EXAMPLE. Step 1 Draw a ray with endpoint C. Quick Check


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1 7. lan 7 asic onstructions Objectives o use a compass and a straightedge to construct congruent segments and congruent angles o use a compass and a straightedge to bisect segments and angles Examples onstructing ongruent egments onstructing ongruent ngles 3 onstructing the erpendicular isector Finding ngle easures 5 onstructing the ngle isector What ou ll Learn o use a compass and a straightedge to construct congruent segments and congruent angles o use a compass and a sraightedge to bisect segments and angles... nd Why o construct the bisector of an angle to illustrate angles of incidence and reflection, as in Exercise 8 onstructing egments and ngles heck kills ou ll Need GO for Help In Exercises 6, sketch each figure. 6. ee back of book. *.. GH 3.. line m 5. acute & E = 0. oint is the midpoint of E. Find E Use a protractor to draw a 608 angle ee back of book. 9. Use a protractor to draw a 08 angle. New Vocabulary construction straightedge compass perpendicular lines angle bisector perpendicular bisector Lessons 5 and 6 ath ackground onstruction methods are justified by postulates such as Euclid s Fourth ostulate, that a circle can be drawn with any center and any positive radius, and by, for example, triangle congruency theorems. ompassandstraightedge constructions provide a handson introduction to these postulates and theorems. In a construction you use a straightedge and a compass to draw a geometric figure. straightedge is a ruler with no markings on it. compass is a geometric tool used to draw circles and parts of circles called arcs. Four basic constructions involve constructing congruent segments, congruent angles, and bisectors of segments and angles. ELE onstructing ongruent egments onstruct a segment congruent to a given segment. Given: onstruct: so that > ore ath ackground: p. tep raw a ray with endpoint. Lesson lanning and esources ee p. E for a list of the resources that support this lesson. oweroint ell inger ractice heck kills ou ll Need For intervention, direct students to: Finding egment Lengths Lesson 3: Example Extra kills, Word roblems, roof ractice, h. Finding ngle easures Lesson : Examples and 5 Extra kills, Word roblems, roof ractice, h.. uick heck hapter ools of Geometry tep Open the compass to the length of. tep 3 With the same compass setting, put the compass point on point. raw an arc that intersects the ray. Label the point of intersection. > pecial Needs L For Example 3 on constructing a perpendicular bisector, ask: Why does the opening of the compass need to be greater than half the length of the segment? the arcs must intersect Use a straightedge to draw. hen construct so that =. learning style: verbal elow Level L emonstrate the construction steps in Examples,, 3, and 5 by using a large demonstration compass. hen ask student volunteers to demonstrate the constructions. learning style: visual
2 ELE onstructing ongruent ngles. each For: onstruction ctivity Use: Interactive extbook, 7 onstruct an angle congruent to a given angle. Given: & onstruct: & so that & > & tep raw a ray with endpoint. tep With the compass point on point, draw an arc that intersects the sides of &. Label the points of intersection and. tep 3 With the same compass setting, put the compass point on point. raw an arc and label its point of intersection with the ray as. Guided Instruction eaching ip If students have problems using a compass and straightedge, have them work in pairs to share the construction steps. ELE Error revention Have students make sure that their compasses are not so loose that they slip during use and make an incorrect arc. oint out that some compasses can be tightened. tep Open the compass to the length. Keeping the same compass setting, put the compass point on. raw an arc to locate point. tep 5 raw. oweroint dditional Examples onstruct W congruent to K. K uick heck & > & onstruct &F with m&f = m&. onstruct & so that & &G. W F onstructing isectors erpendicular lines are two lines that intersect to form right angles. he symbol ' means is perpendicular to. In the diagram at * * * the right, ' and '. * perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments. G ealworld onnection erpendicular hands signal ime out. s you will learn in hapter 5, there is just one line that is the perpendicular bisector of a segment in a given plane. Here is a way to construct the perpendicular bisector. Lesson 7 asic onstructions 5 dvanced Learners L Have students investigate ancient and modern attempts to trisect an angle. learning style: verbal English Language Learners ELL Help students understand the difference between drawing and constructing. Emphasize that when drawing, a ruler and protractor can be used. In constructions, only a compass and straightedge can be used. learning style: verbal 5
3 Guided Instruction eaching ip oint out that the symbol for perpendicular resembles perpendicular lines. sk: What other symbol resembles what it represents? ample: the symbol for parallel 3 ELE onnection to Language rts he word bisector contains the prefix bi, which means two. oweroint dditional Examples 3 Open the compass less than in step of Example 3. Explain why the construction is not possible. ample: When the opening is less than, the arcs drawn in steps and do not intersect. o, the perpendicular bisector cannot be drawn. W bisects &W. m&w = x and m&w = x  8. Find m&w. 3 5 onstruct, the bisector of &. esources aily Notetaking Guide 7 L3 aily Notetaking Guide 7 dapted Instruction L ealworld onnection areers rchitects use construction tools to work with their designs. uick heck uick heck 3 ELE onstructing the erpendicular isector onstruct the perpendicular bisector of a segment. Given: onstruct: so that ' at the midpoint of. tep ut the compass point on point and draw a long arc as shown. e sure the opening is greater than. tep With the same compass setting, put the compass point on point and draw another long arc. Label the points where the two arcs intersect as and. *. tep 3 raw he point of intersection of and is, the midpoint of. * * ' at the midpoint of, so is the perpendicular bisector of. 3 raw. onstruct its perpendicular bisector. n angle bisector is a ray that divides an angle into two congruent coplanar angles. Its endpoint is at the angle vertex.within the ray, a segment with the same endpoint is also an angle bisector.ou may say that the ray or segment bisects the angle. ELE Finding ngle easures lgebra KN bisects &JKL so that m&jkn = 5x  5 and m&nkl = 3x + 5. olve for x and find m&jkn. m&jkn = m&nkl efinition of angle bisector 5x  5 = 3x + 5 ubstitute. 5x = 3x + 30 dd 5 to each side. x = 30 ubtract 3x from each side. x = 5 ivide each side by. m&jkn = 5x  5 = 5(55 = 50 ubstitute 5 for x. m&jkn = 50 * * * Find m&nkl and m&jkl. 50; 00 J N K L losure 6 hapter ools of Geometry Write steps in your own words for bisecting an angle and bisecting a segment. Use drawings that show the arcs in each step. heck students work. 6 uick heck. 5. a. Z Exercises.. V W
4 uick heck Example (page Example (page 5 Example 3 (page 6 Example (page 6 5 onstruct the bisector of an angle. onstructing the ngle isector Given: & onstruct:, the bisector of & tep ut the compass point on vertex. raw an arc that intersects the sides of &. Label the points of intersection and. tep ut the compass point on point and draw an arc. With the same compass setting, draw an arc using point. e sure the arcs intersect. Label the point where the two arcs intersect as. tep 3 raw. is the bisector of &. 5 a. raw obtuse &Z. hen construct its bisector. ee margin, p. 6. b. Explain how you can use your protractor to check your construction. easure l and lz to see that they are O. x ELE EEIE For more exercises, see Extra kill, Word roblem, and roof ractice. ractice and roblem olving GO ractice by Example for Help In Exercises 8, draw a diagram similar to the given one. hen do the construction. heck your work with a ruler or a protractor. 8. ee margin, pp onstruct congruent to.. onstruct VW so that VW =. 3. onstruct E so that E = +.. onstruct J so that J = onstruct & so that & > &. 6. onstruct &F so that m&f = m&. 7. onstruct the perpendicular bisector of. 8. onstruct the perpendicular bisector of. 9. lgebra GH bisects &FGI. a. olve for x and find m&fgh. ; 30 b. Find m&hgi. 30 c. Find m&fgi. 60 G F (3x  3 H (x  I Lesson 7 asic onstructions 7 3. ractice ssignment Guide 6, 5, , 93, 3 7, 6, 58, 33 hallenge est rep 370 ixed eview 50 Homework uick heck o check students understanding of key skills and concepts, go over Exercises,, 8, 6, 8. echnology ip Have students investigate what software can model compass and straightedge constructions. ome programs use a mouse and pointer to model the action of compass, straightedge, and pencil. Exercises If necessary, discuss ways that students can copy the segment lengths. G Enrichment Guided roblem olving eteaching dapted ractice ractice earson Education, Inc. ll rights reserved. Name lass ate ractice 7 Find the area of each rectangle with the given base and height. erimeter, ircumference, and rea. base: 3 ft. base: 60 in. 3. base: m height: in. height:.5 yd height: 0 cm Find the circumference of each circle in terms of π Find the perimeter and area of each rectangle with the given base and height. 7. b = 7 cm, h = 6 cm 8. b = cm, h = cm 9. b = in., h = 0.5 in. 0. b = 7 ft, h = 3 ft. b = m, h = 9 m. b = 3 m, h = 7 m Find the perimeter and area of each figure. ll angles in the figures are right angles Find the area of each circle in terms of π Find the area and perimeter of rectangle with vertices (3, 7, (9, 7, (9, , and (3, Find the perimeter of with vertices (, 9, (7, 3, and (, 3.. he circumference of a circle is 6p. Find the diameter and the radius p 8 L L 7 L3 L L F E. J 7
5 Visual Learners Exercises 0 Have students draw each angle and its bisector before beginning to solve the exercise. onnection to lgebra Exercise 5 his exercise is important to assign. iscuss why both Lani and enyse are correct. Exercise 3 ome students may think the answer choices are the same. Encourage them to read the answer choices carefully looking for how each choices differs. 3.. Example 5 (page 7 pply our kills x lgebra For Exercises 0, bisects &. olve for x and find m&. 0. m& = 5x, m& = 3x + 0 5; 50. m& = x , m& = 5; 8. m& = x  6, m& = x + 6 ; raw acute &. hen construct its bisector. ee margin.. raw right &UV. hen construct its bisector. ee margin. 5. Use your protractor and draw &W with m&w = 0. onstruct &Z > &W. hen construct the bisector of &Z. ee margin. ketch * the * figure described. Explain how to construct it. hen do the construction. 6. ' Z 6 7. ee margin. 7. bisecting right & 8. Optics beam of light and a mirror G can be used to study the behavior of light. Light that strikes the mirror is reflected so that the angle of reflection and the angle of incidence are congruent. In the diagram, is ngle of incidence ngle of reflection perpendicular to the mirror, and E & has a measure of 8. F a. Name the angle of reflection and find its measure. l; b. Find m&. 8 c. Find m&e and m&f. 9; 9 9. Use a straightedge and protractor. a b. ee back of book. a. raw a mirror and a light beam striking the mirror and reflecting from it. b. onstruct the bisector of the angle formed by the incoming and reflected light beams. Label the angles of incidence and reflection. 5. U Z 6. Find a segment on so that you can construct Z as its # bisector. Z V Z 0 W b. Infinitely many; there s only midpt. but there exist infinitely many lines through the midpoint. segment has exactly one # bisecting line because there can be only one line # to a segment at its midpt. c. here are an infinite number of lines in space that are # to a segment at its midpt. he lines are coplanar. GO nline Homework Help Visit: Hchool.com Web ode: aue OpenEnded noopy can draw squares with his compass. ou can only draw circles. ou can, however, construct a square. Explain how to do this. Use sketches if needed. hen do the construction. ee back of book.. nswer these questions about a segment in a plane. Explain each answer. a. How many midpoints does the segment have? ee back of book. b. How many bisectors does it have? How many lines in the plane are its perpendicular bisectors? b c. ee left. c. How many lines in space are its perpendicular bisectors? For Exercises, copy & and &.. ee back of book.. onstruct & so that m& = m& + m&. 3. onstruct & so that m& = m&  m&.. onstruct & so that m& = m&. 8 hapter ools of Geometry 8 7. Find a segment on so that you can construct as its # bisector. hen bisect /. U
6 5. hey are both correct. If you mult. each side of Lani s eq. by, the result is enyse s eq hallenge est rep ultiple hoice 5. easoning When bisects &, & > &. Lani claims there is always a related equation, m& = m&. enyse claims the related equation is m& = m&. Which equation is correct? Explain. 6. Writing escribe how to construct the midpoint of a segment. ee back of book. 7. onstruct a 58 angle. ee left. 8. a. raw a large triangle with three acute angles. onstruct the bisectors of the three angles. What appears to be true about the three angle bisectors? b. epeat the constructions with a triangle that has one obtuse angle. c. ake a onjecture What appears to be true about the three angle bisectors of any triangle? a c. ee back of book. Use a ruler to draw segments of cm, cm and 5 cm. hen construct each triangle, if possible. If not possible, explain ee back of book. 9. with cm, cm, and 5cm sides 30. with cm, 5cm, and 5cm sides 3. with cm, cm, and 5cm sides 3. with cm, cm, and cm sides 3 3. Impossible; the short segments are not long enough to form a n. 33. a. raw a segment,. onstruct a triangle with sides congruent to. b. easure the angles of the triangle. a c. ee margin. c. Writing escribe how to construct a 608 angle; a 308 angle. 3. ultiple hoice Which steps best describe how to construct this pattern? Use a straightedge to draw the segment and then a compass to draw five half circles. Use a straightedge to draw the segment and then a compass to draw six half circles. Use a compass to draw five half circles and then a straightedge to join their ends. Use a compass to draw six half circles and then a straightedge to join their ends. 35. a. Use your compass to draw a circle. a c. ee back of book. Locate three points,, and on the circle. b. onstruct the perpendicular bisectors of and. c. ritical hinking Label the intersection of the two perpendicular bisectors as point O. ake a conjecture about point O. 36. tudy the figures. omplete the definition of a line perpendicular to a plane: line is perpendicular to a plane if it is 9 to every line in the plane that 9. #; the line intersects 37. What must you do to construct the midpoint of a segment?. easure half its length.. easure twice its length.. onstruct an angle bisector.. onstruct a perpendicular bisector. r Line r plane. Line t is not plane. t. ssess & eteach oweroint Lesson uiz Use the figure above. For problems, check students work.. onstruct so that N.. onstruct the perpendicular bisector of. 3. onstruct & so that & &N.. onstruct the bisector of &. N bisects &N. 5. Find x Find m&n. 88 lternative ssessment Have each student construct a right triangle using the methods learned in this lesson. tudents should write a set of steps that other students could use to complete the construction. est rep x + 0 N 3x 7 esources For additional practice with a variety of test item formats: tandardized est rep, p. 75 estaking trategies, p.70 estaking trategies with ransparencies lesson quiz, Hchool.com, Web ode: aua007 Lesson 7 asic onstructions a. b. hey are all 60. c. nswers may vary. ample: ark a pt.,. wing a long arc from. From a pt. on the arc, swing another arc the same size that intersects the arc at a second pt.,. raw l. o construct a 30 l, bisect the 60 l. 9
7 38. Which of these is the first step in constructing a congruent segment? F F. raw a ray. G. raw a line. H. Label two points. J. easure the segment. hort esponse 39. Explain how to do each construction using a compass and a straightedge. a b. a. raw an acute angle, &. onstruct an angle congruent to &. ee margin. b. onstruct an angle whose measure is twice that of &. Extended esponse 0. Explain how to do each construction using a compass and a straightedge. a. ivide a segment into two congruent segments. a c. ee margin. b. ivide a segment into four congruent segments. c. onstruct a segment that is.5 times as long as a given segment. ixed eview 39. [] a. raw. With the compass pt. on swing an arc that intersects and. Label the intersections and, respectively. With the compass point on, swing a O arc intersecting. Label the intersection K. Open the compass to. With compass pt. on K, swing an arc to intersect the arc that already passes through K. Label the intersection. raw. b. With the compass open to K, put the compass point on and swing an arc intersecting. With the compass on and open to K, swing an arc to intersect the first arc. Label the intersection. raw. 50 [] one part correct 0. [] a. onstruct its # bisector. b. onstruct the # bisector. hen construct the # bisector of two new segments. GO for Help Lesson 6. Use a protractor to draw a 7 angle. ee left.. &EF is a straight angle. m&eg = 80. Find m&gef m&uv = 00 and m&vuw = 80. Find possible values of m&uw.. 0 and 80 Lesson 5 Use the number line at the right. Find the length of each segment Lesson  50 hapter ools of Geometry c. raw. o constructions as in parts a and b. Open the compass to the length of the shortest segment in part b. With the * 8. raw. Use your drawing from Exercise 8. nswer and explain. 9. re and opposite rays? 50. re and the same segment? No; they do not have the same endpt. es; they both represent a segment with endpts. and. Geometry at Work abinetmaker abinetmakers not only make cabinets but all types of wooden furniture. he artistry of cabinetmaking can be seen in the beauty and uniqueness of the finest doors, shelves, and tables. he craft is in knowing which types of wood and tools to use, and how to use them. he carpenter s square is one of the most useful of the cabinetmaker s tools. It can be applied to a variety of measuring tasks. he figure shows how to use a carpenter s square to bisect &O. Hchool.com For: Information about cabinetmaking Web ode: aub03 pt. of the compass on, swing an arc in the opp. direction from intersecting at..5 (. O First, mark equal lengths O and O on the sides GEO_3eE005ta3 of the angle. hen position the square so that = to locate point. Finally, draw O. O bisects &O. [3] explanations are not thorough [] two explanations correct [] part (a correct
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