Unit 3 Quadrilaterals. Isosceles Right Triangle Reflections

Transcription

1 Triner/Instructor Notes: Qudrilterls Isosceles Right Tringles Reflections Unit 3 Qudrilterls Isosceles Right Tringle Reflections Overview: Objective: Prticipnts develop the properties of squres through reflections of isosceles right tringles. TExES Mthemtics ompetencies III.012.D. The beginning techer uses properties of congruence nd similrity to explore geometric reltionships, justify conjectures, nd prove theorems. III.013.A. The beginning techer nlyzes the properties of polygons nd their components. III The beginning techer uses geometric ptterns nd properties (e.g., similrity, congruence) to mke generliztions bout two- nd three-dimensionl figures nd shpes (e.g., reltionships of sides, ngles). III The beginning techer uses the properties of trnsformtions nd their compositions to solve problems. III.014.D. The beginning techer pplies trnsformtions in the coordinte plne. V The beginning techer pplies correct mthemticl resoning to derive vlid conclusions from set of premises. V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.019.F. The beginning techer uses pproprite mthemticl terminology to express mthemticl ides. Geometry TEKS b.2.. The student mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. c.1. The student uses numeric nd geometric ptterns to mke generliztions bout geometric properties, including properties of polygons, rtios in similr figures nd solids, nd ngle reltionships in polygons nd circles. d.2.a. The student uses one- nd two-dimensionl coordinte systems to represent points, lines, line segments, nd figures. e.2.. sed on explortions nd using concrete models, the student formultes nd tests conjectures bout the properties nd ttributes of polygons nd their component prts. e.3.a. The student uses congruence trnsformtions to mke conjectures nd justify properties of geometric figures. Geometry Module 3-1

2 Triner/Instructor Notes: Qudrilterls Isosceles Right Tringles Reflections f.1. The student uses similrity properties nd trnsformtions to explore nd justify conjectures bout geometric figures. ckground: Mterils: New Terms: Prticipnts need to hve knowledge of trnsformtions. colored pencils, esel pper, colored mrkers, centimeter ruler, trnsprency rottionl symmetry Procedures: In the Qudrilterls unit, the properties of tringles nd trnsformtions re used to develop the properties of qudrilterls. Prticipnts work the ctivity individully but verify solutions informlly with group members. Fcilitte by providing miniml prompts to help prticipnts clrify their thinking. Try not to nswer questions directly. ertrnd Russell sid, wht mtters in mthemtics is not the intrinsic nture of our terms but the logicl nture of their interreltions. The qudrilterl unit explores the interreltions of tringles nd qudrilterls, using trnsformtions s tool to construct qudrilterls. Mrk isosceles right A with the known properties in terms of ngles nd sides, using tick mrks nd colors. Verify the properties of ngles nd sides with your group members. The isosceles right tringle hs right ngle,, two congruent sides, A nd, A, nd two congruent ngles A nd, ech 45 o. Reflect A cross the line contining. Lbel imge vertices nd properties ppropritely. 1. Wht type of tringle is AA? Justify your nswer. AA is n isosceles right tringle. A = A m AA' = m A + m A' = o A A' Geometry Module 3-2

3 Triner/Instructor Notes: Qudrilterls Isosceles Right Tringles Reflections 2. Reflect AA' nd its component prts cross the line contining AA '. All of the properties of AA' pply to the reflected tringle. Lbel the properties of qudrilterl AA'' A A' ' 3. lssify the qudrilterl AA'', formed from the composite reflections of n isosceles right tringle. AA'' is squre. The figure hs four right ngles nd four congruent sides. 4. List the properties of qudrilterl AA'' in terms of the sides, ngles, digonls nd symmetry in the tble below. Possible properties follow. Sides: All four sides re congruent. Opposite sides re prllel becuse lternte interior ngles re congruent. Using symbols, A A' ' nd A ' A '. onsecutive sides re perpendiculr, becuse vertex ngles of the figure re ll right ngles, for exmple, A ' A' '. Vertex ngles: All four ngles re congruent right ngles. The vertex ngles re bisected by the digonls. Opposite ngles re congruent nd supplementry. onsecutive ngles re congruent nd supplementry. Digonls: There re two digonls. Digonls re congruent to ech other. Digonls bisect ech other t right ngles. Digonls bisect the vertex ngles of the squre. Digonls lie on two of the lines of symmetry for the figure. Geometry Module 3-3

4 Triner/Instructor Notes: Qudrilterls Isosceles Right Tringles Reflections Symmetry: There re four lines of symmetry. The digonls lie on two of the lines of symmetry. The other two lines of symmetry pss through the midpoints of the sides of the squre. 90 o (4-fold) rottionl symmetry exists. The figure cn be rotted 90 o so tht the resulting imge coincides with the originl imge. When the figure is rotted four times through 90 o, the originl vertices coincide. In generl, figure hs rottionl symmetry if there is rottion tht results in the imge superimposing on the pre-imge. Remind prticipnts to dd the new term rottionl symmetry to their glossries. To complete the ctivity, ech group drws the figure nd lists its properties on sheet of esel pper. The sheet of esel pper is plced on the wll for gllery wlk. Pirs of groups view ech other s work. Allow groups bout 5 minutes to meet nd discuss ny differences or errors on the posters. ring prticipnts together for whole clss discussion. Summrize the properties of squres. Leve the posters on the wll. At the conclusion of the qudrilterl unit, prticipnts cn compre the properties of the squre, rhombus, kite, rectngle, prllelogrm nd trpezoid. Prticipnts re performing t the vn Hiele Descriptive Level becuse they develop properties of squres. Geometry Module 3-4

5 Activity Pge: Qudrilterls Isosceles Right Tringles Reflections Isosceles Right Tringle Reflections ertrnd Russell sid, wht mtters in mthemtics is not the intrinsic nture of our terms but the logicl nture of their interreltions. The qudrilterl unit explores the interreltions of tringles nd qudrilterls, using trnsformtions s tool to construct qudrilterls. Mrk isosceles right A with the known properties in terms of ngles nd sides, using tick mrks nd colors. A Verify the properties of ngles nd sides with your group members. Reflect A cross the line contining, nd lbel imge vertices nd properties ppropritely. 1. Wht type of tringle is AA? Justify your nswer. 2. Reflect AA' nd its component prts cross the line contining AA '. All of the properties of AA' pply to the reflected tringle. Lbel the properties of qudrilterl AA''. Geometry Module 3-5

6 Activity Pge: Qudrilterls Isosceles Right Tringles Reflections 3. lssify the qudrilterl AA'', formed from the composite reflections of n isosceles right tringle. 4. List the properties of qudrilterl AA'' in terms of the sides, ngles, digonls nd symmetry in the tble below. Sides Vertex Angles Digonls Symmetry Geometry Module 3-6

7 Triner/Instructor Notes: Qudrilterls Sclene Right Tringle Reflections Sclene Right Tringle Reflections Overview: Objective: Prticipnts develop the properties of rhombi through reflections of sclene right tringles. TExES Mthemtics ompetencies III.012.D. The beginning techer uses properties of congruence nd similrity to explore geometric reltionships, justify conjectures, nd prove theorems. III.013.A. The beginning techer nlyzes the properties of polygons nd their components. III The beginning techer uses geometric ptterns nd properties (e.g., similrity, congruence) to mke generliztions bout two- nd three-dimensionl figures nd shpes (e.g., reltionships of sides, ngles). III The beginning techer uses the properties of trnsformtions nd their compositions to solve problems. III.014.D. The beginning techer pplies trnsformtions in the coordinte plne. V The beginning techer pplies correct mthemticl resoning to derive vlid conclusions from set of premises. V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.019.F. The beginning techer uses pproprite mthemticl terminology to express mthemticl ides. Geometry TEKS b.2.. The student mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. b.3.d. The student uses inductive resoning to formulte conjecture. c.1. The student uses numeric nd geometric ptterns to mke generliztions bout geometric properties, including properties of polygons, rtios in similr figures nd solids, nd ngle reltionships in polygons nd circles. d.2.a. The student uses one- nd two-dimensionl coordinte systems to represent points, lines, line segments, nd figures. e.2.. sed on explortions nd using concrete models, the student formultes nd tests conjectures bout the properties of prllel nd perpendiculr lines. e.3.a. The student uses congruence trnsformtions to mke conjectures nd justify properties of geometric figures. Geometry Module 3-7

8 Triner/Instructor Notes: Qudrilterls Sclene Right Tringle Reflections f.1. The student uses similrity properties nd trnsformtions to explore nd justify conjectures bout geometric figures. ckground: Mterils: Prticipnts need to hve knowledge of trnsformtions. colored pencils, esel pper, grph pper, colored mrkers, centimeter ruler New Terms: Procedures: Distribute the ctivity sheet. Ask prticipnts to complete the ctivity individully but informlly verify with group members. While prticipnts work, wlk round the room listening nd fcilitting. Try not to directly nswer questions, but rther provide miniml prompts to help prticipnts clrify their thinking. When the group members complete the ctivity, ech group drws the figure nd lists its properties on sheet of esel pper. The sheet of esel pper is plced on the wll for gllery wlk. Allow pirs of groups bout 5 minutes to meet nd discuss ny differences or errors on the posters. In this ctivity, sclene right tringle is reflected over the line contining one of the legs of the tringle, then the composite figure is reflected over the line contining the other leg. Predict which qudrilterl will be creted. In the middle of sheet of grph pper, drw sclene right tringle, A, with the right ngle t. Mke sure tht the legs of the tringle lie long grid lines, nd tht the vertices re locted on grid line intersections. The legs of the tringle should be 2 to 3 inches long. Mrk the sides nd ngles with known properties of sclene right tringles. Use different tick mrks to indicte non-congruency. Reflect A cross the line contining. Lbel imge vertices with prime mrks. 1. Wht type of tringle is AA? Why? Discuss with your group to mke sure there is greement. AA is n isosceles cute tringle or n isosceles obtuse tringle. The congruent legs nd congruent bse ngles re formed s result of the reflection. The bse ngles re not 45 o ngles, nd the vertex ngle t is not right ngle. Geometry Module 3-8

9 Triner/Instructor Notes: Qudrilterls Sclene Right Tringle Reflections A c c Isosceles cute tringle A A' c c Isosceles obtuse tringle A' 2. Reflect AA' nd its component prts cross the line contining AA '. Lbel the properties of qudrilterl AA'. Discuss with your group to mke sure there is greement. 3. lssify the qudrilterl AA''. Justify. Qudrilterl AA'' is rhombus becuse it hs four congruent sides. c c A A A' c c c c ' c c ' A' 4. List the properties of qudrilterl AA'' in terms of the sides, ngles, digonls nd symmetry in the tble below. Sides: All four sides re congruent. Opposite sides re prllel becuse lternte interior ngles re congruent. Geometry Module 3-9

10 Triner/Instructor Notes: Qudrilterls Sclene Right Tringle Reflections Vertex Angles: Opposite ngles re congruent. onsecutive ngles re supplementry becuse the pre-imge cute ngles re complementry. The consecutive ngles, which re composed of two sets of the cute complementry ngles, must be supplementry. Digonls: There re two digonls. Digonls re not congruent to ech other. Digonls bisect ech other. Digonls intersect t 90 o ngles. Digonls bisect the vertex ngles of the rhombus, becuse the vertex ngles were formed by reflection. Digonls re lines of symmetry of the figure, becuse they lie on the originl reflection lines. Symmetry: There re two lines of symmetry. The digonls lie on the two lines of symmetry, pssing through opposite vertices of the rhombus. ring prticipnts together for whole clss discussion. ompre the properties of the rhombus with the properties of the squre. Possible comprisons follow: oth figures hve four congruent sides. In both qudrilterls opposite vertex ngles re congruent nd consecutive vertex ngles re supplementry. In both qudrilterls the digonls bisect ech other t right ngles, nd bisect the vertex ngles. In both qudrilterls the digonls lie on lines of symmetry. As the properties of ech qudrilterl re listed, the posters remin on the wll so tht the properties of squres, kites, rectngles, prllelogrms nd trpezoids cn be compred nd contrsted. Prticipnts re performing t the vn Hiele Descriptive Level becuse properties of rhombus re being developed. In the discussion compring the properties of the squre nd rhombus prticipnts pproch the Reltionl Level. Geometry Module 3-10

11 Activity Pge: Qudrilterls Sclene Right Tringles Reflections Sclene Right Tringle Reflections In this ctivity, sclene right tringle is reflected over the line contining one of the legs of the tringle, then the composite figure is reflected over the line contining the other leg. Predict which qudrilterl will be creted. In the middle of sheet of grph pper, drw sclene right tringle, A, with the right ngle t. Mke sure tht the legs of the tringle lie long grid lines, nd tht the vertices re locted on grid line intersections. The legs of the tringle should be 2 to 3 inches long. Mrk the sides nd ngles with known properties of sclene right tringles. Use different tick mrks to indicte non-congruency. Reflect A cross the line contining. Lbel imge vertices with prime mrks. 1. Wht type of tringle is AA? Why? Discuss with your group to mke sure there is greement. 2. Reflect AA' nd its component prts cross the line contining AA '. Lbel the properties of qudrilterl AA'. Discuss with your group to mke sure there is greement. 3. lssify the qudrilterl AA''. Justify. Geometry Module 3-11

12 Activity Pge: Qudrilterls Sclene Right Tringles Reflections 4. List the properties of qudrilterl AA'' in terms of the sides, ngles, digonls nd symmetry in the tble below. Sides Vertex Angles Digonls Symmetry Geometry Module 3-12

13 Triner/Instructor Notes: Qudrilterls Sclene Acute/ Obtuse Tringles Reflections Sclene Acute/Obtuse Tringle Reflections Overview: Objective: Prticipnts develop the properties of kites through reflections of sclene cute or sclene obtuse tringles. TExES Mthemtics ompetencies III.012.D. The beginning techer uses properties of congruence nd similrity to explore geometric reltionships, justify conjectures, nd prove theorems. III.013.A. The beginning techer nlyzes the properties of polygons nd their components. III The beginning techer uses geometric ptterns nd properties (e.g., similrity, congruence) to mke generliztions bout two- nd three-dimensionl figures nd shpes (e.g., reltionships of sides, ngles). III.014.D. The beginning techer pplies trnsformtions in the coordinte plne. V The beginning techer pplies correct mthemticl resoning to derive vlid conclusions from set of premises. V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.019.F. The beginning techer uses pproprite mthemticl terminology to express mthemticl ides. Geometry TEKS b.2.. The student mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. d.2.a. The student uses one- nd two-dimensionl coordinte systems to represent points, lines, line segments, nd figures. e.2.. sed on explortions nd using concrete models, the student formultes nd tests conjectures bout the properties nd ttributes of polygons nd their component prts. e.3.a. The student uses congruence trnsformtions to mke conjectures nd justify properties of geometric figures. f.1. The student uses similrity properties nd trnsformtions to explore nd justify conjectures bout geometric figures. ckground: Mterils: Prticipnts need to hve knowledge of trnsformtions for this ctivity. colored pencils, esel pper, colored mrkers, centimeter ruler Geometry Module 3-13

14 Triner/Instructor Notes: Qudrilterls Sclene Acute/ Obtuse Tringles Reflections New Terms: Procedures: Distribute the ctivity sheet. Prticipnts work the ctivity individully, but informlly verify with group members. Fcilitte by providing miniml prompts to help prticipnts clrify their thinking. Try not to nswer questions directly. Ech group drws the figure nd lists its properties on sheet of esel pper. When completed the sheet of esel pper is posted on the wll for gllery wlk. Give pirs of groups bout 5 minutes to meet nd discuss ny differences or errors on the posters. In this ctivity, sclene cute tringle or sclene obtuse tringle is reflected cross the line contining one of its sides. Predict wht qudrilterl will be creted. In your group decide who will drw sclene cute tringle nd who will drw sclene obtuse tringle. In the middle of sheet of grph pper, drw A so tht coincides with grid line nd ll three vertices lie t grid line intersections. The sides of the tringle should be 2 to 3 inches long. Mrk the sides nd ngles using different tick mrks to indicte non-congruency. Possible exmples re shown. A b b c A c b A c Reflect A cross the line contining, nd lbel the imge ppropritely. 1. lssify qudrilterl AA'. Justify. AA' is kite. A kite is qudrilterl with exctly two distinct pirs of congruent consecutive sides. The term dimond is sometimes used t the Visul Level. Geometry Module 3-14

15 Triner/Instructor Notes: Qudrilterls Sclene Acute/ Obtuse Tringles Reflections A b b A' b b c c A c c A' b b A c c A' 2. List the properties of qudrilterl AA' in terms of the sides, ngles, digonls nd symmetry in the tble below. Sides: Two pirs of sides re congruent. Two sets of consecutive sides re congruent. Opposite sides re not congruent. Note: Opposite sides re not prllel becuse lternte interior ngles re not congruent. Vertex ngles: Only one pir of opposite ngles is congruent. One pir of opposite ngles is not congruent. onsecutive ngles re not congruent. The non-congruent vertex ngles re bisected by one of the digonls. The congruent vertex ngles re not bisected by digonl. Is it possible for kite to hve pir of right ngles? The figures for this ctivity were constructed from cute or obtuse tringles, but if sclene right tringle is reflected cross the line contining its hypotenuse, then the congruent pir of vertex ngles re right ngles. Digonls: There re two digonls. One digonl lies on the line of symmetry. Geometry Module 3-15

16 Triner/Instructor Notes: Qudrilterls Sclene Acute/ Obtuse Tringles Reflections The digonl lying on the line of symmetry bisects the other digonl t right ngles. Digonls my not be congruent to ech other. A Is it possible for the kite to hve congruent digonls? Yes. The figure to the right is n exmple of kite with congruent digonls. AA' = 3 cm = 3 cm Note: Ask prticipnts to look t the tringles formed on either side of AA ', the digonl which does not lie on the line of symmetry. These tringles, AA' nd AA', re both isosceles tringles. A' The kites below re both convex. b b A A' b b c c A c c A' The kite below is concve. b b A c c A' Symmetry: There is one line of symmetry. The line of symmetry contins one of the digonls. ring prticipnts together for whole clss discussion. Summrize the properties of kites. Prticipnts re performing t the vn Hiele Descriptive Level becuse they develop properties of kite. Geometry Module 3-16

17 Activity Pge: Qudrilterls Sclene Acute/Obtuse Tringle Reflections Sclene Acute/Obtuse Tringle Reflections In this ctivity sclene cute tringle or sclene obtuse tringle is reflected cross the line contining one of its sides. Predict wht qudrilterl will be creted. In your group decide who will drw sclene cute tringle nd who will drw sclene obtuse tringle. In the middle of sheet of grph pper, drw A so tht coincides with grid line nd ll three vertices lie t grid line intersections. The sides of the tringle should be 2 to 3 inches long. Mrk the sides nd ngles using different tick mrks to indicte noncongruency. Reflect A cross the line contining, nd lbel the imge ppropritely. 1. lssify qudrilterl AA'. Justify. Geometry Module 3-17

18 Activity Pge: Qudrilterls Sclene Acute/Obtuse Tringle Reflections 2. List the properties of qudrilterl AA' in terms of the sides, ngles, digonls nd symmetry in the tble below. Sides Vertex Angles Digonls Symmetry Geometry Module 3-18

19 Triner/Instructor Notes: Qudrilterls Rotte Tringle Rotte Tringle Overview: Objective: Prticipnts discover properties of rectngles by rotting right tringle round the midpoint of its hypotenuse, nd discover the properties of prllelogrms by rotting non-right tringle round the midpoint of one of its sides. TExES Mthemtics ompetencies III.012.D. The beginning techer uses properties of congruence nd similrity to explore geometric reltionships, justify conjectures, nd prove theorems. III.013.A. The beginning techer nlyzes the properties of polygons nd their components. III The beginning techer uses geometric ptterns nd properties (e.g., similrity, congruence) to mke generliztions bout two- nd three-dimensionl figures nd shpes (e.g., reltionships of sides, ngles). III.014.D. The beginning techer pplies trnsformtions in the coordinte plne. V The beginning techer pplies correct mthemticl resoning to derive vlid conclusions from set of premises. V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.019.F. The beginning techer uses pproprite mthemticl terminology to express mthemticl ides. Geometry TEKS b.2.. Mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. c.1. The student uses numeric nd geometric ptterns to mke generliztions bout geometric properties, including properties of polygons, rtios in similr figures nd solids, nd ngle reltionships in polygons nd circles. d.2.a. The student uses one- nd two-dimensionl coordinte systems to represent points, lines, line segments, nd figures. e.2.. sed on explortions nd using concrete models, the student formultes nd tests conjectures bout the properties nd ttributes of circles nd the lines tht intersect them. e.3.a. The student uses congruence trnsformtions to mke conjectures nd justify properties of geometric figures. f.1. The student uses similrity properties nd trnsformtions to explore nd justify conjectures bout geometric figures. Geometry Module 3-19

20 Triner/Instructor Notes: Qudrilterls Rotte Tringle ckground: Mterils: Prticipnts need knowledge of the properties of rottion nd prllel lines. esel pper, grph pper, colored mrkers, ptty pper, centimeter ruler New Terms: Procedures: Distribute the ctivity sheets. Prticipnts work on ll seven items in groups. During the whole clss discussion, sk two volunteers to record properties of rectngles nd prllelogrms on seprte sheets of esel pper. 1. In the middle of sheet of grph pper drw sclene right A, with the right ngle t vertex. Drw the legs long grid lines nd locte the vertices t grid line intersections. The lengths of the legs of the tringles should be 2 to 3 inches long. Locte the midpoint, M, of the hypotenuse. Drw the medin to the hypotenuse. Lbel the figure ppropritely to indicte congruence or non-congruence. 2. Rotte A 180 o round M. Lbel the imge ppropritely. The figures below represent the process. b ' b M M A A b 3. In your group discuss nd list the properties of rectngle A. The following is list of possible properties for the rectngle: Sides: Opposite sides re congruent. (Rottion preserves congruence.) Opposite sides re prllel. (The consecutive ngles re supplementry. If the interior ngles on the sme side of trnsversl re congruent, then the lines intersected by the trnsversl re prllel.) onsecutive sides re perpendiculr to ech other. Vertex ngles: All vertex ngles re congruent right ngles. Opposite ngles re congruent nd supplementry. Geometry Module 3-20

21 Triner/Instructor Notes: Qudrilterls Rotte Tringle onsecutive ngles re congruent nd supplementry. Digonls: Digonls bisect ech other. Digonls re congruent. The point of intersection of the digonls is lso the center of the circumscribed circle. ' b M A b ' M b Symmetry: The rectngle hs two symmetry lines. The rectngle hs 180 o (or 2-fold) rottionl symmetry. A b Prticipnts my rgue tht the digonls re symmetry lines. To cler up this misconception, trce the figure on ptty pper nd fold long the digonls. 4. On clen sheet of grph pper drw obtuse or cute sclene A, with one of its sides long one of the grid lines. Locte the vertices t grid line intersections. The sides of the tringle should be 1.5 to 3 inches long. Locte the midpoint, M, of A. Drw the medin to side A. Lbel the figure ppropritely indicting congruence or non-congruence. 5. Rotte A 180 o round M. Lbel the imge ppropritely. The figures below represent exmples. ' b b M M A A b Geometry Module 3-21

22 Triner/Instructor Notes: Qudrilterls Rotte Tringle 6. In your group discuss nd list the properties of prllelogrm A under the given hedings. Sides: Opposite sides re congruent. (Rottion preserves congruence.) Opposite sides re prllel. (Alternte interior ngles re congruent, becuse rottion preserves congruence.) Vertex ngles: Opposite vertex ngles re congruent. (Rottion preserves congruence.) onsecutive ngles re supplementry. (Interior ngles on the sme side of trnsversl tht intersects prllel lines re supplementry.) Digonls: Digonls bisect ech other. (M is the midpoint of A nd M is mpped to M ', so tht M the midpoint of.) Symmetry: The prllelogrm hs 180 o (or 2-fold) rottion. (The figure ws produced using 180 o rottion.) Prticipnts my rgue tht the digonls re symmetry lines. To cler up this misconception, trce the figure on ptty pper nd fold long the digonls. 7. ompre the properties of the prllelogrm with the properties of the rectngle. In both qudrilterls opposite sides re prllel nd congruent; opposite ngles re congruent; digonls bisect ech other. Appliction problems: 8. lculte the mesure of ech lettered ngle. c b 94 h k j e 52º f g = 38 o b = 48 o c = 90 o d = 48 o e = 90 o f = 142 o g = 38 o h = 38 o j = 71 o k = 109 o d Geometry Module 3-22

23 Triner/Instructor Notes: Qudrilterls Rotte Tringle 9. R is digonl of rectngle RET. Where cn the other two vertices, E nd T, be locted? The digonls re congruent nd intersect ech other t their respective midpoints. The other T digonl cn be ny congruent line segment, whose midpoint is lso the midpoint of the given segment. Find the midpoint of R. Drw segment from the midpoint to point not on R, congruent to one hlf of R. Extend the segment n equl distnce R on the opposite side of the midpoint. E Alterntively: Drw circle using R s the dimeter. ET is lso dimeter. T E R Prticipnts re performing t the vn Hiele Descriptive Level becuse properties of rectngles nd prllelogrms re being developed. The comprison of the properties of the rectngle nd prllelogrm pproches the Reltionl Level. In 7, prticipnts pply the properties of prllelogrms nd rectngles t the Descriptive Level. In 8, the first solution requires the Descriptive Level. In the second solution, prticipnts combine two figures with relted properties, thus pproching the Reltionl Level. Geometry Module 3-23

24 Activity Pge: Qudrilterls Rotte Tringle Rotte Tringle 1. In the middle of sheet of grph pper drw sclene right A, with the right ngle t vertex. Drw the legs long grid lines nd locte the vertices t grid line intersections. The lengths of the legs of the tringle should be 2 to 3 inches long. Locte the midpoint, M, of the hypotenuse. Drw the medin to the hypotenuse. Lbel the figure ppropritely to indicte congruence or non-congruence. 2. Rotte A 180 o round M. Lbel the imge ppropritely. 3. In your group discuss nd list the properties of rectngle A. Sides Vertex Angles Digonls Symmetry Geometry Module 3-24

25 Activity Pge: Qudrilterls Rotte Tringle 4. On clen sheet of grph pper drw obtuse or cute sclene A, with one of its sides long one of the grid lines. Locte the vertices t grid line intersections. The sides of the tringle should be 1.5 to 3 inches long. Locte the midpoint, M, of A. Drw the medin to side A. Lbel the figure ppropritely indicting congruence or non-congruence. 5. Rotte A 180 o round M. Lbel the imge ppropritely. 6. In your group discuss nd list the properties of prllelogrm A under the given hedings. Sides Vertex Angles Digonls Symmetry 7. ompre the properties of the prllelogrm with the properties of the rectngle. Geometry Module 3-25

26 Activity Pge: Qudrilterls Rotte Tringle Appliction problems: 8. lculte the mesure of ech lettered ngle. e 52 k f g b 94 h j c d = b = c = d = e = f = g = h = j = k = 9. R is digonl of rectngle RET. Where cn the other two vertices, E nd T, be locted? R Geometry Module 3-26

27 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex Truncte Tringle s Vertex Overview: Objective: Prticipnts discover the properties of trpezoids. TExES Mthemtics ompetencies II The beginning techer writes equtions of lines given vrious chrcteristics (e.g., two points, point nd slope, slope nd y- intercept). II.006.G. The beginning techer models nd solves problems involving liner nd qudrtic equtions nd inequlities using vriety of methods, including technology. III.011.A. The beginning techer pplies dimensionl nlysis to derive units nd formuls in vriety of situtions (e.g., rtes of chnge of one vrible with respect to nother) nd to find nd evlute solutions to problems. III The beginning techer pplies formuls for perimeter, re, surfce re, nd volume of geometric figures nd shpes (e.g., polygons, pyrmids, prisms, cylinders, cones, spheres) to solve problems. III The beginning techer uses properties of points, lines, plnes, ngles, lengths, nd distnces to solve problems. III The beginning techer pplies the properties of prllel nd perpendiculr lines to solve problems. III.012.E. The beginning techer describes nd justifies geometric constructions mde using compss nd strightedge, reflection devices, nd other pproprite technologies. III.013.A. The beginning techer nlyzes the properties of polygons nd their components. III.014.E. The beginning techer pplies concepts nd properties of slope, midpoint, prllelism, perpendiculrity, nd distnce to explore properties of geometric figures nd solve problems in the coordinte plne. V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.018.F. The beginning techer evlutes how well mthemticl model represents rel-world sitution. V The beginning techer understnds how mthemtics is used to model nd solve problems in other disciplines (e.g., rt, music, science, socil science, business). V.019.D. The beginning techer communictes mthemticl ides using vriety of representtions (e.g., numeric, verbl, grphicl, pictoril, symbolic, concrete). V.019.F. The beginning techer uses pproprite mthemticl terminology to express mthemticl ides. Geometry Module 3-27

28 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex Geometry TEKS b.2.a. The student uses constructions to explore ttributes of geometric figures nd to mke conjectures bout geometric reltionships. b.2.. The student mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. b.3.d. The student uses inductive resoning to formulte conjecture. c.1. The student uses numeric nd geometric ptterns to mke generliztions bout geometric properties, including properties of polygons, rtios in similr figures nd solids, nd ngle reltionships in polygons nd circles. d.2.a. The student uses one- nd two-dimensionl coordinte systems to represent points, lines, line segments, nd figures. d.2.. The student uses slopes nd equtions of lines to investigte geometric reltionships, including prllel lines, perpendiculr lines, nd specil segments of tringles nd other polygons. d.2.. The student develops nd uses formuls including distnce nd midpoint. e.2.. sed on explortions nd using concrete models, the student formultes nd tests conjectures bout the properties nd ttributes of polygons nd their component prts. ckground: Mterils: New Terms: Prticipnts need knowledge of properties of isosceles tringles, prllel lines, nd their ssocited ngles. esel pper, grph pper, colored mrkers, centimeter ruler isosceles trpezoid, midsegment Procedures: Distribute the ctivity sheet. Prticipnts complete 1 8. They my work independently or in groups. 1 nd 2 provide informtion nd vocbulry needed to define trpezoid nd n isosceles trpezoid. 1. Drw lrge sclene tringle. Lbel it MRA. Locte point T on MR. onstruct line prllel to RA, through point T, which intersects MA t point P. TRAP is trpezoid. The prllel sides RA nd TP re clled the bses of the trpezoid. In your figure clerly identify the ngle reltions within the trpezoid. Geometry Module 3-28

29 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex M T 180º - b P 180º - R b A 2. Drw lrge isosceles tringle. Lbel it QSO with vertex ngle Q. Locte point I on SQ. onstruct line prllel to side SO through point I, which intersects QO t point. ISO is n isosceles trpezoid. The congruent pirs of ngles in n isosceles trpezoid re clled bse ngles. In your figure, clerly identify the ngle reltions within the trpezoid. Q I d d 180º - d 180º - d S d d O An isosceles trpezoid is trpezoid with congruent legs. 3. Work with your group to find the ngle nd side properties of trpezoids nd isosceles trpezoids, using your knowledge of prllel lines, isosceles nd sclene tringles. e prepred to shre nd justify the properties during whole clss discussion. Note tht some books define trpezoid s qudrilterl with t lest one pir of prllel sides. Others define trpezoid s qudrilterl with only one pir of prllel sides. This module will use the ltter definition. Possible properties: A trpezoid hs one pir of prllel sides. The sum of the ngles of trpezoid is 360 o. The pirs of consecutive ngles t opposite bses re supplementry. (Interior ngles on the sme side of trnsversl tht intersects prllel lines re supplementry.) In n isosceles trpezoid the bse ngles (two pirs) re congruent. In n isosceles trpezoid the non-prllel sides re congruent. 4. Locte the vertices of qudrilterl AD on the coordinte plne t A ( 7, 3), ( 3, 3), (1, 5), nd D (5, 3). Explin why AD is trpezoid. Geometry Module 3-29

30 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex Sides nd AD re prllel becuse their slopes L D re both 1. The other two 2 sides hve unequl slopes. M A 5. Find the midpoints, M nd L, of A ndd respectively. ML is clled midsegment. M ( 5, 0), L (3, 4). A midsegment is line segment with endpoints tht re the midpoints of the legs of the trpezoid. 6. How do the coordintes of midpoint of segment relte to the coordintes of its endpoints? The coordintes of the midpoint re the verges of the coordintes of the endpoints: M ( 5, 0) =,, ; L (3, 4) = How does the slope of the midsegment relte to the slopes of the prllel sides of the trpezoid? The slope of the midsegment is 1, which is the sme s the slopes of the bses. 2 Therefore the midsegment is prllel to the bses. 8. How does the length of the midsegment relte to the lengths of the prllel sides of the trpezoid? The length of midsegment ML is the verge of the lengths of nd AD. When most groups hve completed 1 8, conduct whole clss discussion on the properties of trpezoids. Prticipnts justify ech property. Possible properties, with justifictions in prentheses follow: A trpezoid hs exctly one pir of prllel sides (shown by the construction). Geometry Module 3-30

31 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex The sum of the ngles is 360 o. (There re two sets of interior supplementry ngles between prllel lines.) The pirs of consecutive ngles t opposite bses re supplementry. (The interior ngles on the sme side of trnsversl tht intersects prllel lines re supplementry.) In n isosceles trpezoid, the two pirs of bse ngles re congruent. (The two bse ngles from the originl isosceles tringle re congruent; the other two ngles re supplementry to the originl two bse ngles, nd must lso be congruent to ech other.) In n isosceles trpezoid, the non-prllel sides re congruent. (Using 2 s n exmple, the bse ngles re congruent nd congruent to the corresponding bse ngles of QI. Therefore, QI is isosceles nd QI Q. Since QS QO, the congruent sides of QSO, then by subtrction, IS O). The midsegment of trpezoid is the segment connecting the midpoints of the nonprllel sides. The midsegment of trpezoid is prllel to the bses. Its length is the verge of the lengths of the two bses. Frequently the bses of trpezoid re designted by the vribles b 1 nd b 2. Write n expression for the length of the midsegment in terms of b 1 nd b 2. + The length of the midsegment is given by b b Drw tringle on coordinte grid. Locte the midpoints of two of the sides. Drw the midsegment. Why is the midsegment prllel to the third side? In the exmple shown, the slopes of the bse nd the midsegment re both 2 3, nd thus the bse nd the midsegment re prllel. ompre the length of the midsegment to the length of the prllel side of the tringle. The length of the midsegment is hlf of the length of the prllel side. Use the formul for the length of the midsegment of trpezoid to justify this reltionship. Geometry Module 3-31

32 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex Since b is the length of the prllel side, or bse, of the tringle, then in the formul for b + b the length of the midsegment,, b1 = 0, b 2 = b. y substitution, the length of the midsegment is 2 b Appliction problems: cm j k h 77 h = j = k = 125 o 77 o 54 cm 60 cm The perimeter of isosceles trpezoid ADEF is 218 in. is the midsegment. Find AD. The non-prllel sides of the trpezoid re D 4x - 1 E congruent. Ech non-prllel side mesures 2(2x + 1). The longer bse is 8 in. longer thn the shorter bse. 2x + 1 Perimeter of ADEF= 2(4x 1) [2(2x+1)] = 8x x + 4 = 16x + 10 = 218 in. A 4 in. F 16x = 208 in. x = 13 in. Therefore, AD = 2(2x + 1) = 4x + 2 = 4(13) + 2 = 54 in. Geometry Module 3-32

33 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex 12. In the two-dimensionl figure, find the ngle mesure x nd y. Explin. x y The two qudrilterls on the left nd right sides of the figure re kites. The lower qudrilterl is trpezoid, so the upper bse ngles re supplementry to the lower bse ngles. The mesures of the upper trpezoid ngles re both 102 o. Therefore x = 360 o 154 o 102 o = 104 o ; y = 360 o 160 o 102 o = 98 o The following problems re tken from Serr, M. (2003). Discovering Geometry: An Investigtive Approch (3 rd ed.). Emeryville, A: Key urriculum Press, pp. 271, 283 with permission from Key urriculum Press. The Romns used the clssicl rch design in bridges, queducts, nd buildings in the erly centuries of the ommon Er. The clssicl semicirculr rch is relly hlf of regulr polygon built with wedge-shped blocks whose fces re isosceles trpezoids. Ech block supports the blocks surrounding it. 13. The inner edge of the rch in the digrm is hlf of regulr 18-gon. lculte the mesures of ll the ngles in the nine isosceles trpezoids mking up the rch. Keystone Voussoir Rise Spn Abutment Imgine tht the isosceles trpezoids become isosceles tringles by regining their truncted vertices. There re nine isosceles tringles, whose vertices meet t the center of semicircle. The nine vertices ech contribute 20 o to the 180 o t the center of the spn. The sum of the bse ngles of ech isosceles tringle is 180 o 20 o = 160 o. The trpezoid s bse ngles on the outer edge of the rch ech mesure 80 o. The bse ngles on the inner edge of the rch re supplementry to the exterior bse ngle, nd mesure 100 o ech. Geometry Module 3-33

34 Triner/Instructor Notes: Qudrilterls Truncte Tringle s Vertex 14. Wht is the mesure of ech ngle in the isosceles trpezoid fce of voussoir in 15-stone rch? As in 13, the vertices of the isosceles tringles creted by the 15 trpezoids spn 180 o. Ech vertex spns 12 o. The sum of the outer bse ngles in ech trpezoid is 180 o 12 o = 168 o. Ech outer bse ngle mesures 84 o. Ech supplementry inner bse ngle mesures 96 o. Remind prticipnts to dd the new terms isosceles trpezoid nd midsegment to their glossries. Prticipnts re performing t the vn Hiele Reltionl Level s they develop properties of trpezoids nd isosceles trpezoids using deductive resoning rther thn observtion nd mesurement. Geometry Module 3-34

35 Activity Pge: Qudrilterls Truncte Tringle s Vertex Truncte Tringle s Vertex 1. Drw lrge sclene tringle. Lbel it MRA. Locte point T on MR. onstruct line prllel to side RA, through point T, which intersects MA t P. TRAP is trpezoid. The prllel sides RA nd TP re clled the bses of the trpezoid. In your figure clerly identify the ngle reltions within the trpezoid. 2. Drw lrge isosceles tringle. Lbel it QSO with vertex ngle Q. Locte point I on SQ. onstruct line prllel to side SO, through point I, which intersects QO t. ISO is n isosceles trpezoid. The congruent pirs of ngles in n isosceles trpezoid re clled bse ngles. In your figure, clerly identify the ngle reltions within the trpezoid. Geometry Module 3-35

36 Activity Pge: Qudrilterls Truncte Tringle s Vertex 3. Work with your group to find the ngle nd side properties of trpezoids nd isosceles trpezoids, using your knowledge of prllel lines, isosceles nd sclene tringles. e prepred to shre nd justify your properties during whole clss discussion. Note tht some books define trpezoid s qudrilterl with t lest one pir of prllel sides. Others define trpezoid s qudrilterl with only one pir of prllel sides. This module will use the ltter definition. 4. Locte the vertices of qudrilterl AD on the coordinte plne t A ( 7, 3), ( 3, 3), (1, 5), nd D (5, 3). Explin why AD is trpezoid. Geometry Module 3-36

37 Activity Pge: Qudrilterls Truncte Tringle s Vertex 5. Find the midpoints, M nd L, of A nd D respectively. ML is clled midsegment. 6. How do the coordintes of midpoint of segment relte to the coordintes of its endpoints? 7. How does the slope of the midsegment relte to the slopes of the prllel sides of the trpezoid? 8. How does the length of the midsegment relte to the lengths of the prllel sides of the trpezoid? 9. Drw tringle on coordinte grid. Locte the midpoints of two of the sides. Drw the midsegment. Why is the midsegment prllel to the third side? ompre the length of the midsegment to the length of the prllel side of the tringle. Appliction problems: cm j k h cm h = º j = º k = cm 55 Geometry Module 3-37

38 Activity Pge: Qudrilterls Truncte Tringle s Vertex 11. The perimeter of isosceles trpezoid ADEF is 218 in. is the midsegment. Find AD. D 4x 1 E 2x + 1 A 4 in. F 12. In the two-dimensionl figure below, find the ngle mesures x nd y. Explin. 154 x y Geometry Module 3-38

39 Activity Pge: Qudrilterls Truncte Tringle s Vertex The following problems re tken from Serr, M. (2003). Discovering Geometry: An Investigtive Approch (3 rd ed.) Emeryville, A: Key urriculum Press, pp. 271, 283 with permission from Key urriculum Press. The Romns used the clssicl rch design in bridges, queducts, nd buildings in the erly centuries of the ommon Er. The clssicl semicirculr rch is relly hlf of regulr polygon built with wedge-shped blocks whose fces re isosceles trpezoids. Ech block supports the blocks surrounding it. 13. The inner edge of the rch in the digrm is hlf of regulr 18-gon. lculte the mesures of ll the ngles in the nine isosceles trpezoids mking up the rch. Keystone Voussoir Rise Abutment Spn 14. Wht is the mesure of ech ngle in the isosceles trpezoid fce of voussoir in 15-stone rch? Geometry Module 3-39

40 Triner/Instructor Notes: Qudrilterls Vesic Pisces Vesic Pisces Overview: Objective: Using properties of the different qudrilterls, prticipnts determine the figures within the vesic pisces. TExES Mthemtics ompetencies III.012.D. The beginning techer uses properties of congruence nd similrity to explore geometric reltionships, justify conjectures, nd prove theorems. III The beginning techer uses geometric ptterns nd properties (e.g., similrity, congruence) to mke generliztions bout two- nd three-dimensionl figures nd shpes (e.g., reltionships of sides, ngles). V.018.D. The beginning techer uses forml nd informl resoning to justify mthemticl ides. V.018.F. The beginning techer evlutes how well mthemticl model represents rel-world sitution. V The beginning techer understnds how mthemtics is used to model nd solve problems in other disciplines (e.g., rt, music, science, socil science, business). V.019.D. The beginning techer communictes mthemticl ides using vriety of representtions (e.g., numeric, verbl, grphicl, pictoril, symbolic, concrete). Geometry TEKS b.2.a. The student uses constructions to explore ttributes of geometric figures nd to mke conjectures bout geometric reltionships. b.2.. The student mkes nd verifies conjectures bout ngles, lines, polygons, circles, nd three-dimensionl figures, choosing from vriety of pproches such s coordinte, trnsformtionl, or xiomtic. b.3.. The student constructs nd justifies sttements bout geometric figures nd their properties. b.3.e. The student uses deductive resoning to prove sttement. b.4. The student uses vriety of representtions to describe geometric reltionships nd solve problems. ckground: Mterils: New Terms: Prticipnts need to be fmilir with tringle nd qudrilterl properties. compss, esel pper, colored mrkers, centimeter ruler vesic pisces Procedures: Geometry Module 3-40

41 Triner/Instructor Notes: Qudrilterls Vesic Pisces Distribute the ctivity sheet. ircles A nd pss through ech others centers nd intersect t nd D. Drw the following segments: A, A, AD,, D, D. P A X Q A intersects circle A t P, nd circle t Q. Drw PQ, P, PD, Q, QD. A intersects D t X. D In your group identify nd clssify tringles nd qudrilterls. Explin why the properties you hve identified re true. For exmple, if rhombus is identified on the bsis of four congruent sides, explin why the four sides re congruent. Allow prticipnts minutes to work together to complete the ctivity. Then sk individuls to shre results, while one prticipnt records on sheet of esel pper. This figure is clled vesic pisces. A vesic pisces is creted by two identicl intersecting circles, the circumference of one intersecting the center of the other. All of the figures within the vesic pisces cn be derived from the congruent rdii A, A, AD,,nd D. Some properties nd figures follow. Mny more cn be discerned. AD is rhombus. ( A AD D.) A D. (Digonls of rhombus re perpendiculr to ech other.) AX = X. (Digonls of rhombus bisect ech other.) PA = Q (Rdii of congruent circles re congruent.) Therefore, PA + AX = PX = X + Q= XQ. PQD is rhombus. (Digonls of rhombus re perpendiculr bisectors of ech other.) A is equilterl. (A = A =.) Similrly, AD is equilterl. m PA =120 o.( m A = 60 o, since A is equilterl.) m AD = 30 o. (Digonl D of rhombus AD bisects A.) m PA= 30 o. ( PA is isosceles since PA = A; m PA = 120 o.) Geometry Module 3-41

42 Triner/Instructor Notes: Qudrilterls Vesic Pisces m PD= 60 o. ( m AD + m PA = m PD.) PD is equilterl. ( PD PD, since the digonls lie on the line of symmetry.) m DA= 120 o = 2 m DP. (The mesure of the rc is two times the mesure of the inscribed ngle intercepting the rc.) onclude the unit with the following discussion: Explin to prticipnts tht the vesic pisces is the concept behind the trditionl compss nd stright edge construction of the perpendiculr bisector of segment, in this cse, A. On clen sheet of pper prticipnts drw segment nd construct the perpendiculr bisector using compss. They then drw congruent circles with centers t the ends of the segments nd rdii equl to the lengths of the segments. Prticipnts re performing t the vn Hiele Reltionl Level becuse they use logicl justifictions to explore reltionships mong properties of qudrilterls. Geometry Module 3-42

43 Activity Pge: Qudrilterls Vesic Pisces Vesic Pisces The vesic pisces on the lid of hlice Well ws designed by the excvtor of Glstonbury Abbey, Frederick ligh ond, resident rcheologist of Glstonbury Abbey in the erly 1900s. It ws given to the hlice Well s thnks-offering for Pece in 1919, t the end of World Wr One, by friends nd lovers of the Well nd of Glstonbury. A ircles A nd pss through ech others centers nd intersect t nd D. Drw the following segments: A, A, AD,, D, D. A intersects circle A t P nd circle t Q. Drw PQ, P, PD, Q, QD. A intersects D t X. In your group identify nd clssify tringles nd qudrilterls using side lengths nd ngle mesurements. Explin why the properties you hve identified re true. For exmple, if rhombus is identified on the bsis of four congruent sides, explin why the four sides re congruent. D Geometry Module 3-43