lines cross
They are called "Vertical" because they share the same Vertex (or corner point)
vertex
Vertical angles are congruent and their measures are equal:
In this example, a° and b° are vertical
angles.
a° = b°
http://www.mathwarehouse.com/geometry/angle/interactive-vertical-angles.php
Complementary Angles
Two Angles are Complementary if they add up to 90 degrees (a Right Angle).
These two angles (40° and 50°) are
Complementary Angles, because they add
up to 90°.
Notice that together they make a right angle.
But the angles don't have to be together.
These two are complementary because 27° +
63° = 90°
HINT: C
Corner
or C
Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html
looks like a corner
Linear Pairs
Angles on one side of a straight line will always
add to 180 degrees.
If a line is split into 2 and you know one angle you can always
find the other one by subtracting from 180
25°
A°
A° = 180 – 25°
A° = 155°
Right Angles
A right angle is equal to 90°
Notice the special symbol like a box in the angle. If you see this, it
is a right angle. 90˚ is rarely written.
If you see the box in the corner, you are being told it is a right
angle.
90°
90°
Notice: Two right angles make a straight line
Properties of Equality
• Addition Property: If a=b, then a+c=b+c
• Subtraction Property: If a=b, then a-c=b-c
• Multiplication Property: If a=b, then ac=bc
• Division Property: if a=b and c doesn’t
equal 0, then a/c=b/c
• Substitution Property: If a=b, you may
replace a with b in any equation containing a
and the resulting equation will still be true.
Properties of Equality
Reflexive Property:
For any real number a, a=a
Symmetric Property:
For all real numbers a and b, if a=b, then
b=a
Transitive Property:
For all real numbers a, b, and c, if
a=b
b=c
a=c a=c
EXAMPLES:
IF today is Saturday, THEN we have no school.
“IF-THEN ” statements like the ones above are called CONDITIONALS.
To make a bi-conditional, take off the IF and replace the THEN with
“IF AND ONLY IF”
Today is Saturday, IF AND ONLY IF we have no school.
Conditional statements have two parts…
The part following the word IF is the HYPOTHESIS
The part following the word THEN is the CONCLUSION
IF today is Saturday, THEN we have no school.
Hypothesis: today is Saturday
Conclusion: we have no school
The
of a conditional statement is formed by
exchanging the HYPTHESIS and the CONCLUSION.
CONDITIONAL: IF it is snowing, THEN we will have a snow day.
IF we will have a snow day, THEN it is snowing.
A Counterexample is an example that proves a statement false.
Conditional Statement: IF an animal lives in water,
THEN it is a fish.
* This conditional statement would be false.
You can show that the statement is false because you can
give one counterexample. *
Counterexample: Whales live in water, but whales
are mammals, not fish.
Given
If A then B, and if B
then C.
If sirens
shriek,
then dogs
howl
If dogs
howl,
then cats
freak.
You can conclude:
If A then C.
If sirens
shriek,
then cats
freak.
( 4 sides )
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
Isosceles Trapezoid
All angles are congruent (90 ˚ )
Congruent Sides
Congruent Angles
Parallel Sides
Diagonals are
congruent
Congruent Sides
Congruent Angles
Parallel Sides
Opposite sides
Opposite angles
Consecutive angles are
supplementary
Opposite sides parallel
Diagonals bisect each
other
Congruent Sides
Congruent Angles
Parallel Sides
Diagonals are perpendicular
All sides are congruent
Diagonals bisect angles
Diagonals are perpendicular
and congruent
Diagonals bisect each other
All sides are congruent
All angles are congruent
Angles = 90°
Congruent Sides
Congruent Angles
Parallel Sides
Diagonals are congruent
Congruent Sides
Congruent Angles
Parallel Sides
Congruent Sides
Congruent Angles
Parallel Sides
Diagonals are perpendicular
Re f l e c t i o n
Rotation
Geometry in Motion
A translation "slides" an object a
fixed distance in a given
direction. The original object and its
translation have the same shape and
size, and they face in the same
direction.
Translations
Move up/down
Move right/left
Let's examine some translations
related to coordinate geometry.
In the example, notice how
each vertex moves the same
distance in the same direction.
6 units to the right
Translations
In this next example, the "slide" moves the figure
7 units to the left and 3 units down.
There are 3 different ways to describe a
translation
1.
description:
7 units to the left and 3 units down.
2.
mapping:
(This is read: "the x and y coordinates will become x-7
and y-3". Notice that movement left and down is
negative, while movement right and up is positive just as it is on coordinate axes.)
3.
symbol:
(The -7 tells you to subtract 7 from all of your xcoordinates, while the -3 tells you to subtract 3 from
all of your y-coordinates.)
This may also be seen as T-7,-3(x,y) = (x-7,y-3).
Reflecting over the y-axis:
When you reflect a point across
the y-axis, the y-coordinate
remains the same,
the x-coordinate changes!
The reflection of the point
(x,y) across the y-axis is the
point (-x,y).
Reflecting over the x-axis:
When you reflect a point across the xaxis, the x-coordinate remains the
same, and the y-coordinate changes!
The reflection of the
point (x,y) across
the x-axis is the
point (x,-y).
Examples of the Most Common Rotations
Counterclockwise
rotation by 180°
about the origin:
A is rotated to its
image A'. The general
rule for a rotation by
180° about the origin is
(x,y)
(-x, -y)
Examples of the Most Common
Rotations
Counter clockwise
rotation by 90° about
the origin:
A is rotated 90° to its
image A'. The general
rule for a rotation by
90° about the origin is
(x,y)
(-y, x)
Dilations always involve a change in size.
Dilations
Dilations
Dilations
Dilations
Dilations
Dilations
Dilations
Dilations
Dilations
Dilations
Dilation is the same shape as the original, but is a
different size. The description of dilation includes
the scale factor and the center of the dilation.
A dilation of scalar factor k whose center of
dilation is the origin
may be written: Dk(x,y) = (kx,ky).
.
You are probably familiar with
the word "dilate" as it relates to
the eye. The pupil of the eye
dilates (gets larger or smaller)
depending upon the amount of
light striking the eye.
Dilations - Example 1: If the scale factor
is greater than 1, the image is an
enlargement (bigger).
PROBLEM: Draw the dilation
image of triangle ABC with scale
factor of 2.
OBSERVE: Notice how EVERY
coordinate of the original triangle
has been multiplied by the scale
factor (2).
HINT: Dilations involve multiplication!
Dilations Example 2: If the scale factor is
between 0 and 1, the image is a reduction
(smaller).
PROBLEM: Draw the dilation
image of pentagon ABCDE with
a scale factor of 1/3.
OBSERVE: Notice how EVERY
coordinate of the original
pentagon has been multiplied
by the scale factor (1/3).
HINT: Multiplying by 1/3 is the same as dividing by 3!
Parallel Lines
Angles
Exterior
1
3
2
4
Interior
6
5
7
8
Linear Pairs
Supplemental Add up to 180
‹1,‹2 ‹3,‹4 ‹2,‹4 ‹1,‹3 ‹5,‹6
‹7,‹8 ‹5,‹7 ‹8,‹6
Vertical Angles
Congruent
‹1,‹4 ‹3,‹2 ‹5,‹8 ‹6,‹7
Alternate Interior Angles
Congruent
‹3,‹6 ‹4,‹5
Alternate Exterior Angles
Congruent
‹1,‹8
Corresponding Angles
Congruent
‹1,‹5 ‹3,‹7
Same Side Interior
Supplemental add up to 180
‹3,‹5 ‹4,‹6
‹2,‹7
‹2,‹6 ‹4,‹8
y2 – y1
x2 – x1
Slopes are Negative
Reciprocal
Flip and Change Sign
lines
or
slope
y = mx + b
Slope – Intercept Form
y = mx + b
Slope- directions Y Intercept
where to
Rise
start
Run
It’s a line address
If the slope is a whole number, put it on a stick
To Graph:
Example 1
m=2
slope is 2/1
Example 2
y = -3X+ 0
y = 2X + 1
y = -3X
Starts at 1
Starts at 0
Rise/run = 2/1
rise/run = 3/-1
Directions are up 2, over 1
Directions are up 3, over -1
Thanks to http://www.mathsisfun.com/equation_of_line.html
Linear Equations, Standard Form ax + by = c
Solving for y, It’s a football Game
Y VS Everybody Else
Follow football rules
Example: Solve for Y
2x – 7y = 12
Play Football
Letters vs Numbers
Just 3 easy steps
1. -7y = 12 – 2x
X is offside, Penalty change signs
2. -7y = (12-2x)
Huddle up ( )
3. y = (12-2x) / -7 Man on man defense
Now you are ready to enter it into the calculator or graph it
WATCH YOUR SIGNS!!
Find Equation of the Line:
y = mx + b
I need slope
(m) & the yintercept (b)
To find m – Solve
the equation for y
and use m
or use the
y –y
x – x formula
2
1
2
1
To find b - Plug x, y
and m into the line
equation and solve for
b.
MY ANSWER:
y=
x +
Formulas
Slope:
m=
Midpoint: (x, y) = (
,
)
Distance: d =
Sum of the interior measures:
Sum of the exterior measures:
360°
Measure of the interior angle in a regular polygon:
Measure of the exterior angle in a regular polygon:
360°
Sum of the Angles of a Polygon.
Sum of Exterior
Angles is 360
Figure
# of
Sides
Triangle
Rectangle
Pentagon
Hexagon
Octagon
n -gon
3
4
5
6
8
n
# of
Triangle
s
1
2
3
4
6
n-2
Sum of
(# of
Interior Angles Triangles)(18
0)
180
1 * 180
360
2 * 180
540
3 * 180
720
4 * 180
1080
6 * 180
180 (n-2)
Floor Rugs
Floor Plan
Tiles or floors
Acres
Examples of things
you’d find the area
of.
Perimeter – Path around the
Outside
No Trespassing – Go all the way Around!
Area Formulas
h
Triangle
Square
Area = a2
A = ½b×h
a
Area of Plane Shapes
b
h
Rectangle
Area = b×h
b
h
Parallelogram
Area = b×h
b
b2
h
Trapezoid (US)
Area = ½(b1+b2)h
Circle
Area = πr2
r
b1
Perimeter Formulas
a
Triangle
Square
P = 4a
b
a
a + Shapes
b+c
AreaPof=Plane
c
Rectangle
P = 2b + 2h
b
a
h
Parallelogram
P = 2b + 2a a
b
b2
Trapezoid (US) d
P = a + b1 + b2 + d
b1
Circle
r
Circumference=2πr
r = radius
Trigonometry for Any Triangle
A
b
c
B
a
Law of Sines
sin(A) = sin(B) = sin(C)
a
b
c
Law of Cosines
a² = b² + c² – 2bc * cos(A)
b² = a² + c² - 2ac * cos(B)
c² = a² + b² – 2ab * cos(C)
To convert from:
Degrees to radians – multiply by
C
π
180
Radians to degrees – multiply by 180
π
cos(A) = (a² – b² – c²)
(-2ab)
SOH
SinΘ =
CAH
opp
hyp
Cos Θ=
TOA
adj
hyp
Tan Θ =
opp
adj
(Be sure your calculator is in degrees)
Trigonometry is the study of how the sides and angles of a right triangle are related to each other.
Hyp is always
across from right
angle. Adj and Opp
change depending
on Θ
3 Sides:
1. Hypotenuse - Across from right angle.
2. Opposite - Across from angle Θ.
3. Adjacent – Next to angle.
Θ
adj
hyp
hyp
opp
Θ
opp
adj
Ex 1
Hyp
•Use chart to organize
information
•Set up ratio
•Cross multiply
•Solve for X
Ex 2
31
Adj
x
Opp
10
23
Θ
41
Θ
Trig Func.
tan
Sin-1