Integrated Math 2 Notes

​Chapter​​1​

,​1.1.1​
​Symmetry​

​ hen​​there​​is​​a​​rigid​​transformation​​(reflection,​​rotation,​​or​
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​translation)​​that​​appears​​to​​map​​a​​polygon​​back​​onto​​itself,​​the​
​figure​​is​​said​​to​​have​​symmetry​​.​

​ or​​example,​​an​​isosceles​​trapezoid​​(drawn​​to​​scale​​at​​right)​​has​
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​reflection​​symmetry​​.​​Line​​ι​​is​​the​​line​​of​​symmetry.​​The​​triangle​
​at​​right​​(also​​drawn​​to​​scale)​​does​​not​​have​​reflection​​symmetry,​
​because​​it​​is​​not​​possible​​to​​draw​​a​​reflection​​line​​that​​maps​​the​
​triangle​​back​​onto​​itself.​

​ he​​regular​​pentagon​​at​​right​​has​​rotation​​symmetry​​,​​because​​a​
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​72°,​​144°,​​216°,​​or​​288°​​rotation,​​clockwise​​or​​counterclockwise​
​about​​the​​center​​point​​C​,​​maps​​the​​pentagon​​back​​onto​​itself.​

​ here​​are​​no​​polygons​​with​​translation​​symmetry,​​but​​a​​line​​has​​translation​​symmetry​​,​
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​because​​it​​can​​be​​translated​​onto​​itself.​

​ raphs​​can​​also​​have​​symmetry.​​The​​graph​​on​​the​​left​​below​​has​​reflection​​and​​rotation​
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​symmetry.​​The​​graph​​on​​the​​right​​below​​has​​translation​​and​​rotation​​symmetry.​​Try​
​verifying​​these​​symmetries​​with​​a​​piece​​of​​tracing​​paper!​

,​1.1.2​
​Naming​​Parts​​of​​Geometric​​Figures​

​ ​​point​​is​​named​​using​​a​​single​​capital​​letter.​ ​A​​line​​,​​which​
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​extends​​infinitely​​in​​two​​directions,​​is​​named​​using​​any​​two​
​points​​on​​the​​line.​​For​​example,​​the​​line​​at​​right​​can​​be​​named​
​DE​​,​​ED​​or​​EJ​​,​​among​​other​​ways.​

​ ​​line​​segment​​is​​the​​portion​​of​​a​​line​​between​​two​​points,​​called​​endpoints.​ ​In​​the​
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​diagram​​above,​​the​​line​​segment​​between​​points​​E​​and​​J​​is​​referred​​to​​as​​EJ​​or​​JE​​.​

​ he​​point​​on​​a​​polygon​​where​​two​​line​​segments​​meet​​to​​form​​a​​“corner”​​is​​called​​a​
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​vertex​​.​ ​(The​​plural​​form​​of​​“vertex”​​is​​vertices​​.)​ ​Polygons​​are​​named​​by​​their​​vertices.​
​For​​example,​​the​​vertices​​of​ ∆​ABC​​are​​points​​A​,​​B​,​​and​​C​.​

∠
​ he​​angle​​at​​vertex​​B​​is​​written​​as​​ ​B​,​​and​​the​​measure​​of​​ ​
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∠
​B​​is​​written​​as​​m​​ ​B​.​​The​​length​​of​​side​​BC​​is​​written​​using​
∠
​the​​notation​​BC​​.​​So​​in​∆​ABC​​above​​m​ ​B​​=​​80​​°​,​​and​​BC​​=​
​2cm.​

​ ​​ray​​is​​a​​part​​of​​a​​line​​that​​starts​​at​​one​​point​​and​​extends​
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​infinitely​​in​​only​​one​​direction.​​A​​ray​​is​​named​​by​​its​​endpoint​
​followed​​by​​one​​other​​point​​on​​the​​ray.​​In​​the​​diagram​​at​​right,​
​GF​​,​​GH​​,​​and​​GI​​are​​rays.​​For​​lines​​and​​segments,​​the​​order​​of​​the​
​letters​​in​​the​​name​​do​​not​​matter,​​but​​when​​naming​​rays,​​the​
​endpoint​​must​​always​​come​​first.​

​ n​​angle​​is​​formed​​by​​two​​rays​​joined​​at​​their​​endpoints.​​The​​measure​​of​​an​​angle​
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​represents​​the​​number​​of​​degrees​​of​​rotation​​from​​one​​ray​​to​​the​​other​​about​​their​
​common​​endpoint,​​known​​as​​the​​vertex.​​In​​the​​diagram​​above,​​there​​are​​several​​angles​
​with​​vertex​​G​.​ ​When​​this​​happens,​​the​​angle​​is​​named​​with​​three​​letters,​​and​​the​​vertex​
​must​​be​​the​​second​​letter​​in​​the​​name.​​For​​example,​​the​​angle​​measuring​​10​​°​​can​​be​
∠ ∠
​named​​ ​HGI​​or​​ ​IGH​​.​

, ​1.2.1​
​The​​Investigative​​Process​

​ he​​investigative​​process​​is​​a​​way​​to​
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​study​​and​​learn​​new​​mathematical​​ideas.​
​Mathematicians​​have​​used​​this​​process​​for​
​many​​years​​to​​make​​sense​​of​​new​
​concepts.​

I​ n​​general,​​this​​process​​begins​​with​​a​
​question​​that​​helps​​you​​frame​​what​​you​
​want​​to​​investigate.​

​ or​​example,​​a​​question​​such​​as,​​“​If​​the​​Möbius​​strip​​has​​two​​half-twists,​​what​​will​
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​happen​​when​​that​​strip​​is​​cut​​in​​half​​down​​the​​middle?​​”​​can​​begin​​an​​investigation​​to​​find​
​out​​what​​happens​​when​​the​​Möbius​​strip​​is​​slightly​​altered.​

​ nce​​a​​question​​is​​asked,​​you​​can​​make​​a​​prediction​​or​​a​​conjecture​​based​​on​​the​
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​information​​you​​have​​so​​far.​ ​(A​​conjecture​​is​​a​​mathematical​​statement​​based​​on​
​incomplete​​evidence​​that​​has​​not​​yet​​been​​proven.)​

​ ext,​​the​​exploration​​begins.​ ​This​​part​​of​​the​​process​​may​​last​​a​​while​​as​​you​​gather​
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​enough​​information​​to​​make​​a​​conclusion​​or​​change​​your​​conjecture.​ ​For​​example,​​you​
​may​​make​​a​​conjecture​​about​​the​​angles​​in​​a​​polygon​​based​​on​​a​​few​​examples,​​but​​as​
​you​​draw​​and​​measure​​more​​polygons​​on​​graph​​paper,​​you​​may​​discover​​that​​your​
​conjecture​​is​​incorrect.​ ​When​​this​​happens,​​investigate​​some​​more​​until​​you​​have​​a​​new​
​conjecture​​to​​test.​

​ hen​​a​​conjecture​​seems​​to​​be​​true,​​the​​final​​step​​is​​to​​prove​​that​​the​​conjecture​​is​​always​
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​true.​