Unit D Parallel and Perpendicular Lines

Transcription

1 Baltimore Count Public Schools Unit D Essential Question Parallel and Perpendicular Lines How can vocabular and proofs associated with parallel lines improve logical and critical thinking skills? Sections D-1 Lines and Angles (Section -1) Enduring Knowledge D- Angles Formed b Parallel Lines and Transversals (Section -) D- Proving Lines Parallel (Section -) D-4 Perpendicular Lines (Section -4) D-5 Slopes of Lines (Section -5) Lines in the Coordinate Plane (Section -6) Big Ideas Transformations in the plane can be used to show congruence. Alignment D-1 Similarit transformations can be used to show similarit relationships between figures. D-1, D- D- Definitions, postulates, and theorems are used to prove theorems involving similarit. D-4 Visualizing relationships between two-dimensional and three-dimensional objects can help ou connect geometric concepts to real objects. D-5 Vocabular Prerequisite Skills Estimated Unit Length alternate eterior angles alternate interior angles parallel planes perpendicular bisector same-side interior angles skew lines transversal Solve linear equations. Coping angles using a compass and straight edge. Solve quadratic equations (honors and GT). Write basic geometric proofs including vertical angles and linear pairs. A: 16 (8) das H: 16 (8) das GT: 16 (8) das D-1

2 Baltimore Count Public Schools Tet Holt Geometr, 011 Materials Supplements Technolog Defined STEM Resources and Materials objects of varing shape (i.e. book, cone, jar) response boards dr erase markers and erasers chart paper and markers sentence strips patt paper straightedges protractors Serra, Michael. Patt Paper Geometr Acces 4 Module Chapter Resources, Volume 1 Vocabular Bingo WS D-1 Lines and Transversals Foldable, WS D-1a Drawing a Parallel, RS D- Angle Relationships - Parallel Lines GSP D-a ProofBlock Templates; Template Samples, RS D-b. Paving the Wa, RS D-4 Parallel Postulate, RS D-4a Classifing Lines, RS D-5 Classifing Lines, WS D-5a The Geometer s Sketchpad Microsoft Office PowerPoint ActivInspire The Science of Sound Production in Different Instruments (Connections), Section D- D-

3 Baltimore Count Public Schools D-1 Lines and Angles (Section -1) Objective(s) Identif parallel, perpendicular, and skew lines. Identif the angles formed b two lines and a transversal. Alignment CCSC: G.CO.9, G.CO.1 AIM: O-7 Essential Understanding Assessment Level 1 The relationships between the angles formed from coplanar lines cut b a transversal are further eplored as the coplanar lines are made parallel. Level Identif one pair of each of the following: 1. Parallel planes. Perpendicular planes. Skew lines Identif one pair of each of the following: Level [Plane PQR Plane ABC; Plane PQR Plane SRC; QB and DC 1. Corresponding angles. Alternate interior angles. Same side interior angles Level 4 [angles 1,9;,5; 7,10] t 1 4 k l m Given non-coplanar lines l, m, and n such that l m n, and line k is parallel and coplanar to line l. What is the relationship between line k and lines m and n? Given the diagram above, which of the following cannot be true? A. Line l is parallel to line m B. and 10 are corresponding angles C. 5 and 8 are same-side interior angles D. and 1 are alternate eterior angles [C] [k will be skew to one of the lines and perpendicular to the other.] D- D-1 Lines and Angles

4 Baltimore Count Public Schools Teacher Preparation Background Information Core Instructional Strategies Materials Tet Holt Geometr, 011 pp Materials objects of varing shape (i.e. book, cone, jar) response boards dr erase markers and erasers chart paper and markers sentence strips patt paper straightedges protractors Supplements Vocabular Bingo RS D-1 Lines and Transversals Foldable, RS D-1a Half Plane Model of Hperbolic Geometr Technolog Geometer s Sketchpad Teaching Suggestions Students ma struggle in geometr because of the plethora of vocabular terms the need to know. Refer the students to Stud Strateg: Take Effective Notes p. 145, for an overview of Cornell Notes. This tpe of organized note taking ma benefit the students as the learn more geometric terms, postulates, theorems, etc. Provide the students with sufficient practice in associating angle relationships. Give the opportunit to practice identifing and naming pairs of angles with parallel and non-parallel lines cut b a transversal. The students identified and described geometric relationships between the angles formed when parallel lines are cut b a transversal in grades 7 and 8. These angle pairs included alternate interior, alternate eterior, and corresponding angles. The students are familiar with the words parallel and perpendicular as well as their respective notations. The term same-side-interior angles (consecutive interior angles) will be new learning for the students. Skew lines were covered in section A-1, but should be reviewed since it is a relativel new concept. Level 1 Motivate the students b several displaing objects of varing shape, such as a ball, book, cone, jar, tissue bo, and a tube of toothpaste. Review the vocabular terms parallel segments, perpendicular segments, skew segments, parallel planes, and perpendicular plane, b inviting several students to the front of the room to point out eamples of each of these terms on the objects. Connect the terms to other real world representations with p. 150 (41). Begin a vocabular list on a piece of large poster paper for the students to reference. D-4 D-1 Lines and Angles

5 Baltimore Count Public Schools Project the rectangular prism from p. 146 for the students to view. Identif parallel and perpendicular lines and planes, modeling the correct wa to write these relationships smbolicall using and. Remind the students to write out the word skew because there is no smbol to represent the word. Give the students response boards, and call out various pairs of lines and planes (i.e. parallel, perpendicular, intersecting, skew) shown in the rectangular prism. Ask the students to write down the relationship between each pair using the correct notation on the response boards and to hold them up for assessment. Displa the diagram shown on the right. Ask the students to represent the relationship of lines j and k t smbolicall. Allow time for the students to conclude that the lines are not parallel or 1 j perpendicular and that there is no smbol for 4 intersecting onl. What is the name of line t, which 5 6 intersects lines j and k? Trace over the transversal with a different color, label it transversal, and add 8 7 k this term to the vocabular list. Write each of the names of the Angle Pairs Formed b a Transversal, p. 147, on separate sentence strips. Ask the students to list the interior angles on the left side of their response boards and the eterior angles on the right. Post onl the sentence strips with the terms alternate interior and alternate eterior on the board. Which pairs of angles do ou think are alternate interior and which are alternate eterior? Record the angles under the appropriate sentence strips. Repeat this with corresponding and same side interior angles. Add each term to the vocabular list. Present the diagram shown on the right. Ask the students to list all the A F corresponding angles on their response boards. Check for B C G I understanding as the hold up their D H boards. Repeat this process with the remaining angle relationships. (field independent, active) Provide practice identifing relationships between lines and angle pairs formed b a transversal with Vocabular Bingo - Planes, Lines, and Angles, RS D-1. Give the students a game card and a highlighter. Have the students complete the game card as eplained on the resource sheet. Project a cop of the diagrams from the game sheet for the students to view during the game. Pla the game b calling out a pair of lines, angles, or planes. Instruct the students to highlight a column that represents what is named. Continue to pla until a student has five in a row highlighted. Allow the students to trade game boards for multiple rounds of pla. (field dependent, auditor, active) D-5 D-1 Lines and Angles

6 Baltimore Count Public Schools Differentiation Strategies Level Etend the recognition of angles and transversals b presenting a diagram similar to Eample on p Name a pair of angles and ask the students to identif the transversal and the relationship of the angle pair. Repeat this with several angle pairs. Emphasize the importance of choosing the correct transversal. Switch to naming a transversal and angle relationship for which the students must identif an angle pair that fits the description. Asses the students b instructing them to displa their answers on response boards. (field independent, auditor) Level Prepare the students for proofs with -1 Practice C (1 4). Allow the students to compare diagrams and conclusions to verif correctness. Accelerate-Review-Reteach Consider using a cardboard bo separator as a three dimensional model of parallel and perpendicular planes. Use Lesson -1 Reading Strategies, Chapter Resources to reinforce the meanings of smbols and diagram markings. Encourage the students to use the color coding and letter references as shown on p. 149 (7-9). Create a life-sized diagram of two lines and a transversal on the floor. Pair the students and begin b naming an angle relationship. Have student pairs stand in the appropriate spaces. Add a second transversal when the students are read. Distribute Lines and Transversals Foldable, RS D-1a, to assist with organizing and identifing the geometr terms. Help them fill in the correct angle pairs in each bo. Enrichment-Etension Allow the students to eplore in the same wa as above using Geometer s Sketchpad instead of patt paper and a protractor. Level 5 Introduce the students to spherical geometr with Lesson -1 Challenge, Chapter Resources. Allow the students to eplore the basics of Hperbolic Geometr using The Geometer s Sketchpad Resource Center, Half Plane Model of Hperbolic Geometr at Half Plane Model of Hperbolic Geometr. Identif and name points, lines, and planes, and classif angles Identif parallel and perpendicular lines and angles formed b lines and a transversal Prove and use theorems about angles formed b parallel lines and a transversal D-6 D-1 Lines and Angles

7 Baltimore Count Public Schools D- Angles Formed b Parallel Lines and Transversals (Section -) Objective(s) Prove and use theorems about the angles formed b parallel lines and a transversal. Alignment CCSC: G.CO.9 AIM: O-7 Essential Understanding Assessment Level 1 The construction and design of man real world objects depends on the congruenc of the angles formed b parallel lines and transversals. Level If a b, state the postulate or theorem that supports the conclusion 6 8. Given CE BF, find m ABF. A a b B (6 + ) C F D (5-7) E [Corresponding angles postulate] Level Draw line q. Draw two lines r and s so the are both perpendicular and coplanar to q. Prove r s. Level 4 Given: 5 and l m Prove: l n 1 m [See SA] [See SA] D-7 D- Parallel and Perpendicular Lines

8 Baltimore Count Public Schools Teacher Preparation Background Information Materials Tet Holt Geometr, 011 pp Materials patt paper rulers protractors colored pencils transparencies transparenc pens inde cards tape Supplements Drawing a Parallel, RS D- Angle Relationships Parallel Lines, GSP D-a ProofBlock Templates, RS D-b Technolog Geometer s Sketchpad PowerPoint Teaching Suggestions As the students advance through the geometr course, the number of postulates, theorems, converses, definitions, etc. continues to grow at a stead rate. Keeping track of these statements, remembering them, and then appling them in problem-solving and proof settings is crucial for success. Encourage the students to keep an ongoing notebook of postulates, theorems, converses, definitions, etc., and allow the students to reference this notebook whenever necessar. Understanding the difference between a theorem and its converse becomes ke in this section. The distinction between the two statements needs to be made clear, so that the students are using the statements appropriatel in proofs. This distinction will need to be made even more clear in later units as the students work with parallelograms and triangles. The students identified and described geometric relationships between the angles formed when parallel lines are cut b a transversal in grades 7 and 8. These angle pairs included alternate interior, alternate eterior, and corresponding angles. The term same-side-interior angles (consecutive interior angles) was new learning in the previous section. The students applied the relationships between the angles to solve for missing angle measures in the previous middle school grades. However, working with algebraic epressions to solve for missing angle measures will be new. The students ma be comfortable identifing the vocabular and setting up the appropriate equations, but ma arrive at incorrect answers due to algebraic errors. It is important to provide algebraic eamples to help these students polish their algebra skills; et, avoid allowing the algebra to overshadow the geometr. D-8 D- Parallel and Perpendicular Lines

9 Baltimore Count Public Schools Core Instructional Strategies Level 1 Motivate the students b discussing situations in which objects must be parallel in order to function properl. What would happen if the chalk ledge was not parallel to the floor? What if the wheel ales on a car pointed in different directions? Eplain that the functionalit of the objects rel on the angle relationships associated with parallel lines. Begin the lesson b projecting Drawing a Parallel, RS D-, and present the questions at the bottom of the page. Use the responses to remind the students that the angle pair vocabular reviewed in the last section is the same when the lines are parallel. Emphasize the importance of the markings that indicate parallel lines. (field independent, visual) Distribute patt paper to pairs of students. Allow for free eploration on how to construct parallel lines using folding techniques. Instruct the students to draw a transversal to the parallel lines using a straightedge. Give each pair of students a protractor and have the students measure the special angle relationships and make a conjecture about the relationships between each tpe of angle pair. Level Open the Geometer s Sketchpad file, Angle Relationships-Parallel Lines, GSP D-a. Present the first Sketchpad page, Corresponding Angles. Ask the students to predict the relationship between the angles, based on their appearance. Ask, How will moving the transversal affect the measures of the angles? Click the Move Transversal button and let the students observe the changes. Click the button a second time to pause the movement. Poll the class for conjectures before clicking the button angle JCD. Move the transversal again and direct the students attention to the changes of the angle measures as the transversal moves. Allow the class to formulate their own theorem, as a conditional statement. Choose a volunteer to record the class s theorem on the board or on chart paper to be displaed throughout the class period. Reveal the formal postulate b clicking the button. Ask one student to read the formal statement aloud. Discuss as a class the differences between the formal postulate and the class s statement and make an necessar adjustments. Displa Eample 1A on p. 155 and ask, How can we use the Corresponding Angles Postulate to find the measure of? Displa Eample 1B and have the students work in a Think-Pair-Share stle to discuss the correct wa to set up an equation and find the measure of the angle. Invite a student to share the correct equation and eplain the process used to arrive at this equation. Present a third algebraic eample similar to 1B. Create a third eample for the Honors and GT students where the students must factor the equation to solve for the variable and missing angle measures. D-9 D- Parallel and Perpendicular Lines

10 Baltimore Count Public Schools Continue with the Geometer s Sketchpad activit, Angle Relationships- Parallel Lines, GSP D-a. Follow the same procedure to introduce alternate interior, alternate eterior, and same-side interior angles b clicking the tabs located at the bottom of the Sketchpad file. Dela clicking the Angle Sum button on the Same Side Interior Angles page. Allow the students to analze the measurements of the angle pair at an given time during the movement of the transversal. (field independent, visual) Use the patt paper activit, Eplorations Transparencies, Eploration, Alternate Openers: -, to introduce the angle relationships and corresponding theorems associated with parallel lines. Present algebraic eamples for each tpe of angle pair, incorporating factoring skills for Honors and GT. Consider using Eample on p. 157 for highl able students. Present several eercises as mied practice. Consider creating a PowerPoint Presentation with each eercise on a separate slide. Require the students to identif the postulate or theorem needed in each eercise, and then set up the appropriate equation to solve for variables and angle measures. Etend the angle relationships to applications using eercises 5, 1, 4, and 0 on pp Use Application Practice p. S0 for additional application eercises. Level Transition to proofs given parallel lines b posing the question, Which of the angle relationships could be used to prove the other three angle relationships? Remind the students that postulates are statements that are accepted as fact. Design a ProofBlock for the Corresponding Angles Postulate using ProofBlock Templates, RS D-b. Present the given and proof statements for the Alternate Interior Angles Theorem proof, p Provide a Transitive Propert ProofBlocks from ProofBlocks, RS D-5a, from the previous unit C, and the Vertical Angles ProofBlock available from the ProofBlocks Web site, Model how to write the input and output lines between the ProofBlocks to complete the proof. Invite two students to the board, one to complete a two-column proof and one to complete a flowchart proof. Compare the three proof stles so that the students can use an method when completing proofs. Assign p. 159 (5, 6), where the students complete a proof for the Alternate Eterior Angles Theorem and the Same-Side Interior Angles Theorem. Encourage the students to use an stle of proof the feel the most comfortable with. Select several students to present their proofs with the intent of having various proof stles and proof solutions demonstrated. (field independent, sequential) D-10 D- Parallel and Perpendicular Lines

11 Baltimore Count Public Schools Differentiation Strategies Level 4 Assign problems that appl the theorems and postulate from this section in a proof tpe setting, such as pp (7 9,, 6). Accelerate-Review-Reteach Reinforce the angle relationships using the Teaching Tips and the Reaching All Learners Activit, from TE, p Recall the letter references on p. 149 (7 9), and encourage the students to find these letters within the parallel lines. Displa two parallel lines cut b a transversal with all angles labeled. Give each student an inde card with a pair of angles written on it. Put the headings Congruent and Supplementar on the board. Instruct the students to tape their inde card under the correct heading. Have the class verif that all angles are placed under the correct heading and, as a class, identif the postulate or theorem that eplains their placements. Ease the students into parallel line proofs b providing several proofs with a few statements and reasons left blank. Allow the students time to become comfortable with filling in the blanks before asking them to develop a complete proof. Enrichment-Etension Present diagrams including a second or third transversal with algebraic epressions in which the students must determine the measure of multiple angles. Use p. 161 (9) as a reference. Introduce the proof of the Alternate Interior Angle Theorem b having the students draw an appropriate diagram and identif, from the formal statement, the given information and what needs to be proved. Consider assigning - Practice C to pairs of students. Allow groups of students to collaborate to complete the ten-step proof. Identif relationships of angles formed b lines and transversals Use and appl theorems about the angles formed b parallel lines and a Prove lines parallel given angle relationships D-11 D- Parallel and Perpendicular Lines

12 Baltimore Count Public Schools D- Proving Lines Parallel (Section -) Objective(s) Use the angles formed b a transversal to prove two lines are parallel. Alignment CCSC: G.CO.9 AIM: O-7 Essential Understanding Assessment Level 1 Establishing lines parallel b the converse theorems is essential in architecture, landscaping, and carpentr to ensure structures are appropriatel designed. Level Write the converse of the Corresponding Angles Postulate. Given 11, m 6 1, and m 10, show that a b. Identif the theorem used to justif our answer. a b [If two lines are cut b a transversal so that corresponding angles are congruent, then the two lines are parallel.] Level Justif each step in the flowchart proof , m 6 m, a b b Alt.Int. 's Thm. Level 4 Given: m 1 m Prove: l n Given: Prove: m 8 m5 180 l m l m t l m n [See SA] [See SA] D-1 D- Proving Lines Parallel

13 Baltimore Count Public Schools Teacher Preparation Background Information Core Instructional Strategies Materials Tet Holt Geometr, 011 pp Materials Supplements Technolog uncooked spaghetti response boards dr erase markers and erasers Acces 4 Module ProofBlocks N/A Teaching Suggestions As converses of theorems and postulates are introduced, the students tend to think the content of the lesson is a repeat of the previous section. The do not differentiate between the two distinctl different statements. This error becomes evident when the students are completing proofs. To help them choose the correct postulates and/or theorems, point out that the given information corresponds to the hpothesis in a conditional statement. Often times the students have difficult moving from the given information to the second step in the proof. The students ma incorrectl assume relationships based on the lines looking parallel. Emphasize that lines are not parallel until the are proven parallel. The students identified and described geometric relationships between the angles given parallel lines cut b a transversal, in grades 7 and 8. The students applied the relationships between the angles to solve for missing angle measures in the previous middle school grades. The students did not consider the idea of the converses of the parallel line theorems and postulate. This section will be new learning for the students, although it ma not appear to be new learning to them. Continuall stress the differences between the parallel line theorems, postulate, and their converses. Level 1 Begin b reviewing the tpes of conditional statements. Present the Vertical Angles Theorem. Have the students write the theorem as a conditional statement, and then write the converse, inverse, and contrapositive. Which statements are true? Which statements will alwas have the same truth values? Point out that a conditional statement and its converse are not logicall equivalent and therefore will not alwas have the same truth values. (field independent, auditor) D-1 D- Proving Lines Parallel

14 Baltimore Count Public Schools Level Displa the Corresponding Angles Postulate, and have the students write the converse. Identif this converse as a postulate, and that it is a true statement. While postulates do not have to be proven true, how do ou know that the converse of the Corresponding Angles Postulate must be true? How can ou show this? La three pieces of uncooked spaghetti under the document camera, and invite a student up to tr and disprove the Converse of the Corresponding Angles postulate. Demonstrate how the congruenc of the corresponding angles forces the lines to be parallel. Present Eample 1B, p. 16, and discuss the steps needed to show that the lines are parallel. Allow time for the students to conclude that both corresponding angles equal 140. What can we conclude about the lines? Wh? Stress the importance of stating that the angles are congruent in order to correctl use the converse of the Corresponding Angles Postulate. Use the Additional Eamples Transparenc and Power Presentation, Eample 1, for etra practice. Split the class into three groups and assign each group a parallel line theorem. Instruct the students in each group to write the converse of their theorem and determine its truth value. Invite one student from each group to the board to write each converse. Remind the students that converses are not guaranteed true until proven so. Emphasize that the Converse of the Corresponding Angles is a postulate, and is accepted as true. Summarize the converses using a chart similar to the one on p. 16. La three pieces of uncooked spaghetti under the document camera, and invite students up to attempt to disprove the converses. Demonstrate how the congruenc or supplementar of the angle pairs force the lines to be parallel. Note that more rigorous proofs are necessar to prove converses true, and will be done later in this section. Work through more eamples that appl the converses, such as Eamples and 4, pp Remind the students that the must state that the angles are congruent or supplementar before using converses to conclude that the lines are parallel. Etend the level of difficult in the eercises b adding multiple parallel lines and transversals. Refer to pp (0 5, 46 5). Consider using response boards for a quick assessment of student understanding and to provide immediate feedback to the students. (field dependent, tactile) Level Present the ProofBlock the students created in section D- for the Corresponding Angles Theorem. How will this block need to be changed for the Converse of the Corresponding Angles Theorem? Allow the students time to build the ProofBlocks for each of the converses of the parallel line theorems. D-14 D- Proving Lines Parallel

15 Baltimore Count Public Schools Differentiation Strategies Model the proof of the Converse of the Alternate Eterior Angles Theorem using the ProofBlocks. Invite students to the board to complete a two-column proof, a paragraph proof, and a flowchart proof. Compare and contrast the proof methods, correcting an errors as a class. Guide the students through the two-column proof of Eample, p. 164, andflowchart proof for Check It Out!, p. 164 (). Assign the two-column proof on p. 167 () for the students to complete independentl. Allow the students to modif the format of the proof to a paragraph, flowchart or ProofBlock proof. Have the students partner with another student who chooses the same method of proof to verif their proof. (field dependent, sequential) Level 4 Assign pp (10, 8, 9) as independent practice with writing proofs. Consult the Acces 4 Module, Geometr Database, Parallel Lines section and Geometric Proofs section for additional practice with proofs and situations involving multiple transversals and parallel lines. Accelerate-Review-Reteach Use the Think and Discuss at the bottom of p. 165 to help the students understand the difference between the theorems and their converses. Distribute Lesson - Reading Strategies, Chapter Resource to summarize the four was to prove lines parallel. Have the students work through eercises 1 6. Provide several proofs that are partiall completed. Allow the students to practice correctl filling in the blanks before assigning independent proofs. Have the students analze proofs in which there are errors. Allow the students to work in pairs to identif the errors and rewrite the proofs correctl. Enrichment-Etension Introduce proofs with several parallel lines and transversals, such as p. 169 (4651, 54 56). Encourage critical thinking b assigning Lesson - Challenge, Chapter Resources, Volume 1. Appl postulate and theorems to write proofs given parallel lines Verif that lines are parallel using algebra and formal proofs Eplore and prove relationships involving perpendicular lines D-15 D- Proving Lines Parallel

16 Baltimore Count Public Schools D-4 Perpendicular Lines (Section -4) Objective(s) Prove and appl theorems about perpendicular lines. Alignment CCSC: G.CO.9, G.CO.1 AIM: O-7 Essential Understanding Assessment Level 1 The relationships of two lines to a third line can be used to determine if the two lines are parallel or perpendicular to each other. Level Name the shortest segment from point V to WZ. Find and. V W X Y Z () (10 + 4) VY 0, 15 Level Use construction techniques to verif each theorem below: 1. Through a point not on a line, there is one and onl one line perpendicular to the given line. Level 4 Given: Prove: m 1 m 180, c a b a a. Through a point on a line, there is one and onl one line perpendicular to the given line. d 1 b c [Construction tools ma var; See SA] [See SA] D-16 D-4 Perpendicular Lines

17 Baltimore Count Public Schools Teacher Preparation Background Information Core Instructional Strategies Materials Tet Holt Geometr, 011 pp Materials compass straightedge patt paper Supplements Paving the Wa, RS D-4 Parallel Postulate, RS D-4a Serra, Michael. Patt Paper Geometr Acces 4 Module Technolog Geometer s Sketchpad Teaching Suggestions Read Math Background TE p. 146B for a historical development of the Parallel Postulate. As done in section D-4 of this curriculum, this Math Background shows the connections between the Parallel Postulate, perpendicularit, and constructions. Review this reading as a professional learning guide to gain insight on how to effectivel present these interrelated concepts to the students. In Grade 7, the students learned how to construct the perpendicular bisector of a given line segment. In Grade 8, the students constructed segments perpendicular to a given segment through a given point. The students also copied angles using a compass and straightedge in order to construct congruent triangles. Level 1 Present the scenario from Paving the Wa, RS D-4. Provide the students with construction tools such as protractor, straightedge, patt paper, and Mira. Let the students to puzzle through the situation without providing guidance or instructions. Divide the students into pairs to share their approach to the construction. Facilitate a class discussion outlining the possible constructions. Bring about the various techniques for constructing parallel lines, using patt paper and/or a compass, pp. 16, Make connections to translations, pointing out that the angles translate along the horizontal line to create congruent corresponding angles. Which of the converses discussed earlier support our constructions, and prove that the lines constructed are definitel parallel? Etend the students understanding of constructions and parallel lines with the Parallel Postulate, RS D-4a. Note that the Parallel Postulate guarantees that for an line l, a parallel line can alwas be constructed through a point that is not on l. Wh can there be onl one parallel line through the point? Which converse supports this postulate and the construction? D-17 D-4 Perpendicular Lines

18 Baltimore Count Public Schools Transition to perpendicular lines b modifing the parking lot eample to include parking spaces that are perpendicular to the given horizontal line. How can we be sure that the parking lines alread created are parallel? Guide the students to the conclusion that the right angles formed b the perpendicular lines are congruent corresponding angles, which makes the lines parallel. Have the students use this application to develop the theorem, If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. Introduce the perpendicular bisector construction as a tool that can be used to create parallel lines. Guide the students through the construction as shown on p. 17. Connect this to the patt paper construction the students used in Unit A. Ask, How could we add to this construction to get parallel lines? Demonstrate the additional steps necessar to construct two parallel lines with a perpendicular transversal, p (field independent, tactile, active) Have the students complete Open Investigations..5 from Patt Paper Geometr, to discover the Parallel Postulate and various perpendicular line theorems. Use the results of the constructions to develop the various perpendicular theorems in the Know It Note at the top of p. 17. Present the hpotheses of each theorem, and have the students make the conclusions based on their constructions. Give the students a blank chart in which to record and summarize the theorems. Distribute patt paper and instruct the students to complete Perpendicular Lines Eploration -4. Summarize the activit and appl the definition of the distance from a point to a line using Eample 1 and Check it Out p. 17 (1). Level Present the given information and diagram from the Perpendicular Transversal Theorem Proof on p. 17. What is the shortest segment from BC to DE? Is there enough information to conclude that AB DE? Informall prove that AB DE. Model Eample on p. 17 to introduce the students to formal proofs involving perpendicular lines. Present p. 175 (4, 8) and allow the students to work with the person beside them to complete the proof. (field dependent, sequential) Level 4 Give the students the opportunit to write a proof using onl the given information with Lesson -4 Additional Eample, Check it Out () on p. 17, eercise on p. 176, and eercise 18 on p Consult the Acces 4 Module, Geometr Database, Geometric Proofs section for additional proofs involving perpendicular lines. D-18 D-4 Perpendicular Lines

19 Baltimore Count Public Schools Differentiation Strategies Accelerate-Review-Reteach Help the students complete the graphic organizer from Think and Discuss on p. 174 to help the students translate the theorems from words to smbols. Reinforce solving inequalities and the definition of the distance between a point and a line with Lesson -4 Reteach, Chapter Resources. Create several short proofs and place the diagram with the statements and reasons in an envelope. Distribute one envelope to each student and instruct them to put the statements in logical order and match each statement with the appropriate reason. Check their answers before allowing them to switch envelopes with another student. Review basic vocabular and the converses from the previous section using eercises on p Enrichment-Etension Introduce the circumcenter of a triangle using the Lesson -4 Challenge, Chapter Resources. Consider allowing the students to complete the previous activit using Geometer s Sketchpad, instead of a compass and straightedge. Prove lines parallel given angle relationships Appl theorems relating parallel and perpendicular lines Identif parallel and perpendicular lines from their slopes D-19 D-4 Perpendicular Lines

20 Baltimore Count Public Schools D-5 Slopes of Lines, Lines in the Coordinate Plane (Sections -5, -6) Objective(s) Find the slope of a line. Use slopes to identif parallel and perpendicular lines. Graph lines and write their equations in slope-intercept and point-slope form. Classif lines as parallel, intersecting, or coinciding. Alignment CCSC: G.GPE.5 AIM: O-8 Essential Understanding Assessment Level 1 Understanding the slopes of parallel and perpendicular lines is essential for completing coordinate proofs. Level Determine whether lines with the given slopes represent parallel lines or perpendicular lines and and 4 [parallel, perpendicular] Write the equation of the line with slope of through the point (0, -1), in slope-intercept form. 1 Level Find the equation of the line perpendicular to the line 5through the given point P(, ). 1 4 Teacher Preparation Materials Tet Holt Geometr, 011 pp Materials Supplements response boards dr erase markers and erasers patt paper coordinate grid equation posters Power Presentations with PowerPoint Chapter Resources, Volume 1 Classifing Lines, RS D-5 Classifing Lines Eploration Sheet, WS D-5a Classifing Lines Verification Sheet, WS D-5b D-0 D-5 Slopes of Lines, Lines in the Coordinate Plane

21 Baltimore Count Public Schools Background Information Core Instructional Strategies Technolog Microsoft Office PowerPoint ActivInspire Student Response Sstem Teaching Suggestions Sections -5 and -6 have been combined to form the guide section D-5. The students have studied the majorit of the content in this section in previous mathematics courses. The intent of this section is to refresh the students knowledge, not reteach it. Calculating slope and writing equations of lines are covered in previous algebra courses. B the end of the eighth grade, all students have identified the slope and - intercept of lines when given a graph. Students ma still struggle with finding the slope and writing the equation of a line. All calculations and graphing should be done using the graphing calculator. Avoid calculations and graphing b hand since this content is a review.note that the work in this section is to prepare the students for coordinate proofs in later sections, such as Section E-7 (Introduction to Coordinate Proofs) and Unit G (Polgons and Quadrilaterals). Understanding the slopes of parallel and perpendicular lines is essential for this future work with coordinate proofs. Level 1 Review the slope formula with Power Presentations Warm Up -5. Do these equations look familiar? Does anone remember what this equation represents? Distribute response boards to the students. Use the Additional Eamples -5 Power Presentation to assess the students proficienc calculating slope. Summarize the tpes of slopes b displaing onl the graphs from the graphic organizer on p. 18, and have the students appropriatel name the tpe of slope shown. (field independent, global) Level Have the students find the slope of each line in Eample A, p What does it mean when lines have equal slopes? Have the students sketch the graphs of the lines for a visual reminder that parallel lines have equal slopes. Displa Eample B, p. 184, and ask the students to identif the relationship between the lines using onl their slopes. How can ou tell what kind of lines the are if their slopes are not the same? Does anone remember how to tell if lines are perpendicular b their slopes? Distribute pieces of patt paper and a coordinate grid. Have the students construct a pair of perpendicular lines. La the patt paper on top of a coordinate grid so that one of the lines has a positive slope. Have the students identif two points on each line, and calculate the slopes of the lines. Discuss the relationship between the slopes as a D-1 D-5 Slopes of Lines, Lines in the Coordinate Plane

22 Baltimore Count Public Schools Differentiation Strategies class. Etend the concept of opposite reciprocals to the Perpendicular Lines Theorem. Require the students to multipl the slopes of the lines on the patt paper to verif that their product is -1. Use Check it Out p. 184 (a, b, c) to reinforce the concepts reviewed. Transition to writing equations of lines with Power Presentations Warm Up -6. What information is necessar to use the equation m b? What is the name of this equation? Review the other forms of linear equations using the Know it Note on p Review writing equations of lines, graphing lines, and classifing pair of lines using Eamples 1 on pp Allow the students to perform all calculations and all graphing using the graphing calculator. Avoid calculations and graphing b hand since this content is a review from previous mathematics courses. Assess the students with Power Presentations, Additional Eamples 1. (field independent, tactile) Merge the Power Presentations into an ActivInspire Flipchart Use the Student Response Sstem to assess student proficienc. Use the graphing calculator to graph linear equations, and to eplore the slopes of parallel and perpendicular lines using Eplore Parallel and Perpendicular Lines, Technolog Lab -6, pp Level Provide independent practice using Classifing Lines, RS D-5. Transfer the equations from Section A onto green sheets of paper and the equations from Section B onto white sheets of paper. Post the equations around the room in random order. Instruct the students to begin at a sheet of green paper and to find the slope and the intercept of the line printed on that paper. Direct the students to then go to at least four white sheets of paper to find lines that are perpendicular, parallel, intersecting, and coinciding with the line from the green paper the chose. Make clear to the students that the ma need to visit several white papers, and find the slope and -intercept of several equations, before finding the four lines that satisf the required given conditions. Require the students to show the work for each equation tested on Classifing Lines Eploration Sheet, WS D-5a, and to graph the pairs of lines as indicated on Classifing Lines Eploration Sheet, WS D-5b. (field dependent, active) Accelerate-Review-Reteach Ask, What movements are necessar for ou to leave our seat? Connect this to the slope ratio b emphasizing the importance of rising from our chair before taking a step or running. D- D-5 Slopes of Lines, Lines in the Coordinate Plane

23 Baltimore Count Public Schools Use Lesson -5 Practice A in Chapter Resources, for a review of slope basics. Assign Lesson -6 Practice B in Chapter Resources for additional review of point-slope and slope-intercept forms of equations. Review and relate concepts from Algebra using Connecting Geometr to Data Analsis, Scatter Plots and Lines of Best Fit on p Enrichment-Etension Use the Lesson -5 Challenge, Chapter Resources, Volume 1 to connect slopes to classifing coordinate quadrilaterals. Divide the class into small groups and assign eercises 6-66 on p Have each group present their solution to eercises 6 and 64. Appl theorems involving parallel and perpendicular lines Use slopes to identif parallel and perpendicular lines Write the equations of lines perpendicular or parallel to a given line D- D-5 Slopes of Lines, Lines in the Coordinate Plane

24 Vocabular BINGO - Planes, Lines, and Angles Directions: Fill in the game board with words from the Word Bo. Words ma be repeated. Word Bo Words for Lines Column Words for Planes Column Words for Angles Columns Pairs for Name Me Column Skew Parallel Same-Side Interior 4 and 8 4 and 5 Parallel Perpendicular Alternate Eterior and 7 and 6 Perpendicular Corresponding and 6 and 5 // Alternate Interior 1 and 5 4 and 6 // and 8 1 and 7 GAME BOARD LINES PLANES ANGLES NAME ME RS D-1 D-4

25 Vocabular BINGO - Planes, Lines, and Angles Use the diagrams below to choose the correct name or pair of figures from the Game Board. Mark the corresponding bo on the Game Board with a highlighter or a slash. Five in a row wins, horizontall, verticall, or diagonall. Q R P S B C A D RS D-1 D-5

26 D-6 RS D-1a

27 D-7 RS D-1a

28 Drawing a Parallel m a k l b c Figure 1 Figure Name a pair of corresponding angles in Figures 1 and. Name a pair of alternate interior angles in Figures 1 and. How are Figure 1 and Figure similar? How are the different? RS D- D-8

29 ProofBlock Templates Postulate []// [] Corresponding Angles Postulate [] [] Theorem []// [] Alternate Interior Angles Theorem [] [] D-9 RS D-b

30 Paving the Wa Leslie has been hired to help finish the resurfacing of a parking lot. Her job is to paint the lines that designate parking spaces. Before she arrives to the job site, two spaces are painted as guidelines. How can she complete a row of eight parking spaces, and guarantee that all the lines are parallel? RS D-4 D-0

31 Parallel Postulate Through a point P not on line l, there is eactl one line parallel to l. Use construction techniques to verif the truthfulness of this postulate. D-1 RS D-4a

32 RS D-5 D- Classifing Lines Section A: (Place each equation onto a green sheet of paper. Note that all equations have a slope of.) ) ( 7 ) ( 4 1 1) ( 6 5 4) ( Section B: (Place each equation onto a white sheet of paper.) 1) ( 4 1) 6( 1 4) ( 9 ) ( 6 6 6

33 Classifing Lines Eploration Sheet M equation: slope = -intercept = Use the space below to show our work for each equation tested. Circle the smbol at the bottom of the bo to indicate the line s relation to our original line. (intersecting:, coincinding: ) Equation: Equation: Equation: slope = -int = slope = -int = slope = -int = Equation: Equation: Equation: slope = -int = slope = -int = slope = -int = Equation: Equation: Equation: slope = -int = slope = -int = slope = -int = D- RS D-5a

34 Classifing Lines Verification Sheet Directions: Graph each pair of lines as indicated below. Be sure to label the lines on each graph. 1. Graph the original line and the line parallel to the original. Equation of parallel line:. Graph the original line and the line perpendicular to the original. Equation of perpendicular line:. Graph the original line and the line intersecting the original. Equation of intersecting line: 4. Graph the original line and the line coinciding with the original. Equation of coinciding line: D-4 RS D-5b

35 Unit D: Parallel and Perpendicular Lines Supplementar Answers to Assessment Questions D-: Level Answers ma var. Sample answer: 1 s r 1. q r 1. Given q s. m1 90. Definition of Perpendicular m 90. m1 m. Substitution Definition of Congruent Angles 5. r s 5. Converse of Corresponding Angles Postulate D-: Level 4 Answers ma var. Sample answer: Statements Reasons Given. n m. Converse of Alternate Interior Angle Theorem. l m. Given Alternate Eterior Angle Theorem Transitive Propert of Congruence D-: Level 8 Vertical Thm. 5 Vertical Thm. m 8 m Def. of m 5 m Def. of m 8 m5 180 m m 180 l m Given Substitution Conv. Same-Side Int. Thm. D-5 Unit D Supplementar Answers

36 D-: Level 4 Answers ma var. Sample answer: Statements Reasons 1. m 1 m Given. m1 m 180. Linear Pair Theorem. m1 m m1 m 10. Substitution 4. m m Subtraction Propert of Equalit 5. l n 5. Converse of Corresponding Angles Postulate D-4: Level A B D-4: Level 4 Statements Reasons 1. m 1 m Given. b c. Same-Side Interior Angle Theorem. c a. Given 4. b a 4. Perpendicular Transversal Theorem D-6 Unit D Supplementar Answers