2.1 Slope and Parallel Lines

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1 Name Class ate.1 Slope and Parallel Lines Essential Question: How can ou use slope to solve problems involving parallel lines? Eplore Proving the Slope Criteria for Parallel Lines Resource Locker The following theorem states an important connection between slope and parallel lines. Theorem: Slope Criteria for Parallel Lines Two nonvertical lines are parallel if and onl if the have the same slope. Follow these steps to prove the slope criteria for parallel lines. First prove that if two lines are parallel, then the have the same slope. Suppose lines m and n are parallel lines that are neither vertical nor horizontal. Let and be two points on line m, as shown. You can draw a horizontal line through and a vertical line through to create the slope triangle, C. You can etend _ C to intersect line n at point and then etend it to point F so that C = F. Finall, ou can draw a vertical line through F intersecting line n at point E. Mark the figure to show parallel lines, right angles, and congruent segments. m E n Houghton Mifflin Harcourt Publishing Compan When parallel lines are cut b a transversal, corresponding angles are congruent, so C. C b the Triangle Congruence Theorem. CPCTC, _ C and C =. The slope of line m = _ C, and the slope of line n = _ F. The slopes of the lines are equal because C F Module 55 Lesson 1

2 C Now prove that if two lines have the same slope, then the are parallel. Suppose lines m and n are two lines with the same nonzero slope. You can set up a figure in the same wa as before. Let and be two points on line m, as shown. You can draw a horizontal line through and a vertical line through to create the slope triangle, C. You can etend _ C to intersect line n at point and then etend it to point F so that C = F. Finall, ou can draw a vertical line through F intersecting line n at point E. Mark the figure to show right angles and congruent segments. m E n C F Since line m and line n have the same slope, _ C = _ F. ut F = C, so b substitution, _ C = _ C. Multipling both sides b C shows that C =. Now ou can conclude that C b the Triangle Congruence Theorem. CPCTC, C. Line m and line n are two lines that are cut b a transversal so that a pair of corresponding angles are congruent. You can conclude that. Reflect 1. Eplain wh the slope criteria can be applied to horizontal lines.. Eplain wh the slope criteria cannot be applied to vertical lines even though all vertical lines are parallel. Houghton Mifflin Harcourt Publishing Compan Module 56 Lesson 1

Eplain 1 Using Slopes to Classif Quadrilaterals b Sides You can use the slope criteria for parallel lines to analze figures in the coordinate plane.

3 Eplain 1 Using Slopes to Classif Quadrilaterals b Sides You can use the slope criteria for parallel lines to analze figures in the coordinate plane. Eample 1 Show that each figure is the given tpe of quadrilateral. Show that C is a trapezoid. Step 1 Find the coordinates of the vertices of quadrilateral C. ( 1, 1), (, 3), C (3, 1), ( 3, 3) Step Use the slope formula to find the slope of _ and the slope of _ C. slope of _ = _ = _ (-1) = _ C slope of C _ - 1 = _ - = _ 1 - (-3) (-3) = _ 6 = _ 3 Step 3 Compare the slopes. Since the slopes are the same, _ is parallel to _ C. Quadrilateral C is a trapezoid because it is a quadrilateral with at least one pair of parallel sides. Show that PQRS is a parallelogram. Houghton Mifflin Harcourt Publishing Compan Image Credits: uildpi/ Construction Photograph/lam P - - S Q R Step 1 Find the coordinates of the vertices of quadrilateral PQRS. P (-3, ), Q (1, ), R (, ), S (, ) Step Use the slope formula to find the slope of each side. _ - 1 PQ : _ - = _ (-3) = _ - = QR : _ - = _ = _ = 1-1 _ RS : _ - = _ = _ = - SP :_ = -_ = _ = Module 57 Lesson 1

4 Step 3 Compare the slopes. Since the slope of PQ is _ the same as the slope of _, PQ is parallel to. Since the slope of QR is _ the same as the slope of _, QR is parallel to. Quadrilateral PQRS is a parallelogram because. \ Reflect 3. What If? Suppose ou know that the lengths of _ PQ and _ QR in the figure in Eample 1 are each _ 0. What tpe of parallelogram is quadrilateral PQRS? Eplain. Your Turn Show that each figure is the given tpe of quadrilateral.. Show that JKLM is a trapezoid. J M - - L 0 - K - 5. Show that C is a parallelogram C Houghton Mifflin Harcourt Publishing Compan Module 58 Lesson 1

5 Eplain Using Slopes to Find Missing Vertices Eample Find the coordinates of the missing verte in each parallelogram. C with vertices (1, ), (, 3), and (5, 1) Step 1 Graph the given points. Step Find the slope of _ b counting units from to. The rise from - to 3 is 5. The run from 1 to - is Step 3 Start at and count the same number of units. rise of 5 from 1 is. run of 3 from 5 is. Label (, ) as verte C. Step Use the slope formula to verif that _ C slope of C _ = _ (-) = _ 1 slope of _ -1 - (-) = _ = _ The coordinates of verte C are (, ). _ C PQRS with vertices P ( 3, 0), Q (, ), and R (, ) Step 1 Graph the given points. Step Find the slope of _ PQ b counting units from Q to P. The rise from to 0 is. The run from to 3 is. Houghton Mifflin Harcourt Publishing Compan Step 3 Start at R and count the same number of units. rise of from is. run of from is. Label (, ) as verte S. Step Use the slope formula to verif that _ QR slope of _ QR = - _ - _ PS. = - _ slope of PS _ - = _ = - _ The coordinates of verte S are (, ). Module 59 Lesson 1

6 Reflect 6. iscussion In Part, ou used the slope formula to verif that C. escribe another wa ou can check that ou found the correct coordinates of verte C. Your Turn Find the coordinates of the missing verte in each parallelogram. 7. JKLM with vertices J ( 3, ), K (0, 1), and M (1, 3) 8. EFG with vertices E (, ), F (, 1), and G (3, ) Elaborate 9. Suppose ou are given the coordinates of the vertices of a quadrilateral. o ou alwas need to find the slopes of all four sides of the quadrilateral in order to determine whether the quadrilateral is a trapezoid? Eplain. 10. student was asked to determine whether quadrilateral C with vertices (0, 0), (, 0), C (5, 7), and (0, ) was a parallelogram. Without plotting points, the student looked at the coordinates of the vertices and quickl determined that quadrilateral C could not be a parallelogram. How do ou think the student solved the problem? Houghton Mifflin Harcourt Publishing Compan 11. Essential Question Check-In What steps can ou use to determine whether two given lines on a coordinate plane are parallel? Module 60 Lesson 1

7 Evaluate: Homework and Practice 1. Jodie draws parallel lines p and q. She sets up a figure as shown to prove that the lines must have the same slope. First she proves that JKL RST b the S Triangle Congruence Theorem. What should she do to complete the proof? Online Homework Hints and Help Etra Practice K p S q J R L T Show that each figure is the given tpe of quadrilateral.. Show that C is a trapezoid C - 3. Show that KLMN is a parallelogram. L Houghton Mifflin Harcourt Publishing Compan K N M Module 61 Lesson 1

8 Find the coordinates of the missing verte in each parallelogram. Use slopes to check our answer.. C with vertices (3, 3), ( 1, ), and (5, 1) 5. STUV with vertices S (-3, -1), T (-1, 1) and V (0, 0) Show that quadrilateral C is not a trapezoid. 7. Show that quadrilateral FGHJ is a trapezoid, but is not a parallelogram. F G C J H Houghton Mifflin Harcourt Publishing Compan Module 6 Lesson 1

9 etermine whether each statement is alwas, sometimes, or never true. Eplain our reasoning. 8. If quadrilateral C is a trapezoid and the slope of is 3, then the slope of C is parallelogram has vertices at (0, 0), (, 0), (0, ), and at a point on the line =. 10. If the slope of PQ is _ 1 3 and the slope of RS _ is - 1, the quadrilateral PQRS is a 3 parallelogram. 11. If line m is parallel to line n and the slope of line m is greater than 1, then the slope of line n is greater than If trapezoid JKLM has vertices J (-, 1), K ( 3, 3), and L ( 1, ), then the coordinates of verte M are (, ). Houghton Mifflin Harcourt Publishing Compan Module 63 Lesson 1

10 Eplain whether the quadrilateral determined b the intersections of the given lines is a trapezoid, a parallelogram, both, or neither Line Equation Line l = + 3 Line m = + 6 Line n = - 3 Line p + = Line Equation Line l = + 3 Line m - = 0 Line n + = 6 Line p = Line Equation Line l = + Line m + 5 = Line n - + = Line p + = Line Equation Line l 3 + = Line m + 3 = 0 Line n = Line p = 3 lgebra Find the value of each variable in the parallelogram. 17. p q w z - 5 z + 1 w + 3 Houghton Mifflin Harcourt Publishing Compan Module 6 Lesson 1

0. Use the slope-intercept form of a linear equation to prove that if two lines are parallel, then the have the same slope. (Hint: Use an indirect proof.

11 0. Use the slope-intercept form of a linear equation to prove that if two lines are parallel, then the have the same slope. (Hint: Use an indirect proof. ssume the lines have different slopes, m 1 and m. Write the equations of the lines and show that there must be a point of intersection.) 1. Critique Reasoning Maumi was asked to determine whether quadrilateral RSTU is a trapezoid given the vertices R (, 3), S (1, ), T (1, -), and U (, 1). She noticed that the slopes of _ RU and _ ST are undefined, so she concluded that the quadrilateral could not be a trapezoid. o ou agree? Eplain.. Kaitln is planning the diagonal spaces for the parking lot at a mall. Each space is a parallelogram. Kaitln has alread planned the spaces shown in the figure and wants to continue the pattern to draw the net space to the right. What are the endpoints of the net line segment she should draw? Eplain our reasoning. Houghton Mifflin Harcourt Publishing Compan Image Credits: WC Images/lam Module 65 Lesson 1

3. Multi-Step Two carpenters are using a coordinate plane to design a tabletop in the shape of a trapezoid. The have alread drawn the two sides of the tabletop shown in the figure.

12 3. Multi-Step Two carpenters are using a coordinate plane to design a tabletop in the shape of a trapezoid. The have alread drawn the two sides of the tabletop shown in the figure. The want side to lie on the line =. What is the equation of the line on which side _ C will lie? Eplain our reasoning. = - C Quadrilateral PQRS has vertices P( 3, ), Q( 1, ), and R(5, 0). For each of the given coordinates of verte S, determine whether the quadrilateral is a parallelogram, a trapezoid that is not a parallelogram, or neither. Select the correct answer for each lettered part. a. S (0, 0) Parallelogram Trapezoid but not Neither parallelogram b. S (3, ) Parallelogram Trapezoid but not Neither parallelogram c. S (, 1) Parallelogram Trapezoid but not Neither parallelogram d. S (6, -) Parallelogram Trapezoid but not Neither parallelogram e. S (5, 3) Parallelogram Trapezoid but not Neither parallelogram Houghton Mifflin Harcourt Publishing Compan Image Credits: Jetta Productions/lend Images/Corbis Module 66 Lesson 1

H.O.T. Focus on Higher Order Thinking 5. Eplain the Error Tariq was given the points P (0, 3), Q (3, 3), R (0, -), and S (, 1) and was asked to decide whether quadrilateral PQRS is a trapezoid.

13 H.O.T. Focus on Higher Order Thinking 5. Eplain the Error Tariq was given the points P (0, 3), Q (3, 3), R (0, -), and S (, 1) and was asked to decide whether quadrilateral PQRS is a trapezoid. Eplain his error. slope of SP = _ 3 - (-1) _ 0 - (-) = _ = slope of QP = _ 3 - (-3) _ 3-0 = 6_ 3 = Since at least two sides are parallel, the quadrilateral is a trapezoid. 6. nalze Relationships Four members of a marching band are arranged to form the vertices of a parallelogram. The coordinates of three band members are M ( 3, 1), G (1, 3), and Q (, 1). Find all possible coordinates for the fourth band member. Houghton Mifflin Harcourt Publishing Compan Image Credits: Kell-Moone Photograph/Corbis 7. Make a Conjecture Plot an four points on the coordinate plane and connect them to form a quadrilateral. Find the midpoint of each side of the quadrilateral and connect consecutive midpoints to form a new quadrilateral. What tpe of quadrilateral is formed? Repeat the process b starting with a different set of four points. o ou get the same result? State a conjecture about our findings. Module 67 Lesson 1

14 Lesson Performance Task Suppose archeologists uncover an ancient cit with the foundations of 16 houses. The locations of the houses are as follows: (, ) (-5, 6) (3, -6) (-1, 0) (5, -8) (3, 5) (-3, 3) (0, 5) (-8, 1) (, -1) (1, -3) (-, -3) (8, -7) (-5, -) (-, 8) (6, -) a. esides graphing the points, how could ou show that the streets are parallel? Eplain. b. re the streets parallel? Houghton Mifflin Harcourt Publishing Compan Image Credits: mitr Vasilev/P Images Module 68 Lesson 1

4.5 Equations of Parallel and Perpendicular Lines

4.5 Equations of Parallel and Perpendicular Lines Name Class Date.5 Equations of Parallel and Perpendicular Lines Essential Question: How can ou find the equation of a line that is parallel or perpendicular to a given line? Resource Locker Eplore Eploring