Read each question carefully Read the directions for

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 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 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 Factors (6 ) (4 ) 3 0 = 4 0 5 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 0 7 0 8 9 10 11 12 13 14 15 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, ) • A Ray (Name with 2 capital letter, ) • A Angle (Name with 3 letters. Middle letter is vertex • B C ) 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 Split s Thanks to http: //www. mathsisfun. com/geometry/complementary-angles. html

Vertical 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° 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

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.

Conditionals 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

Converse The CONVERSE 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. CONVERSE: IF we will have a snow day, THEN it is snowing.

Counterexample 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.

If-Then Transitive Property 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.

Quadrilaterals ( 4 sides ) Parallelogram Rectangle Rhombus Square Trapezoid Kite Isosceles Trapezoid

Rectangle 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 Parallelogram

Rhombus Congruent Sides Congruent Angles Parallel Sides Diagonals are perpendicular All sides are congruent Diagonals bisect angles Square 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

Isosceles Trapezoid Congruent Sides Congruent Angles Parallel Sides Diagonals are congruent Trapezoid Kite Congruent Sides Congruent Angles Parallel Sides Diagonals are perpendicular

R e Reflection f l f e c l t e c Rotation Geometry in Motion i t i o n Transformation

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

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!

Tran sv ersal 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

PARALLEL LINES Same Slope y 2 – y 1 x 2 – x 1 Slopes are Negative Reciprocal Flip and Change Sign lines or slope y = mx + b

Slope – Intercept Form y = mx + b Slope- directions Rise Run Y Intercept where to start 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 = -3 X+ 0 y = 2 X + 1 Starts at 1 Rise/run = 2/1 Directions are up 2, over 1 Thanks to http: //www. mathsisfun. com/equation_of_line. html y = -3 X Starts at 0 rise/run = 3/-1 Directions are up 3, over -1

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 2 x – 7 y = 12 Play Football Letters vs Numbers Just 3 easy steps 1. -7 y = 12 – 2 x X is offside, Penalty change signs 2. -7 y = (12 -2 x) Huddle up ( ) 3. y = (12 -2 x) / -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 To find b - Plug x, y and m into the line equation and solve for b. MY ANSWER: y= x +

Formulas tuff S e n Li Slope: m= Midpoint: (x, y) = ( , ) Distance: d = o Polyg ns: 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 (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 = a 2 A = ½b×h a Area of Plane Shapes b h Rectangle Area = b×h h b Parallelogram Area = b×h b b 2 h Trapezoid (US) Area = ½(b +b )h 1 Circle Area = πr 2 2 r b 1

Perimeter Formulas a Triangle Square P = 4 a b a a + Shapes b+c Area. Pof=Plane c Rectangle P = 2 b + 2 h b a h Parallelogram P = 2 b + 2 a a b b 2 Trapezoid (US) d P = a + b 1 + b 2 + d b 1 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 – 2 bc * cos(A) b² = a² + c² - 2 ac * cos(B) c² = a² + b² – 2 ab * 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) (-2 ab)

3 Trig Functions: SOH SinΘ = CAH opp hyp Cos Θ= TOA adj hyp TanΘ = opp adj

Trigonometry Functions (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 opp Θ opp adj

To Solve: • Use chart to organize information • Set up ratio • Cross multiply • Solve for X Ex 1 Hyp Ex 2 31 Adj x Opp 10 23 Θ 41 Θ Trig Func. tan Sin-1