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QUANDRANTS ANGLES

TOPIC OUTLINE i FOUR QUADRANTS ii UNIT CIRCLE REFERENCE ANGLES iii DEGREES RADIANS

i FOUR QUADRANTS when including negative values the x and y axis divide the space up into FOUR QUADRANTS Quadrant II S 180o 90o y A x T Quadrant III Quadrant I y 270o C x 0o Quadrant VI

Quadrant I A ALL trigonometric functions are positive Quadrant II S SINE COSECANT functions are positive Quadrant III T TANGENT COTANGENT functions are positive Quadrant IV C COSINE SECANT functions are positive

ii UNIT CIRCLE REFERENCE ANGLES The UNIT CIRCLE is a circle with a radius of 1 from the origin 0 0 and used understand Sines and Cosines of angles in right triangles A REFERENCE ANGLE is the smallest angle between the terminal side and the x axis

UNIT CIRCLE REFERENCE ANGLES 90o 60o 150o e gl 45o An An ce 120o n e er 180135o f Re 0 1 e nc re fe Re e l g 30o 180o 0o 1 0 1 0 210o 330o 240o An ce 300o n re 60 e o f 3 270 Re 315o e l g e gl An e 0 nc 1 8 re fe Re 225o 0 1

iii DEGREES RADIANS Degrees is a measure for angles out of 360o A Radian is the angle formed by taking the radius and stretching it around the circle

90o 0o 180o Degrees __ 4 Radian 270o Radians into Degrees Radians x 180 Degrees into Radians Degrees x 180

ANGLES

TOPIC OUTLINE i TYPES OF ANGLES ii SUPPLEMENTARY COMPLEMENTARY ANGLES iii INTERIOR EXTERIOR ANGLES iv DEGREES RADIANS

i TYPES OF ANGLES Acute Angle Less than 90 Right Angle exactly 90 Obtuse Angle greater than 90 but less than 180

Reflex Angle greater than 180 Congruent Angles two angles with the same measure Straight Angle exactly 180

ii Supplementary Complementary Angles Complementary Angles are two angles with a sum of 90o a a b 90o b Supplementary Angles are two angles with a sum of 180o a b a b 180o

iii Interior Exterior Angles Exterior Angle Interior Angle Interior Angles are angles inside of a shape Exterior Angles are between the side of the shape and an adjacent line

iv DEGREES RADIANS Degrees is a measure for angles out of 360o A Radian is the angle formed by taking the radius and stretching it around the circle

90o 0o 180o Degrees __ 4 Radian 270o Radians into Degrees Radians x 180 Degrees into Radians Degrees x 180

CIRCLE PARTS ANGLES

TOPIC OUTLINE CIRCLES i PARTS OF A CIRCLE ii CENTRAL ANGLES iii INSCIRBED ANGLES iv SEMI CIRCLE ANGLES v INTERIOR ANGLES vi TANGENT CHORD ANGLES vii EXTERIROR ANGLES

i PARTS OF A CIRCLE N A T T N GE Center S U I D RA DIAMETER CHO RD T N CA E S CIRCUMFERENCE

CIRCUMFERENCE is the distance around the edge of an object RADIUS is a line from the center of the circle to a point on the circle DIAMETER a chord that contains the center CHORD is a segment that runs from one point on the circle to another point on the circle SECANT is a line that intersects the circle at two points TANGENT is a line that intersects circle at one point

ii CENTRAL ANGLES have a vertex at the center of the circle by two intersecting radii A X B C ABC X

iii INSCRIBED ANGLES have a vertex on the circle by two intersecting chords A X B C 1 ABC X 2 __

iv SEMI CIRCLE ANGLE is an inscribed angle inside a semi circle and is always a right angle B C A 1 ABC 180o 90o 2 __

v INTERIOR ANGLES are formed by two intersecting chords inside a circle X1 C A D E X2 B 1 AEC X1 X2 2 __

vi TANGENT CHORD ANGLE is formed by an intersecting chord and tangent line whose vertex is on the circle B A xo C 1 o ABC 2 x __

vii EXTERIOR ANGLES have a vertex outside of a circle 1 ABC 2 X1 X2 __ Types of Exterior Angles TWO TANGENTS TWO SECANTS TANGENT AND SECANT

PLANE SOLID GEOMETRY

TOPIC OUTLINE i TYPES OF PLANE SHAPES ii PLANE SHAPE AREAS iii SOLID SHAPE VOLUMES iv SURFACE AREA

i TYPES OF PLANE SHAPES POLYGON closed shape with straight sides PARALLELOGRAM 4 sided plane with parallel opposite sides QUADRILATERAL flat shape with 4 straight sides TESSELLATION pattern of identical shapes that don t have any gaps or overlap

ii PLANE SHAPE AREAS FORMULAS AREA measures how much space is on a flat surface TRIANGLE 1 2 Base x Height h b TRAPEZOID 1 2 Base x Height h b CIRCLE r radius 2 PARALLELOGRAM Base x Height h b

iii SOLID SHAPE VOLUME VOLUME measures the amount of space inside an object Measured in Cubic Meters c3 SPHERE 1 2 Base x Height r

CUBE Side3 s CYLINDER h Radius 2 x Height RECTANGULAR PRISM Length x Width x Height w l PYRAMID Base x Height 3 h h b

iv SURFACE AREA SURFACE AREA measures the total area of the surface of a 3D object measured in sq units SPHERE 4 x Radius r

CUBE s 6 x Side CYLINDER h 2 Radius 2 Radius x Height RECTANGULAR PRISM 2 W x L H x L H x W l h w

TRIANGLES

TOPIC OUTLINE i TYPES OF TRIANGLES ii SIMILAR TRIANGLES iii ANGLE ANGLE AA iv SIDE SIDE SIDE SSS v SIDE ANGLE SIDE SAS vi RIGHT ANGLED TRIANGLES vii PYTHAGORAS THEOREM

i TYPES OF TRIANGLES SCALENE TRIANGLE has all Unequal sides ISOSCELES TRIANGLE has Two equal sides EQUILATERAL TRIANGLE has all Equal sides

ii SIMILAR TRIANGLES SIMILAR TRIANGLES are two triangles that only differ in size Types of SIMILAR TRIANGLES Angle Angle AA Side Side Side SSS Side Angle Side SAS

iii Angle Angle AA AA have corresponding angles that are congruent E B D F A C A D If B E Then ABC DEF

iv Side Side Side SSS SSS have three sides of one triangle are congruent to three sides of another triangle E B F D A C If AB __ AC __ BC __ DE Then ABC DF EF DEF

v Side AngleSide SAS SAS have two triangles that are congruent and have proportional angles A E D B C If AE AB AD AC Then ABC AED

vi RIGHT ANGLED TRIANGLES A Right Angled Triangle has a Right Angle inside of it To solve for missing sides of a Right Angled Triangle we use the Pythagoras Theorem

vii PYTHAGORAS THEOREM Opposite Pythagoras Theorem is the square of the hypotenuse is equal to the sum of the squares of the other two sides a2 Hy po c2 ten us e Adjacent b2 a2 b2 c2

Equation of a Straight Line

Topic Outline i SLOPE ii SLOPE INTEREPT iii POINT SLOPE

i Slope RISE y2 y1 ____ _______ m RUN m RISE y x2 x1 RUN x RISE measures the Y Axis RUN measures the X Axis

ii Slope Intercept y mx b Slope y b Y intercept m x Slope measures how steep the line is The Y Intercept is where the line crosses the Y axis

iii Point Slope POINT SLOPE is used to find other points on a line by using the formula y y1 m x x1

TRANSFORMATIONS SYMMETRY

TOPIC OUTLINE i TRANSFORMATIONS SYMMETRY DEFINED ii ROTATIONS iii DILATIONS iv TRANSLATIONS

i WHAT ARE TRANSFORMATIONS AND SYMMETRY Transformations are the changing a shape using by Rotating Flipping Resizing or Moving the shape Symmetry means that a shape becomes exactly like another by Rotating Flipping or Moving the shape

ii ROTATION a ROTATION turns object from a center point in a circle

A Rotation R of 90o x y y x 180o x y x x 270o x y y x 360o x y x y Returns to normal position at R360

iii REFLECTION a REFLECTION flips an image over an imaginary line without changing shape or size

A Reflection r in x axis x y x y y axis x y x y y x x y x y y x x y y x

iv DILATION a DILATION resizes a shape to be bigger or smaller Dk x y kx ky

v TRANSLATION a TRANSLATION moves an object without resizing or rotating it Th k x y x h y k

PROOFS

WHAT IS A PROOF A PROOF is a step by step explanation for the thought process used to reach a conclusion about a geometric statement

Proofs are written in two columns where the statements are listed in one column and the reasons for each statement s truth in the other Statement Reason 1 Statement about the problem 1 Reason for that statement

A well written proof has an argument that is clearly developed with each step supported by Theorems Postulates and Axioms

Theorems statements that can be proved to be true Postulates statements that are assumed to be true without proof Axioms self evident truths or the basic facts that are accepted without any proof

If we had the Example ABC and CBD are supplementary If CBD is 145o what is ABC C A xo 145o B D

SOLUTION To find x you would set ABC CBD equal to 180o and find that x 35o 180 145 x x 35o

Now if this same problem was written as a proof it would look like ABC and CBD are supplementary Given CBD 145o Prove ABC 35o C A 35o 145o B D

Statement Reason 1 ABC and CBD are supplementary 1 Given 2 CBD 145o 2 Given 3 ABC CBD 180o 3 Definition of supplementary 4 ABC CBD 180o 4 substitution 5 ABC 35o 5 Subtraction Axiom

GEOMETRY PRACTICE PROBLEMS

Question 1 What is the length of AD 2 A A B C D 9 7 2 14 5 B C D

ANSWER AB BC CD AD 2 5 2 9 Answer A 9

Question 2 The equation of the x axis is A B C D y y x y x 0 0 1

ANSWER y 0 is the equation of the x axis Answer B y 0

Question 3 Which equation represents a circle whose center is 1 5 and radius of 7 units A B C D x 1 2 y 5 2 49 x 1 2 y 5 2 49 x 1 2 y 5 2 7 x 1 2 y 5 2 7

ANSWER Plug 1 5 in to the formula x h x k r2 x 1 x 5 72 Answer B x 1 2 y 5 2 49

Question 4 ABC and CBD are complementary and the measurement of CBD is 60o What is ABC C A x B 60o D A B C D 90o 60o 30o 180o

ANSWER ABD ABC CBD 90 60 x Answer C X 30o

Question 5 ABC and CBD are Supplementary and the measurement of CBD is 80o What is the ABC C A x 80o B D A B C D 100o 90o 10o 180o

ANSWER ABD ABC CBD 180 80 x Answer A x 100o

Question 6 Which angles are alternate interior angles 1 2 3 4 5 6 7 8 A B C D 1 8 2 6 3 6 1 2

ANSWER Alternate Interior Angles are two interior angles that lie on different parallel lines and on opposite sides of a transversal Answer C 3 6

Question 7 Which angles are alternate exterior angles 1 2 3 4 5 6 7 8 A B C D 1 8 2 6 3 6 1 2

ANSWER Alternate Exterior Angles are two exterior angles on opposite sides of a transversal that lie on different parallel lines Answer A 1 8

Question 8 Which angles are corresponding angles 1 2 3 4 5 6 7 8 A B C D 1 8 2 6 3 6 1 2

ANSWER Corresponding Angles are angles that occupy the same relative position at each intersection where a straight line crosses two others B 2 6

Question 9 If 1 is 120o what is the value of 5 1 2 3 4 5 6 7 8 A B C D 120o 80o 140o 180o

ANSWER Since 1 5 are corresponding and over two parallel lines they both have the same angle Answer A 120o

Question 10 Classify this angle according to its sides A Equilateral B Isosceles C Scalene

ANSWER Equilateral triangles have all equal sides Answer A Equilateral

Question 11 Classify this angle according to its sides A Equilateral B Isosceles C Scalene

ANSWER Isosceles triangles have two equal sides Answer B Isosceles

Question 12 Classify this angle according to its sides A Equilateral B Isosceles C Scalene

ANSWER Scalene triangles don t have any equal sides Answer C Scalene

Question 13 What is the value of BAC and BCA if ABC is 100o B 100o A x x C A B C D 180o 40o 60o 80o

ANSWER Sum of interior angles 180 and since its an Isosceles triangle BAC BCA 180 100 x X 80 2 X 40o Answer B 40o

Question 14 What is the value of x x 4 3 A B C D 5 25 10 12

ANSWER Use the Pythagoras theorem a2 b2 c2 42 32 x2 25 x2 x 5 Answer A 5

Question 15 What is the value of x 10 8 x A B C D 5 25 10 6

ANSWER Use the Pythagoras theorem a2 b2 c2 82 x2 102 64 x2 100 x2 36 x 6 Answer D 6

Question 16 Find the perimeter of the rectangle with a length of 120ft and a width of 60ft 120ft 60ft A B C D 7200ft 360ft 180ft 1200ft

ANSWER To find the perimeter add up all of the sides 120 120 60 60 360 Answer B 360 ft

Question 17 Find the area of a parallelogram with a base of 10ft and a height of 5ft 5ft 10ft A B C D 15ft 25ft 50ft 100ft

ANSWER Area Base x Height 5 x 10 50 Answer c 50ft

Question 18 Find the circumference of this circle that has a diameter of 10ft 10ft A B C D 3 14ft 100ft 31 4ft 360ft

ANSWER Circumference x Diameter x 10 31 4 Answer C 31 4ft

Question 19 Find the circumference of this circle with a radius of 5in 5in A B C D 78 5in 25in 31 4in 360in

ANSWER Area x Radius 2 x 5 2 78 5 Answer A 78 5 in

Question 20 What is the volume of this cube A 16cm B 12cm C 64cm 3 4 4 4 3 3 D 4cm 3

ANSWER Plug into cube volume formula Volume Side 3 V 4 3 V 64 Answer C 64 cm3

Question 22 What is the volume of this rectangular prism 10 7 5 A B C D 35cm 199cm 22cm 50cm 3 3 3 3

ANSWER Plug into rectangular prism volume formula Volume Length x Width x Height V 7 x 5 x 10 V 199 cm3 Answer B 199 cm3

Question 23 The measure of one supplementary angle is twice the measure of the second What is the measure of each angle A B C D 45o 60o 80o 90o 90o 120o 160o 180o

ANSWER First let x the first angle and 2x second angle Then since a supplementary angle is 180o set the two angles equal to 180 x 2x 180 Now solve for the value of the first angle x x 2x 180 3x 180 x 60 Since the first angle is 60o multiply it by two to find the value of the second angle 60 x 2 120o So the value of the two angles are 60o 120o Answer B 60o 120o