Parallel and Perpendicular Lines. What are the slope and y-intercept of each equation?
|
|
- Kelly Long
- 6 years ago
- Views:
Transcription
1 6 6-6 What You ll Learn To determine whether lines are parallel To determine whether lines are And Wh To use parallel and lines to plan a bike path, as in Eample Parallel Lines Parallel and Perpendicular Lines Check Skills You ll Need G for Help Lessons - and 6- What is the reciprocal of each fraction? 7 7 What are the slope and -intercept of each equation? = + 6 = ; - 8 ; 8 7 = 6 6; 0 8 = 6 + 6; New Vocabular parallel lines lines negative reciprocal 6-6 Plan bjectives To determine whether lines are parallel To determine whether lines are Eamples Determining Whether Lines Are Parallel Writing Equations of Parallel Lines Writing Equations for Perpendicular Lines Real-World Problem Solving In the graph at the right, the red and blue lines are parallel Parallel lines are lines in the same plane that never intersect The equation of the red line is = + The equation of the blue line is = - Ke Concepts Propert Slopes of Parallel Lines Nonvertical lines are parallel if the have the same slope and different -intercepts An two vertical lines are parallel Eample The equations = + and = - have the same slope,, and different -intercepts The graphs of the two equations are parallel Math Background Vertical lines have an undefined, or infinite, slope because the denominator in the ratio is zero This is wh the Propert of Slopes of Parallel Lines includes the word nonvertical More Math Background: p 06D Lesson Planning and Resources You can use slope-intercept form to determine whether the lines are parallel See p 06E for a list of the resources that support this lesson Real-World Connection The lanes for competitive swimming are parallel Determining Whether Lines Are Parallel Are the graphs of = + and + 6 = parallel? Eplain Write + 6 = in slope-intercept form Then compare with = + 6 =- + Subtract from each side 6 = 6 6 Divide each side b 6 = + Simplif The lines are parallel The equations have the same slope, different -intercepts Are the graphs of =- and = es; same slope, different -intercepts, and - 7 parallel? Eplain Lesson 6-6 Parallel and Perpendicular Lines Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Multipling and Dividing Lesson -: Eample 7 Etra Skills and Word Problem Practice, Ch Slope-Intercept Form Lesson 6-: Eample Etra Skills and Word Problem Practice, Ch 6 Special Needs L Some students ma confuse the terms parallel and Have them underline the ll in the word parallel Ask: Do the l s in the word parallel look parallel or? learning stle: visual Below Level L To demonstrate that the signs of the slopes of parallel lines are the same, have students graph = - Then have them compare their graphs with the one on p learning stle: visual
2 Teach Guided Instruction Alternative Method Write = + and = - on the board Ask: What number do the equations have in common? What does this number represent in each equation? the slope Have students find the -coordinates of the point on each line when = Then find the difference between the -coordinates Tell students that the difference is the vertical distance between the two lines Repeat for = 00, = 0,000 and = 00,000 Compare the vertical distance between the lines for each pair of points The vertical distances are all the same Ask: What geometric term describes lines that are alwas the same distance apart? parallel What might ou conclude about lines that have the same slope? Lines that have the same slope are parallel Additional Eamples Are the graphs of =- - and + = 6 parallel? es Write an equation for the line that contains (-, ) and is parallel to = - ± 8 nline Visit: PHSchoolcom Web Code: ate-077 You can use the fact that the slopes of parallel lines are the same to write the equation of a line parallel to a given line To write the equation, ou use the slope of the given line and the point-slope form of a linear equation Writing Equations of Parallel Lines Write an equation for the line that contains (, ) and is parallel to = - Step Identif the slope of the given line = - c slope Step Perpendicular Lines Write the equation of the line through (, ) using slope-intercept form - = m( - ) point-slope form - = ( ) Substitute (, ) for (, ) and for m - = - () Use the Distributive Propert - = - Simplif = - Add to each side Write an equation for the line that contains (, -6) and is parallel to = + 9 The lines at the right are Perpendicular lines are lines that intersect to form right angles The equation of the red line is = - The equation of the blue line is = + Ke Concepts Propert Slopes of Perpendicular Lines Two lines are if the product of their slopes is - A vertical and a horizontal line are also Eample The slope of is The slope of = + is Since? =-, the graphs of the two equations are The product of two numbers is - if one number is the negative reciprocal of the other Here is how to find the negative reciprocal of a number Start with Find its Write the a fraction: S reciprocal: S negative reciprocal: Since? =-, is the negative reciprocal of Start with Find its Write its an integer: S reciprocal: S negative reciprocal: Since Q =-, R is the negative reciprocal of Chapter 6 Linear Equations and Their Graphs Advanced Learners L Draw a square and its diagonals on a coordinate grid Challenge students to prove that the diagonals are learning stle: verbal English Language Learners ELL In Eample, have students undo negative reciprocal one word at a time Ask: What do ou multipl b to make a fraction negative? Discuss the meaning of reciprocal learning stle: verbal
3 A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Test-Taking Tip After finding the slopes of the lines, multipl the two slopes as a check ( ) You can use the negative reciprocal of the slope of a given line to write an equation of a line to that line Writing Equations for Perpendicular Lines Multiple Choice Which equation describes a line that contains (0, -) and is to = +? = - = = = Step Identif the slope of Step Find the negative reciprocal the given line of the slope = + The negative reciprocal of is c slope Step Use the slope-intercept form to write an equation = m + b = + (-) Substitute for m, and for b = - Simplif The equation is = - So D is the correct answer Write an equation of the line that contains (, 8) and is to = + ± 9 You can use equations of parallel and lines to solve some real-world problems Real-World Problem Solving Urban Planning A bike path for a new cit park will connect the park entrance to Park Road The path will be to Park Road Write an equation for the line representing the bike path Error Prevention Students ma think that a negative reciprocal is alwas negative Tell students that the phrase negative reciprocal actuall means the negative of the reciprocal So the are reall just finding the opposite of the reciprocal Additional Eamples Find an equation of the line that contains (6, ) and is to =- + 7 The line in the graph represents the street in front of a new house The point is the front door The sidewalk from the front door will be to the street Write an equation representing the sidewalk 6 front door street Park park entrance Park Road -intercept new path to Park Road (, ) (, ) Resources Dail Notetaking Guide 6-6 L Dail Notetaking Guide 6-6 Adapted Instruction L Closure Real-World Connection Careers Urban planners use mathematics to make decisions on communit problems such as traffic congestion and air pollution Step Find the slope m of Park Road m = = = = Points (, ) and (, ) are on Park Road Step Find the negative reciprocal of the slope The negative reciprocal of is So the slope of the bike path is The -intercept is The equation for the bike path is = + A second bike path is planned It will be parallel to Park Road and will also contain the park entrance Write an equation for the line representing this bike path ± Lesson 6-6 Parallel and Perpendicular Lines Ask: Compare the equations of non-vertical parallel lines Parallel lines have the same slope, but different -intercepts Compare the equations of lines Perpendicular lines have slopes that are negative reciprocals of each other The will have the same -intercept onl if that is where the two lines intersect
4 Practice Assignment Guide A B -8, 0,,,, -60 A B 9-9,,, 6-0, 6-6 C Challenge 6-70 Test Prep 7-76 Mied Review Homework To check students understanding of ke skills and concepts, go over Eercises 8, 6,, 7-9, Eercise Suggest to students that the graph the line = 6 EXERCISES Practice and Problem Solving A G Practice b Eample for Help Eample (page ) Eample (page ) For more eercises, see Etra Skill and Word Problem Practice Find the slope of a line parallel to the graph of each equation = + = - = = = = 7 Are the graphs of the lines in each pair parallel? Eplain 7 See margin 7 = + 8 = + 9 = = + = 8 - = 6 0 = + =- = = + 7 = - + = 8 Write an equation for the line that is parallel to the given line and that passes through the given point = 6 - ; (0, 0) 8 See margin =-; (, 0) =- + ; (-, ) 6 = 7 + 6; (-, -6) 7 = 0-8; (8, -) 8 = + ; (, -) Eample (page ) Find the slope of a line to the graph of each equation 9 = 0 =- = 7-7 = = =-8undefined Write an equation for the line that is to the given line and that passes through the given point GPS Guided Problem Solving L Enrichment Reteaching Adapted Practice L L L Practice Pearson Education, Inc All rights reserved Name Class Date Practice 6- Parallel and Perpendicular Lines Find the slope of a line parallel to the graph of each equation = + = + =-9 - = = 6 - = = = = = 0 =- - - = 6 Write an equation for the line that is to the given line and that passes through the given point (6, ); = - (-, ); =- + 9 (-, -); = (, ) ; = (, -6); = + 8 (0, -); = Write an equation for the line that is parallel to the given line and that passes through the given point (, ); = - 7 (, ); =- + (, -); = - (,0); = (-8, -); = + 7 (9, -7); -7 - = Tell whether the lines for each pair of equations are parallel,, or neither = =- = + L B Eample (page ) Appl Your Skills = + 7; (0, 0) 6 = - ; (, 6) ± 0 7 = + ; (, ) = 7; (-, ) ± = ; (, ) 0 - = 9; (8, -) ± ± Maps A cit s civil engineer is planning a new parking garage and a new street The new street will go from the entrance of the parking garage to Handel St It will be to Handel St What is the equation of the line representing the new street? ± Entrance Tell whether the lines for each pair of equations are parallel,, or neither = + = + = - 6, =, + 6 parallel =- +, = + =, =- + 7 neither 6 = -, = + parallel 7 =, = =, + = parallel 9 - =, - + = 8 neither Handel St - =- = = = 6 =- 6 = = 6 = =- 0 Multiple Choice Which pair of equations graph as lines? C = + = + = - 9 = - 9 =- + = - = + 8 = + pages 6 9 Eercises 7 no, different slopes 8 es, same slopes and different -intercepts 9 es, same slopes and different -intercepts 6 6 Chapter 6 Linear Equations and Their Graphs 0 no, different slopes es, same slopes and different -intercepts es, same slopes and different -intercepts 6 ± ±
5 Real-World Connection A skier s fastest speed occurs when the skis are parallel Find the equation for each line 9 ; ± ; ± ; ± 7 ± ; Teaching Tip Eercises 9 Ask students to describe equations of lines that are neither parallel nor The slopes are neither the same nor negative reciprocals of each other Error Prevention! Eercise 6 Suggest that students first sketch a graph of the triangle to determine which two sides ma be 6 ; ; ± GPS Maps Use the map below for Eercises 7 9 New Hampshire Ave Massachusetts Ave Pennslvania Ave Washington, DC G nline Homework Help Visit: PHSchoolcom Web Code: ate What is the slope of New Hampshire Avenue? about 8 Show that the parts of Pennslvania Avenue and Massachusetts Avenue near New Hampshire Avenue are parallel Answers ma var Sample: same slope of 9 Show that New Hampshire Avenue is not to Pennslvania Avenue Answers ma var Sample:? Q R u 0 a The graphs of = and =-are shown on the standard screen at the right The product of the slopes is - Eplain wh the lines do not appear to be a b See margin b Graphing Calculator Graph = and =- on a graphing calculator In the feature, choose the square screen What do ou notice? pen-ended Write an equation for a line parallel to the graph of - = Answers ma var Sample: ± Lesson 6-6 Parallel and Perpendicular Lines 7 0a The units on the aes are different, so the screen is not square b The lines appear 7
6 Assess & Reteach Lesson Quiz Find the slope of a line parallel to - = Find the slope of a line to + = 7 Tell whether the lines for each pair of equations are parallel,, or neither = -, =- + = +, + =- neither =- -, + = 6 parallel 6 Write an equation of the line that contains (, ) and is to =- + Alternative Assessment Write an equation in slopeintercept form on the board Have students first write an equation of a line that is parallel to the given line, and then write an equation of a line that is to the given equation Let students compare their answers Repeat No; the slopes are not equal No; the slopes are not neg reciprocals Problem Solving Hint For Eercises 7, sketch a graph to help ou understand the statement in each eercise False; the product of two positive numbers can t be 6 True; ± and ± are parallel 7 False; all direct variations go through the point (0, 0) If the have the same slope, the are the same line, not parallel lines C Challenge Are the graphs of + 7 = 6 and 7 = + 6 parallel? Eplain See left Are the graphs of 8 + = 6 and 8 - = 6? Eplain See left Writing Are all horizontal lines parallel? Eplain See margin Tell whether each statement is true or false Eplain our choice 7 See below left Two lines with positive slopes can be 6 Two lines with positive slopes can be parallel 7 The graphs of two different direct variations can be parallel 8 60 See Geometr A quadrilateral with both pairs of opposite sides parallel is a margin parallelogram Use slopes to determine whether each figure is a parallelogram 8 B 9 K 60 P J A Q C S 6 L D M R Geometr A quadrilateral with four right angles is a rectangle Use slopes to determine whether each figure is a rectangle 6 6 See margin 6 B 6 6 K A L Q P R C N S D M 6 Geometr A quadrilateral with two pairs of parallel sides and with diagonals that are is a rhombus Quadrilateral ABCD has vertices A(-, ), B(, 6), C(6, 6), and D(, ) Show that ABCD is a rhombus See margin 6 Geometr A triangle with two sides that are to each other is a right triangle Triangle PQR has vertices P(, ), Q(, -), and R(0, ) Determine whether PQR is a right triangle Eplain See margin es; same slopes and different -intercepts 8 The slopes of AD and BC are both undefined, so the are parallel The slopes of AB and CD are both, so the are parallel The quadrilateral is a parallelogram 9 The slope of is The JK slope of KL is The slope of LM is The slope of JM 6 is The quadrilateral is a parallelogram ; 8 ± 7 69 ; ± 7 8 Chapter 6 Linear Equations and Their Graphs Tell whether the lines in each pair are parallel,, or neither 66 a - b = ; -a + b = 67 a + b = 8; b- a = parallel Assume the two lines are Find an equation for each line See left For what value of k are the graphs of + = 8 and 6 = k - parallel? Perpendicular? ; 60 The slopes of PQ and RS 6 The slopes of AB and CD 6 The slopes of KL and MN are both The slopes of QR are both The slopes of and SP are both BC are both The slopes and AD are both of LM 6 and KN are both The quadrilateral is The product is, so The product is not, so a parallelogram the quadrilateral is a the quadrilateral is not a rectangle rectangle
7 Test Prep Multiple Choice Short Response Standardized Test Prep 7 Which equation has as its graph a line to a line that has a slope of? D A = + B = - C = - D + = 8 7 Which equation has as its graph a line parallel to the graph of - - =? F F = + G = - 6 H =- + J = - 7 A parallelogram has vertices A(0, ), B(, -), C(6, ), and D(p, q) Which of the following ordered pairs has possible values for (p, q)? C A (0, 6) B (6, ) C (, 6) D (6, ) 7 Which equation has as its graph a line to the graph of the line that passes through the points (0, -) and (8, 9)? F F ( ) G 0 H 0 J 7 7 Which equation has as its graph a line parallel to the graph of the line that passes through the points (-8, ) and (-, )? C A ( ) ( 8) B 0 C 6 D 7 76 Suppose the line through points (, 6) and (, ) is parallel to the graph of + = Find the value of Show our work See margin Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p 69 Test-Taking Strategies, p 6 Test-Taking Strategies with Transparencies Connection to Geometr Eercise 7 Tell students that the can eliminate some possible answers b first sketching the three points in the eercise on a coordinate plane A-head Mied Review G for Help Lesson 6- Write an equation for the line through the given point with the given slope 77 (0, ); m = ± 78 (-, 0); m = (, -); m = ± ( ) 80 (-, -9); m = ± 9 ( ± ) 8 (-6, ); m = ( ± 6) 8 (7, ); m = ( 7) Lesson 6- Lesson - Find the slope of each line lesson quiz, PHSchoolcom, Web Code: ata-0606 Determine whether each relation is a function 86 {(, ), (, ), (, )} es 87 {(, ), (, ), (, )} es 88 {(, ), (, ), (, )} no 89 {(, ), (, ), (, )} es Lesson 6-6 Parallel and Perpendicular Lines 9 6 The slopes of PQ and RS 6 BC and AD both have a BD has a slope of are both The slopes of PS and QR slope of zero and are both AD BC are parallel and The product is, so CD AB The diagonal AC has a slope of The diagonals are both have a slope of the quadrilateral is a rectangle AB and CD ~ABCD is a rhombus are parallel The diagonal 6 RP has a slope of has a slope of RP RQ is the neg reciprocal of RQ, so kpqr is a right triangle 76 [] Find slope of ± : ±, therefore slope is 6 Find :,, ±, (R equivalent eplanation) [] correct value of but no work R minor computation error in work 9
1. A(-2, 2), B(4, -2) 3 EXAMPLE. Graph the line y = Move up 3 units. Quick Check See left.
-. Plan Objectives To graph lines given their equations To write equations of lines Eamples Graphing Lines in Slope- Intercept Form Graphing Lines Using Intercepts Transforming to Slope- Intercept Form
More informationNew Vocabulary parallelogram rhombus rectangle square kite trapezoid isosceles trapezoid
6-. Plan bjectives To define and classif special tpes of quadrilaterals Eamples Classifing a Quadrilateral Classifing Coordinate Methods Using the Properties of Special Quadrilaterals 6- What You ll Learn
More informationGraph each pair of functions on the same coordinate plane See margin. Technology Activity: A Family of Functions
- What You ll Learn To analze translations To analze stretches, shrinks, and reflections...and Wh To analze a fabric design, as in Eample Families of Functions Check Skills You ll Need G for Help Lessons
More information(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2
.5 Equations of Parallel and Perpendicular Lines COMMON CORE Learning Standards HSG-GPE.B.5 HSG-GPE.B. Essential Question How can ou write an equation of a line that is parallel or perpendicular to a given
More information2.5. Verifying Properties of Geometric Figures. LEARN ABOUT the Math. Proving a conjecture about a geometric figure
.5 Verifing Properties of Geometric Figures YOU WILL NEED grid paper and ruler, or dnamic geometr software P( 7, 9) Q(9, ) J - - M - R(9, ) - - - L - - S(, ) K GOAL Use analtic geometr to verif properties
More informationVertical and Horizontal Translations. Graph each pair of functions on the same coordinate plane See margin
- Lesson Preview What You ll Learn BJECTIVE BJECTIVE To analze vertical translations To analze horizontal translations... And Wh To analze a fabric design, as in Eample BJECTIVE Vertical and Horizontal
More informationUnit 6: Connecting Algebra and Geometry Through Coordinates
Unit 6: Connecting Algebra and Geometry Through Coordinates The focus of this unit is to have students analyze and prove geometric properties by applying algebraic concepts and skills on a coordinate plane.
More informationProving Properties of a Parallelogram
Eplain Proving Properties of a Parallelogram You have alread used the Distance Formula and the Midpoint Formula in coordinate proofs As ou will see, slope is useful in coordinate proofs whenever ou need
More information2.4 Coordinate Proof Using Distance with Quadrilaterals
Name Class Date.4 Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance formula in coordinate proofs? Resource Locker Eplore Positioning a Quadrilateral
More informationTo prove theorems using figures in the coordinate plane
6-9 s Using Coordinate Geometr Content Standard G.GPE.4 Use coordinates to prove simple geometric theorems algebraicall. bjective To prove theorems using figures in the coordinate plane Better draw a diagram!
More information25.4 Coordinate Proof Using Distance with Quadrilaterals
- - a a 6 Locker LESSON 5. Coordinate Proof Using Distance with Quadrilaterals Name Class Date 5. Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationProperties of Quadrilaterals
MIAP Chapter 6: Linear functions Master 6.1a Activate Prior Learning: Properties of Quadrilaterals A quadrilateral is a polgon with 4 sides. A trapezoid is a quadrilateral that has eactl one pair of parallel
More informationReteaching O 4 6 x. 6a. 2 6b a b. 3 8a. 0 8b. undefined. 3 10a. undefined 10b. 0 11a. 1 11b. 1
Chapter Answers Alternative Activit -. Check students work.. Check students work.. 80. 7. Angles change. The sum remains 80. 8. Subtract (0 55) from 80 to get 8 Reteaching -. Alternative Activit -. 0.
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More informationACTIVITY: Frieze Patterns and Reflections. a. Is the frieze pattern a reflection of itself when folded horizontally? Explain.
. Reflections frieze pattern? How can ou use reflections to classif a Reflection When ou look at a mountain b a lake, ou can see the reflection, or mirror image, of the mountain in the lake. If ou fold
More informationANGLES See the Math Notes boxes in Lessons and for more information about angle relationships.
CC1 Basic Definitions Defense Practice ANGLES 2.1.1 2.1.5 Applications of geometr in everda settings often involve the measures of angles. In this chapter we begin our stud of angle measurement. After
More informationExample 1: MATH is a parallelogram. Find the values of w, x, y, and z. Write an equation for each and write the property of parallelograms used.
Name: Date: Period: Geometr Notes Parallelograms Fab Five Quadrilateral Parallelogram Diagonal Five Fabulous Facts about Parallelograms: ) ) 3) 4) 5) ***This is the Parallelogram Definition and Theorems!
More informationRotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things
. Rotations object in a plane? What are the three basic was to move an Rotate A biccle wheel can rotate clockwise or counterclockwise. 0 0 0 9 9 9 8 8 8 7 6 7 6 7 6 ACTIVITY: Three Basic Was to Move Things
More informationCoordinate geometry Straight lines are an important part of our environment. We play sport on courts with parallel and perpendicular lines, and
Number and Algebra Coordinate geometr Straight lines are an important part of our environment. We pla sport on courts with parallel and perpendicular lines, and skscrapers would not be standing without
More information8.7 Coordinate Proof with
8.7 Coordinate Proof with Quadrilaterals Goal Eample p Use coordinate geometr to prove properties of quadrilaterals. Determine if quadrilaterals are congruent Determine if the quadrilaterals with the given
More informationPractice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise
Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,
More informationName Date Class. 6. In JKLM, what is the value of m K? A 15 B 57 A RS QT C QR ST
Name Date Class CHAPTER 6 Chapter Review #1 Form B Circle the best answer. 1. Which best describes the figure? 6. In JKLM, what is the value of m K? A regular convex heptagon B irregular convex heptagon
More informationChapter 2 Diagnostic Test
Chapter Diagnostic Test STUDENT BOOK PAGES 68 7. Calculate each unknown side length. Round to two decimal places, if necessary. a) b). Solve each equation. Round to one decimal place, if necessary. a)
More informationhttps://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework_...
Page 1 of Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 > 3-8 Slopes of 3-8 Slopes of Teks Focus TEKS ()(C) Determine an equation of a line parallel or perpendicular to a given
More informationOn Your Own. Applications. Unit 3. b. AC = 8 units y BD = 16 units C(5, 2)
Applications 1 a. If students use uniform and y scales or ZSquare, they should have a shape that looks like a kite. Using the distance formula, DC = AD = 1 and BC = BA = 0. b. AC = 8 units y BD = 16 units
More informationLaurie s Notes. Overview of Section 6.3
Overview of Section.3 Introduction In this lesson, eponential equations are defined. Students distinguish between linear and eponential equations, helping to focus on the definition of each. A linear function
More informationSLOPE A MEASURE OF STEEPNESS through 2.1.4
SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written
More informationName Date Class. Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent.
SKILL 43 Angle Relationships Example 1 Adjacent angles are pairs of angles that share a common vertex and a common side. Vertical angles are opposite angles formed by the intersection of two lines. Vertical
More informationLesson 9: Coordinate Proof - Quadrilaterals Learning Targets
Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of the way along each median
More informationDISTANCE FORMULA: to find length or distance =( ) +( )
MATHEMATICS ANALYTICAL GEOMETRY DISTANCE FORMULA: to find length or distance =( ) +( ) A. TRIANGLES: Distance formula is used to show PERIMETER: sum of all the sides Scalene triangle: 3 unequal sides Isosceles
More informationGeometry Regents Lomac Date 3/2 due 3/4 Coordinate Plane: the power of right triangles
Geometry Regents Lomac 2015-2016 Date 3/2 due 3/4 Coordinate Plane: the power of right triangles 1 Name Per LO: I can find slopes and distances for pairs of points and use them to identify congruent segments
More informationHow can you enlarge or reduce a figure in the coordinate plane? Dilate. ACTIVITY: Comparing Triangles in a Coordinate Plane.
. Dilations How can ou enlarge or reduce a figure in the coordinate plane? Dilate When ou have our ees checked, the optometrist sometimes dilates one or both of the pupils of our ees. ACTIVITY: Comparing
More information5.8 Start Thinking. 5.8 Warm Up. 5.8 Cumulative Review Warm Up
5.8 Start Thinking Use dnamic geometr software to create an ABC in a coordinate plane such that the center of the triangle is the origin. Use the software to manipulate the triangle so it has whole-number
More informationGraphing square root functions. What would be the base graph for the square root function? What is the table of values?
Unit 3 (Chapter 2) Radical Functions (Square Root Functions Sketch graphs of radical functions b appling translations, stretches and reflections to the graph of Analze transformations to identif the of
More information2.1 Length of a Line Segment
.1 Length of a Line Segment MATHPOWER TM 10 Ontario Edition pp. 66 7 To find the length of a line segment joining ( 1 y 1 ) and ( y ) use the formula l= ( ) + ( y y ). 1 1 Name An equation of the circle
More information6-3 Conditions for Parallelograms
6-3 Conditions for Parallelograms Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Justify each statement. 1. 2. Reflex Prop. of Conv. of Alt. Int. s Thm. Evaluate each expression for x = 12 and
More informationInclination of a Line
0_00.qd 78 /8/05 Chapter 0 8:5 AM Page 78 Topics in Analtic Geometr 0. Lines What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and
More informationLines and Their Slopes
8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation
More informationUNIT 5: GEOMETRIC AND ALGEBRAIC CONNECTIONS. Apply Geometric Concepts in Modeling Situations
UNIT 5: GEOMETRIC AND ALGEBRAIC CONNECTIONS This unit investigates coordinate geometry. Students look at equations for circles and use given information to derive equations for representations of these
More informationCHAPTER 6 : COORDINATE GEOMETRY CONTENTS Page 6. Conceptual Map 6. Distance Between Two Points Eercises Division Of A Line Segment 4 Eercises
ADDITIONAL MATHEMATICS MODULE 0 COORDINATE GEOMETRY CHAPTER 6 : COORDINATE GEOMETRY CONTENTS Page 6. Conceptual Map 6. Distance Between Two Points Eercises 6. 3 6.3 Division Of A Line Segment 4 Eercises
More informationAlgebra I. Linear Equations. Slide 1 / 267 Slide 2 / 267. Slide 3 / 267. Slide 3 (Answer) / 267. Slide 4 / 267. Slide 5 / 267
Slide / 67 Slide / 67 lgebra I Graphing Linear Equations -- www.njctl.org Slide / 67 Table of ontents Slide () / 67 Table of ontents Linear Equations lick on the topic to go to that section Linear Equations
More informationParallel lines are lines that never intersect and are always the same distance apart. Parallel Lines
Lesson 4.6 Objectives Determine if two lines are parallel or perpendicular. Write equations of parallel and perpendicular lines. Slopes of Parallel and Perpendicular Lines Parallel and perpendicular lines
More information5 and Parallel and Perpendicular Lines
Ch 3: Parallel and Perpendicular Lines 3 1 Properties of Parallel Lines 3 Proving Lines Parallel 3 3 Parallel and Perpendicular Lines 3 Parallel Lines and the Triangle Angles Sum Theorem 3 5 The Polgon
More informationWRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #1313
WRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #11 SLOPE is a number that indicates the steepness (or flatness) of a line, as well as its direction (up or down) left to right. SLOPE is determined
More informationPeriod: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means
: Analytic Proofs of Theorems Previously Proved by Synthetic Means Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of
More informationACTIVITY 9 Continued Lesson 9-2
Continued Lesson 9- Lesson 9- PLAN Pacing: 1 class period Chunking the Lesson Eample A Eample B #1 #3 Lesson Practice M Notes Learning Targets: Graph on a coordinate plane the solutions of a linear inequalit
More informationGeometry Unit 5 Geometric and Algebraic Connections. Table of Contents
Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing
More informationLesson 1: Slope and Distance
Common Core Georgia Performance Standards MCC8.G.8* (Transition Standard 01 013; asterisks denote Transition Standards) MCC9 1.G.GPE.4 MCC9 1.G.GPE.5 Essential Questions 1. How is the Pythagorean Theorem
More informationSize Transformations in the Coordinate Plane
Size Transformations in the Coordinate Plane I.e. Dilations (adapted from Core Plus Math, Course 2) Concepts 21-26 Lesson Objectives In this investigation you will use coordinate methods to discover several
More informationContent Standards Two-Variable Inequalities
-8 Content Standards Two-Variable Inequalities A.CED. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales.
More information(0, 4) Figure 12. x + 3. d = c. = b. Figure 13
80 CHAPTER EQUATIONS AND INEQUALITIES Plot both points, and draw a line passing through them as in Figure. Tr It # _, 0 Figure Find the intercepts of the equation and sketch the graph: = _ +. (0, (This
More informationGEOMETRY COORDINATE GEOMETRY Proofs
GEOMETRY COORDINATE GEOMETRY Proofs Name Period 1 Coordinate Proof Help Page Formulas Slope: Distance: To show segments are congruent: Use the distance formula to find the length of the sides and show
More informationModule Four: Connecting Algebra and Geometry Through Coordinates
NAME: Period: Module Four: Connecting Algebra and Geometry Through Coordinates Topic A: Rectangular and Triangular Regions Defined by Inequalities Lesson 1: Searching a Region in the Plane Lesson 2: Finding
More informationEducation Resources. This section is designed to provide examples which develop routine skills necessary for completion of this section.
Education Resources Straight Line Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section.
More information3.5 Write and Graph Equations
.5 Write and Graph Equations of Lines Goal p Find equations of lines. Your Notes VOCABULARY Slope-intercept form Standard form Eample Write an equation of a line from a graph Write an equation of the line
More informationMidpoint of a Line Segment Pg. 78 # 1, 3, 4-6, 8, 18. Classifying Figures on a Cartesian Plane Quiz ( )
UNIT 2 ANALYTIC GEOMETRY Date Lesson TOPIC Homework Feb. 22 Feb. 23 Feb. 24 Feb. 27 Feb. 28 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.5 2.1-2.3 2.1-2.3 Mar. 1 2.6 2.4 Mar. 2 2.7 2.5 Mar. 3 2.8 2.6 Mar. 6 2.9 2.7 Mar.
More information9-1. Translations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
9- Translations Vocabular Review. Underline the correct word to complete the sentence. If two triangles are congruent, corresponding angle measures are the same/ different and corresponding side lengths
More informationPolygons in the Coordinate Plane
. Polgons in the Coordinate Plane How can ou find the lengths of line segments in a coordinate plane? ACTIVITY: Finding Distances on a Map Work with a partner. The coordinate grid shows a portion of a
More information6.4 rectangles 2016 ink.notebook. January 22, Page 22. Page Rectangles. Practice with. Rectangles. Standards. Page 24.
6.4 rectangles 2016 ink.notebook Page 22 Page 23 6.4 Rectangles Practice with Rectangles Lesson Objectives Standards Lesson Notes Page 24 6.4 Rectangles Press the tabs to view details. 1 Lesson Objectives
More informationDay 116 Bellringer. 1. Use the triangle below to answer the questions that follow.
Day 116 Bellringer 1. Use the triangle below to answer the questions that follow. 3 in 5 in 4 in a) Find the area of the triangle. b) Find the perimeter of the triangle. 2. Use the distance formula to
More informationName Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =
Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations
More informationMathematics Stage 5 PAS5.1.2 Coordinate geometry
Mathematics Stage PAS.. Coordinate geometr Part Graphing lines Acknowledgments This publication is copright New South Wales Department of Education and Training (DET), however it ma contain material from
More informationSlope Station. 2. When are two lines in the plane perpendicular to one another?
Slope Station 1. When are two lines in the plane parallel to each other? When their slopes are equal 2. When are two lines in the plane perpendicular to one another? When their slopes are opposite reciprocals
More informationAre You Ready? Triangle Sum Theorem
SKILL 30 Triangle Sum Theorem Teaching Skill 30 Objective Use the Triangle Sum Theorem to find the measures of missing angles. Have students read the Triangle Sum Theorem. Point out that the theorem is
More informationStudent Name: Teacher: Date: Miami-Dade County Public Schools. Test: 9_12 Mathematics Geometry Exam 2
Student Name: Teacher: Date: District: Miami-Dade County Public Schools Test: 9_12 Mathematics Geometry Exam 2 Description: GEO Topic 5: Quadrilaterals and Coordinate Geometry Form: 201 1. If the quadrilateral
More informationDistance on the Coordinate Plane
6 7 Distance on the Coordinate Plane What You ll Learn You ll learn to find the distance between two points on the coordinate plane. Wh It s Important Transportation Knowing how to find the distance between
More information6-1 Study Guide and Intervention Angles of Polygons
6-1 Study Guide and Intervention Angles of Polygons Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from
More informationTubes are Fun. By: Douglas A. Ruby Date: 6/9/2003 Class: Geometry or Trigonometry Grades: 9-12 INSTRUCTIONAL OBJECTIVES:
Tubes are Fun B: Douglas A. Rub Date: 6/9/2003 Class: Geometr or Trigonometr Grades: 9-2 INSTRUCTIONAL OBJECTIVES: Using a view tube students will conduct an eperiment involving variation of the viewing
More informationEnhanced Instructional Transition Guide
Enhanced Instructional Transition Guide / Unit 04: Suggested Duration: 6 das Unit 04: Geometr: Coordinate Plane, Graphing Transformations, and Perspectives (9 das) Possible Lesson 0 (6 das) Possible Lesson
More information12.1. Angle Relationships. Identifying Complementary, Supplementary Angles. Goal: Classify special pairs of angles. Vocabulary. Complementary. angles.
. Angle Relationships Goal: Classif special pairs of angles. Vocabular Complementar angles: Supplementar angles: Vertical angles: Eample Identifing Complementar, Supplementar Angles In quadrilateral PQRS,
More informationQuadrilaterals. Polygons Basics
Name: Quadrilaterals Polygons Basics Date: Objectives: SWBAT identify, name and describe polygons. SWBAT use the sum of the measures of the interior angles of a quadrilateral. A. The basics on POLYGONS
More informationIntensive Math-Algebra I Mini Lesson MA.912.A.3.10
Intensive Math-Algebra I Mini Lesson M912.3.10 Summer 2013 Writing Equations of Perpendicular Lines Student Packet Day 10 Name: Date: Benchmark M912.3.10 Write an equation of a line given any of the following
More informationPolygon Interior Angles
Polygons can be named by the number of sides. A regular polygon has All other polygons are irregular. A concave polygon has All other polygons are convex, with all vertices facing outwards. Name each polygon
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationConnecting Algebra and Geometry with Polygons
Connecting Algebra and Geometr with Polgons 15 Circles are reall important! Once ou know our wa around a circle, ou can use this knowledge to figure out a lot of other things! 15.1 Name That Triangle!
More informationGeometric and Algebraic Connections
Geometric and Algebraic Connections Geometric and Algebraic Connections Triangles, circles, rectangles, squares... We see shapes every day, but do we know much about them?? What characteristics do they
More informationGraph Linear Equations
Lesson 4. Objectives Graph linear equations. Identif the slope and -intercept of linear equations. Graphing Linear Equations Suppose a baker s cookie recipe calls for a miture of nuts, raisins, and dried
More information= = The number system. Module. Glossary Math Tools... 33
- > + > < - %. < + a = - = = b in. F - - Module The number sstem Lesson Rational and Irrational Numbers........ 8.NS. Lesson ompare and Order Numbers......... 8 8.NS., 8.NS. Lesson Estimate the Value of
More informationEssential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1.
Locker LESSON 4.4 Graphing Eponential Functions Common Core Math Standards The student is epected to: F-IF.7e Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
More informationMath 3 - Lesson Title: Using the Coordinate Plane for Proofs
Targeted Content Standard(s): Use coordinates to prove simple geometric theorems algebraically. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove
More information14-1. Translations. Vocabulary. Lesson
Chapter 1 Lesson 1-1 Translations Vocabular slide, translation preimage translation image congruent figures Adding fied numbers to each of the coordinates of a figure has the effect of sliding or translating
More informationEnhanced Instructional Transition Guide
Enhanced Instructional Transition Guide Grade / Unit : Suggested Duration: das Unit : Statistics ( das) Possible Lesson 0 ( das) Possible Lesson 0 ( das) POSSIBLE LESSON 0 ( das) This lesson is one approach
More informationName of Lecturer: Mr. J.Agius. Lesson 46. Chapter 9: Angles and Shapes
Lesson 46 Chapter 9: Angles and Shapes Quadrilaterals A quadrilateral is any four-sided shape. Any quadrilateral can be split up into two triangles by drawing in a diagonal, like this: The sum of the four
More informationLesson 11 Skills Maintenance. Activity 1. Model. The addition problem is = 4. The subtraction problem is 5 9 = 4.
Lesson Skills Maintenance Lesson Planner Vocabular Development -coordinate -coordinate point of origin Skills Maintenance ddition and Subtraction of Positive and Negative Integers Problem Solving: We look
More informationApplications. 44 Stretching and Shrinking
Applications 1. Look for rep-tile patterns in the designs below. For each design, tell whether the small quadrilaterals are similar to the large quadrilateral. Explain. If the quadrilaterals are similar,
More information9 CARTESIAN SYSTEM OF COORDINATES You must have searched for our seat in a cinema hall, a stadium, or a train. For eample, seat H-4 means the fourth seat in the H th row. In other words, H and 4 are the
More informationTranslations. Essential Question How can you translate a figure in a coordinate plane? A B
. Translations Essential Question How can ou translate a figure in a coordinate plane? Translating a Triangle in a oordinate Plane USING TOOLS STRTEGILLY To be proficient in math, ou need to use appropriate
More informationReady To Go On? Skills Intervention 9-1 Multiple Representations of Functions
9A Read To Go On? Skills Intervention 9-1 Multiple Representations of Functions Using Multiple Representations to Solve Problems The table shows the sum of the interior angles of polgons and the number
More informationUnit 6: Quadrilaterals
Name: Geometry Period Unit 6: Quadrilaterals Part 1 of 2: Coordinate Geometry Proof and Properties! In this unit you must bring the following materials with you to class every day: Please note: Calculator
More informationPrentice Hall Gold Geometry Teaching Resources Answers Chapter 5
Prentice Hall Gold Geometry Teaching Resources Answers Chapter 5 PRENTICE HALL GOLD GEOMETRY TEACHING RESOURCES ANSWERS CHAPTER 5 PDF - Are you looking for prentice hall gold geometry teaching resources
More information5. Trapezoid: Exactly one pair of parallel sides. 6. Isosceles Trapezoid is a trapezoid where the non-parallel sides are equal.
Quadrilaterals page #1 Five common types of quadrilaterals are defined below: Mark each picture: 1. Parallelogram: oth pairs of opposite sides parallel. 2. Rectangle: Four right angles. 3. Rhombus: Four
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationPrentice Hall Gold Geometry Chapter 3 Test Answer Key
We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with prentice hall gold geometry
More informationWriting Equations of Parallel and Perpendicular Lines
Writing Equations of Parallel and Perpendicular Lines The coordinate plane provides a connection between algebra and geometry. Postulates 17 and 18 establish a simple way to find lines that are parallel
More informationACTIVITY: Forming the Entire Coordinate Plane
.5 The Coordinate Plane How can ou graph and locate points that contain negative numbers in a coordinate plane? You have alread graphed points and polgons in one part of the coordinate plane. In Activit,
More information3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines
3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x + 2 3. y = x 3 4. y = 4 x + 3 3 November 2, 2015 3.4 Proofs with Perpendicular Lines Geometry 3.5 Equations of Parallel and Perpendicular Lines Day
More informationPrerequisite Skills Appendix
Prerequisite Skills Appendi Adding Polnomials To add, add the like terms. 9 1. Add. a) b) 7 6 7 c) 6 d) a a 8 a a 1 e) f) 6a b a b 7 Angle Properties To find the measure of, recall that the sum of the
More informationMidpoint and Distance Formulas
CP1 Math Unit 5: Coordinate Geometry: Day Name Midpoint Formula: Midpoint and Distance Formulas The midpoint of the line segment between any two points (x!, y! ) to (x!, y! ) is given by: In your groups,
More information8 Quadrilaterals. Before
8 Quadrilaterals 8. Find Angle Measures in Polygons 8. Use Properties of Parallelograms 8.3 Show that a Quadrilateral is a Parallelogram 8.4 Properties of Rhombuses, Rectangles, and Squares 8.5 Use Properties
More information