read each question carefully. read the directions for the test carefully

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 14 150 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112120

9 0 9 18 27 36 45 54 63 72 81 90 99 108117 126135

10 0 10 20 30 40 50 60 70 80 90 100 110 120130 140150

11 0 11 22 33 44 55 66 77 88 99 110121 132143 154 165

12 0 12 24 36 48 60 72 84 96 108120 132144 156168 180

13 0 13 26 39 52 65 78 91 104 117130 143156 169182 195

14 0 14 28 42 56 70 84 98 112 126140 154168 182196 210

15 0 15 30 45 60 75 90 105 120 135150 165180 195 210225

Factors Multiples Perfect Squares(6 ) (4 ) = 24

Geometry Basics

Point (Name with 1 capital letter)A

AB�������������� � • •

A BLine (Name with 2 capital letters, )

AB��������������

• •A B

Ray (Name with 2 capital letter, )

ABCAngle (Name with 3 letters. Middle letter is vertex

)B

A

C

•A B

Line Segment (Name with two letters, AB)

Plane (Name with 3 non-collinear points, ABC)A B

C

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 Circle Sum of angles in a 4 sided figure (quadrilateral)

Also called linear pair

Complementary AnglesRight AnglesSymbol (┌ or ┐)Perpendicular ┴A corner

90

Straight Angle (line)Supplementary AnglesLinear PairHalf CircleSum of angles in a triangle

180

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°

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.

Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html

HINT: S Straight or S Splits

In this example, a° and b° are vertical angles.

a° = b°

Vertical Angles

Angles opposite each other when two lines crossThey are called "Vertical" because they share the same Vertex (or corner point)

http://www.mathwarehouse.com/geometry/angle/interactive-vertical-angles.php

Vertical angles are congruent and their measures are equal:

vertex

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°

Complementary Angles

Two Angles are Complementary if they add up to 90 degrees (a Right Angle).

Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html

HINT: C Corner or C 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 AnglesA 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

Conditionals & Bi-conditionals

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

Conditionals

ConverseThe 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.

CONVERSE

CONVERSE:

CounterexampleA 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.

If-Then Transitive PropertyGiven

If A then B, and if B then C.

You can conclude: If A then C.

If sirens shriek,

then dogs howl

If dogs howl,

then cats freak.

If sirens shriek,

then cats freak.

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Kite

Trapezoid

Isosceles Trapezoid

( 4 sides )

Rectangle

Congruent SidesCongruent AnglesParallel Sides

All angles are congruent (90 ˚ )

Diagonals are congruent

Parallelogram

Congruent SidesCongruent AnglesParallel Sides

Opposite sides

Opposite angles

Consecutive angles are supplementary

Opposite sides parallel

Diagonals bisect each other

Rhombus

Congruent SidesCongruent AnglesParallel Sides

All sides are congruent

Diagonals are perpendicular

Diagonals bisect angles

Square

Congruent SidesCongruent AnglesParallel Sides

Diagonals are perpendicular and congruent

Diagonals bisect each other

All sides are congruent

All angles are congruent

Angles = 90°

Isosceles Trapezoid Diagonals are congruent

Trapezoid Congruent SidesCongruent AnglesParallel Sides

Kite Congruent SidesCongruent AnglesParallel Sides

Diagonals are perpendicular

Geometry in Motion

ReflectionReflection

Transformation

Dilation

Rotation

Refl

ectio

n

Refl

ecti

on

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/downMove 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

TranslationsIn this next example, the "slide"  moves the figure7 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 x-coordinates, 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 x-axis, 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

DilationsDilations

DilationsDilations

Dilations

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.

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 originmay be written:  Dk(x,y) = (kx,ky).

.

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!

5 6

7 8

1 2

3 4

Interior

ExteriorParallel Lines Angles

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 ‹2,‹7

Corresponding Angles Congruent ‹1,‹5 ‹3,‹7 ‹2,‹6 ‹4,‹8

Same Side Interior Supplemental add up to 180 ‹3,‹5 ‹4,‹6

Transversal

lines

Same Slope

Slopes are Negative Reciprocal

Flip and Change Sign

PARALLEL LINES

y2 – y1

x2 – x1

or

y = mx + b

slope

Slope – Intercept Form

y = mx + b

Slope- directions

RiseRun

Y Interceptwhere to start

It’s a line address

If the slope is a whole number, put it on a stick m = 2 slope is 2/1

To Graph:

y = 2X + 1 Starts at 1

Rise/run = 2/1

Directions are up 2, over 1

Example 1 Example 2y = -3X+ 0

y = -3X

Starts at 0

rise/run = 3/-1

Directions are up 3, over -1

Thanks to http://www.mathsisfun.com/equation_of_line.html

Example: Solve for Y2x – 7y = 12

Just 3 easy steps1. -7y = 12 – 2x X is offside, Penalty change signs2. -7y = (12-2x) Huddle up ( )3. y = (12-2x) / -7 Man on man defense

WATCH YOUR SIGNS!!

Linear Equations, Standard Form ax + by = cSolving for y, It’s a football Game

Y VS Everybody Else

Follow football rules

Play FootballLetters vs Numbers

Now you are ready to enter it into the calculator or graph it

Find Equation of the Line: y = mx + b

I need slope (m) & the y-intercept (b)

MY ANSWER:

y = x +

To find m – Solve the equation for y and use mor use the y2 – y1

x2 – x1 formula

To find b - Plug x, y and m into the line equation and solve for b.

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°

Formulas

Line Stuff

Polygons:

Figure # of Sides

# of Triangles

Sum of Interior Angles

(# of Triangles)(180)

Triangle 3 1 180 1 * 180Rectangle 4 2 360 2 * 180Pentagon 5 3 540 3 * 180Hexagon 6 4 720 4 * 180Octagon 8 6 1080 6 * 180n -gon n n-2 180 (n-2)

Sum of the Angles of a Polygon.

Sum of Exterior Angles is 360

Tiles or floors

Floor Plan

Floor Rugs

Acres

Examples of things you’d find the area

of.

Perimeter – Path around the Outside

No Trespassing – Go all the way Around!

Area of Plane Shapes

Triangle

A = ½b×h  Square

Area = a2

RectangleArea = b×h  

ParallelogramArea = b×h

Trapezoid (US)Area = ½(b1+b2)h

 Circle

Area = πr2

Area Formulas

b

h

b

h

b1

h

a

b

h

r

b2

Area of Plane Shapes

Triangle

P = a + b + c

  SquareP = 4a

RectangleP = 2b + 2h

  ParallelogramP = 2b + 2a

Trapezoid (US)P = a + b1 + b2 + d  

Circle

Circumference=2πrr = radius

Perimeter Formulas

a b

c

h

b

a d

b1

b2

a

b

a

r

a

bc

A

B C

Law of Sinessin(A) = sin(B) = sin(C) a b c

Law of Cosinesa² = b² + c² – 2bc * cos(A)

b² = a² + c² - 2ac * cos(B)

c² = a² + b² – 2ab * cos(C)

cos(A) = (a² – b² – c²) (-2ab)

To convert from:Degrees to radians – multiply by π

180

Radians to degrees – multiply by 180 π

Trigonometry for Any Triangle

3 Trig Functions:

SOH

ΘSin =hyp

opp

adj

TOA

ΘTan =oppΘ

CAH

Cos =hyp

adj

Trigonometry Functions(Be sure your calculator is in degrees)

Hyp is always across from right angle. Adj and Opp change depending on Θ

Trigonometry is the study of how the sides and angles of a right triangle are related to each other.

3 Sides:

1. Hypotenuse - Across from right angle. 2. Opposite - Across from angle Θ. 3. Adjacent – Next to angle.

Θ

Θadj

adjhyp

hyp

opp

opp

•Use chart to organize information•Set up ratio•Cross multiply•Solve for X

To Solve: Ex 1 Ex 2

Hyp 31

Adj x

Opp 10 23

Θ 41 Θ

Trig Func. tan Sin-1