6.1 Angles in Polygons

54 Pages·2014·1.59 MB·English

www.ck12.org Chapter6. PolygonsandQuadrilaterals
6
C
HAPTER
Polygons and
Quadrilaterals
Chapter Outline
6.1 ANGLES IN POLYGONS
6.2 PROPERTIES OF PARALLELOGRAMS
6.3 PROVING QUADRILATERALS ARE PARALLELOGRAMS
6.4 RECTANGLES, RHOMBUSES AND SQUARES
6.5 TRAPEZOIDS AND KITES
6.6 CHAPTER 6 REVIEW
This chapter starts with the properties of polygons and narrows to focus on quadrilaterals. We will study several
different types of quadrilaterals: parallelograms, rhombi, rectangles, squares, kites and trapezoids. Then, we will
provethatdifferenttypesofquadrilateralsareparallelogramsorsomethingmorespecific.
301
6.1. AnglesinPolygons www.ck12.org
6.1 Angles in Polygons
LearningObjectives
• Extendtheconceptofinteriorandexterioranglesfromtrianglestoconvexpolygons.
• Findthesumsofinterioranglesinconvexpolygons.
• Identifythespecialpropertiesofinterioranglesinconvexquadrilaterals.
ReviewQueue
1. Findxandy.
(a)
(b)
2.
(a) Findw◦,x◦,y◦,andz◦.
(b) Whatisw◦+y◦+z◦?
◦
(c) Whattwoanglesadduptoy ?
(d) Whatare72◦,59◦,andx◦ called? Whatarew◦,y◦,andz◦ called?
KnowWhat? TotherightisapictureofDevil’sPostpile,nearMammothLakes,California. Thesepostsarecooled
lava (called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns.
However,variationsincoolingcausedsomecolumnstoeithernotbeperfectorpentagonal.
First,defineregularinyourownwords. Then,whatisthesumoftheanglesinaregularhexagon? Whatwouldeach
anglebe?
302
www.ck12.org Chapter6. PolygonsandQuadrilaterals
InteriorAnglesinConvexPolygons
Recall from a previous chapter that interior angles are the angles inside a closed figure with straight sides. Even
thoughthisconceptwasintroducedwithtriangles,itcanbeextendedtoanypolygon. Asyoucanseeintheimages
below,apolygonhasthesamenumberofinterioranglesasitdoessides.
In Chapter 1, we learned that a diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that
◦
thesumoftheanglesinatriangleis180 . Whataboutotherpolygons?
Investigation6-1: PolygonSumFormula
ToolsNeeded: paper,pencil,ruler,coloredpencils(optional)
1. Drawaquadrilateral,pentagon,andhexagon.
2. Cuteachpolygonintotrianglesbydrawingallthediagonalsfromonevertex. Countthenumberoftriangles.
Makesurenoneofthetrianglesoverlap.
3. Makeatablewiththeinformationbelow.
303
6.1. AnglesinPolygons www.ck12.org
TABLE 6.1:
NameofPolygon NumberofSides Numberof(cid:3)sfrom (Column 3) × (◦ in Total Number of
onevertex a(cid:3)) Degrees
Quadrilateral 4 2 2×180◦ 360◦
Pentagon 5 3 3×180◦ 540◦
Hexagon 6 4 4×180◦ 720◦
◦
4. Doyouseeapattern? Noticethatthetotalnumberofdegreesgoesupby180 . So,ifthenumbersidesisn,then
thenumberoftrianglesfromonevertexisn−2. Therefore,theformulawouldbe(n−2)×180◦.
PolygonSumFormula: Foranyn−gon,thesumofthemeasuresoftheinterioranglesis(n−2)×180◦.
Example1: Findthesumoftheinterioranglesofanoctagon.
Solution: UsethePolygonSumFormulaandsetn=8.
(8−2)×180◦=6×180◦=1080◦
◦
Example2: Thesumoftheinterioranglesofapolygonis1980 . Howmanysidesdoesthispolygonhave?
Solution: UsethePolygonSumFormulaandsolveforn.
(n−2)×180◦=1980◦
180◦n−360◦=1980◦
180◦n=2340◦
n=13 Thepolygonhas13sides.
Example3: Howmanydegreesdoeseachangleinanequiangularnonagonhave?
Solution: First,findthesumoftheinterioranglesinanonagonbysettingn=9.
(9−2)×180◦=7×180◦=1260◦
◦
Second, because the nonagon is equiangular, every angle’s measure is equal, so divide 1260 by 9 to find each
◦
angle’smeasureis140 .
EquiangularPolygonFormula: Foranyequiangularn−gon,themeasureofeachangleis (n−2)×180◦.
n
RegularPolygon:Apolygonthatisbothequilateralandequiangular.
It is important to note that in the Equiangular Polygon Formula, the word equiangular can be substituted with
regular.
Example4: AlgebraConnectionFindthemeasureofx.
304
www.ck12.org Chapter6. PolygonsandQuadrilaterals
◦
Solution: Fromtheinvestigation,aquadrilateral’sinterioranglemeasurestotal360 . Writeanequationtosolvefor
x.
89◦+(5x−8)◦+(3x+4)◦+51◦=360◦
8x=224◦
x=28◦
ExteriorAnglesinConvexPolygons
Recallthatanexteriorangleisanangleontheoutsideofapolygonandisformedbyextendingasideofthepolygon.
As you can see, there are two sets of exterior angles for any vertex on a polygon. It does not matter which set you
usebecauseonesetisjusttheverticalanglesoftheother,makingthemeasurementequal. Inthepicturetotheleft,
thecolor-matchedanglesareverticalanglesandcongruent.
Inapreviouschapter,weintroducedtheExteriorAngleSumTheorem,whichstatedthatthemeasuresoftheexterior
◦
anglesofatriangleaddupto360 . Let’sextendthistheoremtoallpolygons.
Investigation6-2: ExteriorAngleTear-Up
ToolsNeeded: pencil,paper,coloredpencils,scissors
1. Drawahexagonlikethehexagonsabove. Colorintheexterioranglesaswell.
2. Cutouteachexteriorangleandlabelthem1-6.
3. Fitthesixanglestogetherbyputtingtheirverticestogether. Whathappens?
◦
The angles all fit around a point, meaning that the exterior angles of a hexagon add up to 360 , just like a triangle.
Thisistrueforallpolygons.
◦
ExteriorAngleSumTheorem: Thesumofthemeasuresoftheexterioranglesofanypolygonis360 .
ProofoftheExteriorAngleSumTheorem
305
6.1. AnglesinPolygons www.ck12.org
Given: Anyn−gonwithnsides,ninterioranglesandnexteriorangles.
◦
Prove: nexterioranglemeasuresaddupto360
NOTE:Theinterioranglesarex ,x ,...x .
1 2 n
Theexterioranglesarey ,y ,...y .
1 2 n
TABLE 6.2:
Statement Reason
1. Any n−gon with n sides, n interior angles and n Given
exteriorangles.
◦ ◦
2. x and y arealinearpair Definitionofalinearpair
n n
◦ ◦
3. x and y aresupplementary LinearPairPostulate
n n
4. x◦+y◦=180◦ Definitionofsupplementaryangles
n n
5. (x◦+x◦+...+x◦)+(y◦+y◦+...+y◦)=180◦n Sumofallinteriorandexterioranglesinann−gon
1 2 n 1 2 n
6. (n−2)180◦=(x◦+x◦+...+x◦) PolygonSumFormula
1 2 n
7. 180◦n=(n−2)180◦+(y◦+y◦+...+y◦) SubstitutionPoE
1 2 n
8. 180◦n=180◦n−360◦+(y◦+y◦+...+y◦) DistributivePoE
1 2 n
9. 360◦=(y◦+y◦+...+y◦) SubtractionPoE
1 2 n
Example5: Whatisy?
Solution: y is an exterior angle, as are all the other given angle measures. Exterior angle measures have a sum
◦
of‘360 ,sosetupanequation.
70◦+60◦+65◦+40◦+y=360◦
y=125◦
306
www.ck12.org Chapter6. PolygonsandQuadrilaterals
Example6: Whatisthemeasureofeachexteriorangleofaregularheptagon?
Solution: Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles
areequal. Theexterioranglesaddupto360◦,soeachangleis 360◦ ≈51.429◦.
7
Know What? Revisited A regular polygon has congruent sides and angles. A regular hexagon’s interior angles
total(6−2)180◦=4·180◦=720◦ . Eachinterioranglewouldmeasure720◦ dividedby6or120◦.
ReviewQuestions
1. Fillinthetable.
TABLE 6.3:
#ofsides # of (cid:3)s from one (cid:3)s×180◦ (sum) Each angle in a Sumoftheexterior
vertex regularn−gon angles
◦ ◦
3 1 180 60
◦ ◦
4 2 360 90
◦ ◦
5 3 540 108
◦ ◦
6 4 720 120
7
8
9
10
11
12
2. Whatisthesumofthemeasuresoftheinterioranglesina15-gon?
3. Whatisthesumofthemeasuresoftheinterioranglesina23-gon?
◦
4. Thesumoftheinterioranglesofapolygonis4320 . Howmanysidesdoesthepolygonhave?
◦
5. Thesumoftheinterioranglesofapolygonis3240 . Howmanysidesdoesthepolygonhave?
6. Whatisthemeasureofeachinteriorangleinaregular16-gon?
7. Whatisthemeasureofeachinteriorangleinanequiangular24-gon?
8. Whatisthemeasureofeachexteriorangleofadodecagon?
9. Whatisthemeasureofeachexteriorangleofa36-gon?
10. Whatisthesumoftheexterioranglesofa27-gon?
◦
11. Ifthemeasureofoneinteriorangleofaregularpolygonis160 ,howmanysidesdoesithave?
◦
12. Howmanysidesdoesaregularpolygonhaveifthemeasureofoneofitsinterioranglesis168 ?
◦
13. Ifthemeasureofoneinteriorangleofaregularpolygonis15814 ,howmanysidesdoesithave?
17
◦
14. Howmanysidesdoesaregularpolygonhaveifthemeasureofoneexteriorangleis15 ?
◦
15. Ifthemeasureofoneexteriorangleofaregularpolygonis36 ,howmanysidesdoesithave?
◦
16. Howmanysidesdoesaregularpolygonhaveifthemeasureofoneexteriorangleis32 8 ?
11
Findthevalueofthemissingvariable(s).
17.
307
6.1. AnglesinPolygons www.ck12.org
18.
19.
20.
21.
22.
23.
24.
308
www.ck12.org Chapter6. PolygonsandQuadrilaterals
25.
26.
27. Theinterioranglesofapentagonarex◦,x◦,2x◦,2x◦,and2x◦. Whatisthemeasureofthelargerangles?
28. Theexterioranglesofaquadrilateralarex◦,2x◦,3x◦,and4x◦. Whatisthemeasureofthesmallestangle?
29. Theinterioranglesofahexagonarex◦,(x+1)◦,(x+2)◦,(x+3)◦,(x+4)◦,and(x+5)◦. Whatisx?
30. ChallengeEachinteriorangleformsalinearpairwithanexteriorangle. Inaregularpolygonyoucanusetwo
differentformulastofindthemeasureofeachexteriorangle. Onewayis 360◦ andtheotheris180◦−(n−2)180◦
n n
◦
(180 minusEquiangularPolygonFormula). Usealgebratoshowthesetwoexpressionsareequivalent.
31. AnglePuzzleFindthemeasuresoftheletteredanglesbelowgiventhat
m||n.
ReviewQueueAnswers
1. (a) 72◦+(7x+3)◦+(3x+5)◦=180◦
10x+80◦=180◦
10x=100◦
x=10◦
(b) (5x+17)◦+(3x−5)◦=180◦
8x+12◦=180◦
8x=168◦
x=21◦
2. (a) w=108◦,x=49◦,y=131◦,z=121◦
◦
(b) 360
(c) 59◦+72◦
(d) interiorangles,exteriorangles
309
6.2. PropertiesofParallelograms www.ck12.org
6.2 Properties of Parallelograms
LearningObjectives
• Defineaparallelogram.
• Understandthepropertiesofaparallelogram
• Applytheoremsaboutaparallelogramssides,anglesanddiagonals.
ReviewQueue
1. Drawaquadrilateralwithonesetofparallelsides.
2. Drawaquadrilateralwithtwosetsofparallelsides.
3. Findthemeasuresofthemissinganglesinthequadrilateralsbelow.
(a)
(b)
Know What? A college has a parallelogram-shaped courtyard between two buildings. The school wants to build
twowalkwaysonthediagonalsoftheparallelogramwithafountainwheretheyintersect. Thewalkwaysaregoing
tobe50feetand68feetlong. Wherewouldthefountainbe?
WhatisaParallelogram?
Parallelogram: Aquadrilateralwithtwopairsofparallelsides.
Herearesomeexamples:
310