Segments, Rays, Parallel Lines and Planes Q L R M. Segment AB. Endpoint. Ray YX. Naming Segments and Rays

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1 - egments, ays, arallel ines and lanes -. lan What You ll earn To identify segments and rays To recognize parallel lines... nd Why To identify compass directions that can be represented by opposite rays, as in xercise 6 dentifying egments and ays heck kills You ll Need O for elp udging by appearances, will the lines intersect?. no. yes. Name the plane represented by each surface of the box.. the bottom NM 5. the top 6. the front NK 7. the back 8. the left side KN 9. the right side New Vocabulary segment ray opposite rays parallel lines skew lines parallel planes K N M no esson - Objectives To identify segments and rays To recognize parallel lines xamples Naming egments and ays dentifying arallel and kew egments dentifying arallel lanes Math ackground The undefined terms point, line, and plane form the basis for the definitions of ray, segment, and parallel planes. Together these terms form the beginning vocabulary for the study of geometry. uclid used this approach in ook of The lements. eal-world onnection sunbeam models a ray. The sun is its endpoint. uick heck Many geometric figures, such as squares and angles, are formed by parts of lines called segments or rays. segment is the part of a line consisting of two endpoints and all points between them. ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint. Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line. XM ndpoint Naming egments and ays egment ndpoint Name the segments and rays in the figure at the right. The three segments are,, and. The four rays are or,,, and or. ritical Thinking and form a line. re they opposite rays? xplain. No, they do not have the same endpoint. X esson - egments, ays, arallel ines and lanes ay YX YX Y ndpoint and are opposite rays. More Math ackground: p. esson lanning and esources ee p. for a list of the resources that support this lesson. oweroint ell inger ractice heck kills You ll Need or intervention, direct students to: asic ostulates of eometry esson -: xamples, xtra kills, Word roblems, roof ractice, h. pecial Needs raw line on the board and ask: re and opposite rays? Why or why not? No, and are not opposite rays because they do not have only one endpoint in common, they share many points. learning style: visual elow evel emind students that the different notations for line, line segment, and ray readily identify and distinguish them. learning style: verbal

2 . Teach ecognizing arallel igures uided nstruction Teaching Tip oint out that the first letter naming a ray is always its endpoint. The second letter is any other point on the ray. mphasize that opposite rays are two distinct collinear rays with only their endpoints in common. Vocabulary Tip * * You read 6 as line is parallel to line. ines that do not intersect may or may not be coplanar. arallel lines are coplanar lines that do not intersect. kew lines are noncoplanar; therefore, they are not parallel and do not intersect. * * * * 6 and are skew. XM Visual earners emind students to associate the notations for line, segment, and ray with the actual figures. oweroint dditional xamples Name the segments and rays in the figure. nline egments or rays are parallel if they lie in parallel lines. They are skew if they lie in skew lines. and are skew because and are skew. XM a. Name all labeled segments that are parallel to.,, and are parallel to. * dentifying arallel and kew egments *,,, Visit: hchool.com Web ode: aue-0775 b. Name all labeled segments that are skew to. N,, and are skew to. N uided nstruction XM Tactile earners ome students may have trouble visualizing skew lines from the figure shown. rovide physical models for these students. uick heck Use the diagram in xample. a. Name all labeled segments that are parallel to., N b. Name all labeled segments that are skew to.,, c. Name another pair of parallel segments; of skew segments. N, ; N, arallel planes are planes that do not intersect. line and a plane that do not intersect are also parallel. lane 6 lane. * lane 6. hapter Tools of eometry dvanced earners ave students justify the statement, kew lines are noncoplanar; therefore they are not parallel and do not intersect. They may need to reason indirectly. learning style: verbal nglish anguage earners Use xercises 5- to reinforce the meaning of the new vocabulary in the lesson as well as the terms always, sometimes, and never in the context of mathematical reasoning. learning style: verbal

3 a. WT n VU, UT n VW, n TWVU uick heck XM dentifying arallel lanes Use the diagram at the right to name the figures. a. two pairs of parallel planes plane 6 plane plane 6 plane b. a line that is parallel to plane * is parallel to. Name the figures. a. three pairs of parallel planes b. a line that is parallel to plane UV nswers may vary. ample: W T U V oweroint dditional xamples Use the figure from xample. Name all segments that are parallel to. Name all that are skew to. parallel:,, ; skew:,,, dentify a pair of parallel planes in your classroom. ample: floor and ceiling esources aily Notetaking uide - aily Notetaking uide - dapted nstruction X or more exercises, see xtra kill, Word roblem, and roof ractice. ractice and roblem olving O ractice by xample for elp xample (page Use the figure at the right for xercises., T, W, T, W, TW. Name all the labeled segments.. Name all the labeled rays., T, TW, WT, T,. a. Name a pair of opposite rays with T as an endpoint. T or T, TW b. Name another pair of opposite rays., T T W losure ow are parallel and skew lines alike? ow are they different? oth parallel and skew lines never intersect; parallel lines are coplanar, whereas skew lines are not. xample (page xample (page 5 Name all segments shown in the diagram that are parallel to the given segment , Name all segments shown in the diagram that are skew to the given segment. 7.,, 8.,, 9., 0 and. nswers may vary. amples are given. Use the diagram above and name a pair of figures to match each description. n 0. parallel planes. a line and a plane that are parallel, xercises. nswers may vary. ample:,.,, Use the figure at the right to name * the following.. all lines that are parallel to. two lines that are skew to *. all lines that are parallel to plane 5. the intersection of plane and plane esson - egments, ays, arallel ines and lanes 5 5

4 . ractice ssignment uide -, -6 -, 7-9 hallenge 0-5 Test rep 6-50 Mixed eview 5-66 omework uick heck To check students understanding of key skills and concepts, go over xercises, 6,, 5, 8. rror revention! xercise tudents may think that T and W are opposite rays. sk: ow many points do opposite rays have in common? exactly What is it? the endpoint of both rays nrichment uided roblem olving eteaching dapted ractice ractice Name lass ate ractice - f, find the value of each of the following.. x... ind if the coordinate of is -7 and the coordinate of is ind K if the coordinate of is 7 and the coordinate of K is ind the coordinate of if = 8 and the coordinate of is ind the coordinate of X if XY = and the coordinate of Y is Name the angle at the right in three different ways. f X 5, find the value of each of the following. 9. y 0.. X ind the measure of each angle.. &. &. & 5. & 6. & 7. & 8. Name all acute angles in the figure. 9. Name all obtuse angles in the figure. 0. Name all right angles in the figure.. f = 6, find the value of x. Then find and.. f = 06, find the value of x. Then find and. x x 6 y O y X x Measuring egments and ngles M 7 x 9 earson ducation, nc. ll rights reserved. pply Your kills. alse; they intersect above pt.. Vocabulary Tip lways, sometimes, and never refer to all possible cases, not to intervals of time. O nline omework elp Visit: chool.com Web ode: aue-00 n xercises 6, describe the statement as true or false. f false, explain. * * 6. 6 true 7. 6 alse; they are skew. 8. plane 6 plane true 9. plane 6 plane alse; they intersect above. * * * * * * * * * * 0. and are skew lines. true. and are skew lines. ee left.. and are skew lines.. and are skew lines. alse; they are n. alse; they are n.. The following steps show how to draw planes and intersecting in tep tep tep Use similar steps to draw plane and plane intersecting in. ee margin. omplete xercises 5 with always, sometimes, or never to make a true statement. always never 5. Two parallel lines are 9 coplanar. 6. Two skew lines are 9 coplanar. always * * always 7. Two opposite rays 9 form a line. 8. T and T are 9 the same line. never sometimes 9. and are 9 the same ray. 0. K and are 9 the same ray.. Two planes that do not intersect are 9 parallel. always. Two lines that lie in parallel planes are 9 parallel. sometimes. Two lines in intersecting planes are 9 skew. sometimes. Multiple hoice has endpoints (-, and (,. Which point is also on? (-6, (-, (0, (6, 0 5. oordinate eometry has endpoint (, and contains (, 6. ive possible coordinates for point so that and are opposite rays. raph your answer. nswers may vary. ample: (0, 0; check students graphs. 6. irectional ompass On a directional compass, the directions north and south can be represented by opposite rays. a. Name two other compass directions that can be represented by opposite rays. b. What other pairs of opposite directions, if any, can you find? a b. ee margin. 7. Open-nded ummarize the three ways in which two lines may be related. ive examples from the real world that illustrate the relationships. ee margin. *. *. 6 6 hapter Tools of eometry 6. a. nswers may vary. ample: northeast and southwest b. nswers may vary. ample: northwest and southeast, east and west 7. Two lines can be parallel, skew, or intersecting in one point. amples: Train tracks parallel; vapor trail of a northbound jet and an eastbound jet at different altitudes skew; streets that cross intersecting 9. b. xamples may vary: ample: The floor and ceiling are parallel. wall intersects both. The lines of intersection are parallel.

5 8. nswers may vary. ample: kew lines cannot be contained in one plane. Therefore, they have escaped a plane. hallenge. nswers may vary. ample: V,, 8. Writing The term skew is a Middle nglish word meaning to escape. xplain how this meaning might be appropriate for skew lines. ee left. 9. ritical Thinking uppose two parallel planes and are each intersected by a third plane. a. Make a conjecture about the intersection of planes and and the intersection of planes and. The lines of intersection are parallel. b. ind examples in your classroom. ee margin. 0. a. raw a line. raw points and on the line. ow many different segments do points and determine? Name the segments. ee margin. b. raw another line. raw points,, and on the line. ow many segments do points,, and determine? Name them. ee margin. c. ontinue to draw lines, labeling one more point each time. Make a table showing the number of points and the number of segments determined. ook for and describe a pattern in the data. ee margin. d. Use your pattern to find how many segments are determined if you label 0 points on a line. 5 segments e. f you label n points on a line, how many segments can you name? Use the figure at the right for xercises and.. o planes and have other lines in common that are parallel to? xplain. ee margin. * *. Visualization re there planes that intersect planes and in lines parallel to? raw a sketch to support your answer. ee margin. The figure at the right is a pyramid.. Name three lines that intersect * at one point.. What line could be parallel to? ee left. 5. Visualization onsider a plane through V that is parallel to plane *. an a line * in that plane be parallel * to? an it intersect? an it be skew to? xplain each answer. Yes; no; yes; explanations may vary. V n(n. ssess & eteach oweroint esson uiz Use the figure below for xercises.. Name the segments that form the triangle., T, T. Name the rays that have point T as their endpoint. TO, T, T, T. xplain how you can tell that no lines in the figure are parallel or skew. The three pairs of lines intersect, so they cannot be parallel or skew. Use the figure below for xercises and 5. X T W O Test rep Multiple hoice lesson quiz, chool.com, Web ode: aua-00 Use the figure at the right for xercises ow many labeled segments are in the figure? Which ray is opposite? What is another name for? Which figure could be the intersection of two planes?. line. ray. point. segment esson - egments, ays, arallel ines and lanes 7. Name a pair of parallel planes. plane n plane XW 5. Name a line that is skew to * * * XW. or lternative ssessment rovide each student with a model of a rectangular solid, such as an empty cereal box. ave students describe how to find each of the following on the model: intersecting lines, parallel lines, skew lines, parallel planes, and intersecting planes. 0. a. one segment, b. segments,,, c. Number of points 5 6 Number of segments nswers may vary. ample: or each new point, the number of new segments equals the number of old points.. No; two different planes cannot intersect in more than one line.. yes; plane, for example 7

6 Test rep esources or additional practice with a variety of test item formats: tandardized Test rep, p. 75 Test-Taking trategies, p.70 Test-Taking trategies with Transparencies hort esponse Mixed eview 50. a. Use the diagram to explain how parallel lines and skew lines are alike and how parallel lines and skew lines are different. a b. ee margin. b. oes the diagram suggest other lines that are * * * * parallel to M, besides K,, and? xplain. K M Use this heckpoint uiz to check students understanding of the skills and concepts of essons - through -. esources rab & o heckpoint uiz O for elp esson nswers may vary. amples are given. Use the diagram for xercises 5 58 and name each geometric figure. 5. a line 5. a point * 5. the intersection of and * 5. two planes that intersect in and 55. the plane represented by the top of the box 56. the plane represented by the front of the box 57. the intersection of planes and 58. another point in plane raw the following ee margin. * 59. T NV esson - ind the next two terms in each sequence., 9 6.,.08,.6,.,., c., , -, -, -7, -, -6, c 6.,,,,, c, 65.,,,, M, c, 66. easoning aven conjectured: f you subtract a number from a given number, the result is always less than the given number. s her conjecture true? xplain. No; whenever you subtract a negative number, the answer is greater than the given number. lso, if you subtract 0, the answer stays the same. heckpoint uiz essons - through - ind the next two terms in each sequence..5678, ,.5,, 6.5, c 9,.5..,.5,.56,.567, c 50. [] a. like: They do not intersect. ifferent: arallel lines are coplanar and skew lines lie in different planes. 8 b. No; of the 8 other lines shown, intersect M and are skew to M. [] one likeness, one difference or : dd.5. or : xtend the decimal to one more place with a digit that is more than the one to its left. 8 hapter Tools of eometry. Writing escribe the pattern of each sequence in xercises and. ee left. Use the diagram for xercises 0. n xercises 7, do the points appear to be coplanar? f yes, name the plane. f no, explain. 7. ee margin.. oints,,, and 5. oints,,, and 6. oints,,, and 7. oints,,, and 8. Name all the segments parallel to.,, 9. nswers may vary. 9. Name a pair of skew lines. ample: and ee left. 0. raw a net for the figure. ee margin. N T V heckpoint uiz. yes, plane 5. yes, plane 6. No;,, and are in the front plane, is not. 7. No;,, and are in the top plane, is not. 0.