Lesson 8: Parallel and Perpendicular Lines - EngageNY

1 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 8 GEOMETRY Lesson 8: Parallel and Perpendicular Lines 81 This work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. This file derived from This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike Unported License. Lesson 8: Parallel and Perpendicular Lines Student Outcomes Students recognize Parallel and Perpendicular Lines from slope. Students create equations for Lines satisfying criteria of the kind: Contains a given point and is Parallel / Perpendicular to a given line . Lesson Notes This Lesson brings together several of the ideas from the previous lessons.

2 In places where ideas from certain lessons are employed, these are identified with the Lesson number underlined ( , Lesson 6). Classwork Opening (5 minutes) Students begin the Lesson with the following activity using geometry software to reinforce the theorem studied in Lesson 6, which states that given points ( 1, 2), ( 1, 2), ( 1, 2), and ( 1, 2), if and only if ( 1 1)( 1 1)+( 2 2)( 2 2)=0. Construct two Perpendicular segments, and measure the abscissa (the -coordinate) and ordinate (the -coordinate) of each of the endpoints of the segments. (The teacher may extend this activity by asking students to determine whether the points used must be the endpoints.)

3 This is easily investigated using the dynamic geometry software by creating free moving points on the segment and watching the sum of the products of the differences as the points slide along the segments.) Calculate ( 1 1)( 1 1)+( 2 2)( 2 2). Note the value of the sum, and observe what happens to the sum as students manipulate the endpoints of the Perpendicular segments. NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 8 GEOMETRY Lesson 8: Parallel and Perpendicular Lines 82 This work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. This file derived from This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike Unported License.

4 Example 1 (10 minutes) Let 1and 2 be two non-vertical Lines in the Cartesian plane. 1and 2 are Perpendicular if and only if their slopes are negative reciprocals of each other ( Lesson 5). Explain to students that negative reciprocals also means the product of the slopes is 1. PROOF: Suppose 1 is given by the equation = 1 + 1, and 2 is given by the equation = 2 + 2. Start by assuming 1, 2, 1, and 2 are all not zero. Students revisit this proof for cases where one or more of these values is zero in the practice problems. Let s find two useful points on 1: (0, 1) and (1, 1+ 1). Why are these points on 1? is the -intercept, and is the -intercept plus the slope.

5 Similarly, find two useful points on 2. Name them and . (0, 2) and (1, 2+ 2). Explain how you found them and why they are different from the points on 1. We used the -intercept of 2 to find and then added the slope to the -intercept to find point . They are different points because the Lines are different, and the Lines do not intersect at those points. By the theorem from the Opening Exercise (and Lesson 5), write the equation that must be satisfied if 1 and 2 are Perpendicular . Take a minute to write your answer, and then explain your answer to your neighbor. If 1 is Perpendicular to 2, then we know from the theorem we studied in Lesson 5 and used in our Opening Exercise that this means (1 0)(1 0)+( 1+ 1 1)( 2+ 2 2)=0 1+ 1 2=0 1 2= 1.

6 Summarize this proof and its result to a neighbor ( Lesson 5). If either 1 or 2 are 0, then 1 2= 1 is false, meaning 1 2 is false, that is they cannot be Perpendicular . Students study this case later. NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 8 GEOMETRY Lesson 8: Parallel and Perpendicular Lines 83 This work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. This file derived from This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike Unported License. Discussion (Optional) Working in pairs or groups of three, ask students to use this new theorem to construct an explanation of why ( 1 1)( 1 1)+( 2 2)( 2 2)=0 when ( Lesson 6).

7 As students construct their arguments, circulate around the room listening to the progress that is being made with an eye to strategically selecting pairs or groups to share their explanations with the whole group in a manner that builds from basic arguments that may not be fully formed to more sophisticated and complete explanations. The explanations should include the following understandings: = 2 2 1 1 and = 2 2 1 1 Because we constructed the segments to be Perpendicular , we know that = 1 ( Lesson 7). = 1 2 2 1 1 2 2 1 1= 1 ( 2 2)( 2 2)= ( 1 1)( 1 1) ( 1 1)( 1 1)+( 2 2)( 2 2)=0 ( Lesson 6) Exercise 1 (5 minutes) Exercise 1 1.

8 A. Write an equation of the line that passes through the origin and intersects the line + = to form a right angle. = b. Determine whether the Lines given by the equations + = and = + are Perpendicular . Support your answer. The slope of the first line is , and the slope of the second line is . The product of these two slopes is ; therefore, the two Lines are Perpendicular . c. Two Lines having the same -intercept are Perpendicular . If the equation of one of these Lines is = + , what is the equation of the second line ? = + Scaffolding: If students are struggling, change this example to specific Lines .

9 Give the coordinates of the two Parallel Lines and the Perpendicular line . Calculate the slopes of each and compare. The teacher may also have students construct two Lines that are Parallel to a given line using dynamic geometry software and then measure and compare their slopes. NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 8 GEOMETRY Lesson 8: Parallel and Perpendicular Lines 84 This work is derived from Eureka Math and licensed by Great Minds. 2015 Great Minds. This file derived from This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike Unported License. Example 2 (12 minutes) In this example, students study the relationship between a pair of Parallel Lines and the Lines that they are Perpendicular to.

10 They use the angle congruence axioms developed in Geometry Module 4 for parallelism. Students are investigating two questions in this example. This example can be done using dynamic geometry software, or students can just sketch the Lines and note the angle pair relationships. Example 2 a. What is the relationship between two coplanar Lines that are Perpendicular to the same line ? Have students draw a line and label it . Students then construct line 1 Perpendicular to line . Finally, students construct line 2 not coincident with line 1, also Perpendicular to line . What can we say about the relationship between Lines 1 and 2?