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# Week 2 Notes- 9.1 Figures in the Plane MATH 209

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This 13 page Class Notes was uploaded by Emma Foster on Saturday January 24, 2015. The Class Notes belongs to MATH 209 at University of Alabama - Tuscaloosa taught by Katherine Nichols in Spring2015. Since its upload, it has received 254 views.

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Date Created: 01/24/15

oi in We Chapter 91 Visualization Being able to visualize and manipulate objects is an important part of many professions 0 Here we wish to sharpen those skills First we need to understand the building blocks used to make up shapes and objects Basic Terms 0 an idealized version of a dot having no size or shape 39 m an idealized version of a string having no thickness 39 art of a line lying between two points end points on a line mm part of line lying on one side of a point on the line idealized version of an infinite flat piece of paper having no thickness Activity 1 Visualize a line in a plane The line divides the plane into two disjoint pieces visualize these two pieces as well Visualize a line in space Does a line in space divide the space into disjoint pieces The word disjoint means distinct and not meeting 2 Visualize two lines in a plane How many disjoint pieces do the line divide the plane into 3 Visualize three lines in a plane How many disjoint pieces do the line divide the plane into pom v39 UM 4 Unmeqmem A 8 A a my 7 9amp7 WON W x A mg m mm A 1an m SPOCQ lt How many dieion m 916C687 T H Um many d1830m139 0W3 Q MOPS M0 dig039m was Q imam opuone 61m m 0 mm r A A H000 many 1330 m 916C632 4 DIME H000 many Mm WCCS 0mm pOSRxmu msZ 394 WES 4 WES 0mm9038 01n s 3 Q WES 0 Wm IPI BCCS 90pm QWC S 0 To make shapes we use line segments to break up a plane These segments on the outside of a shape or object are called The point where two or more edges meet are CBHEd I Angles Two Ways to De ne Angles Aw is an amount of rotation about a fixed point e 0 0 e mammal 39 mmol WI Suppose there are two rays in a plane and these two rays have a common endpoint P The two rays and the region between them is called the angle at P formed by two rays 39 pr l Measurement of Angles Angles are commonly measuredi lQ anwa Ma K0 More Terminologies Question Which angle is larger Which angle is a right angle Postulate and Theorem 239 ifquot axiom is a mathematical statement that is considered foundational and is simply r t A assume to be trueg 800 4 j a mathematical statement that has been proven to be true by the use of logical reasning sed on previously proven theorems and or postulates First Fact Angles Formed by Two Lines Angles can be formed by configurations of lines Activity 1 The next figure shows three pairs of lines meeting or you may wish to think of this as showing one pair of lines in three situations when the lines are moved to different positions in each case how do angles a and 6 appear to be related and how do angles b and d appear to be related may ore across from emhomu 4 386mm 06 me some GROW c a d 2 Do you think that the same phenomenon you observed in Problem 1 will hold for any pairs of lines meeting at a point 3 Explain why what you observed in Problem 1 must always be true by using the fact that an angle formed by a straight line is 180 Use this fact I I I 39 quotll agVi 5 1 Q l39 L J 1 a r quotIL i quot i l vll l 1 gt H I l a ll J u a 1 nl fi Jug3 v 1 I 9522 a 33 1 I rquot r 4 39 I 0 v f H IN 1 1 y Iquot l 5 1 3 I l l39 1l quot I 39 lquot V l i if t 5 39l 2 Winn w 0 n 11 39quot 3r 39 391 1 quotx Aglbgto i E 7 7 r R 9 olmommomms 800 9 Old 1800 W omen mm mm are QXOC KN moo ebclwao 0m d lilllop 39 aggmqlelb 9 mm a Qllli llOlNJ orbs bOOgoNl CM 800 volsleoob 0nd C 0040 we 0an 0 0mm 0 Parallel Two lines in a plane that never meet are calle Configurations of Lines in a Plane Three examples amk m E Nag The Parallel Postulate When two parallel lines are cut by a third line sometimes called a 39 re angles are named A V A V thhSU lSCll Wlt 4 V IL Ifg x l ff If I y l y l I Angles A amp H B amp G l y Alternate interior angles are equal Alternate exterior angles are equal Corresponding angles are equal Vertical angles are equal Third Fact Theorem The angles formed by the vertices of a triangle add to 180 Class Activity Show how they add to 180 Practice problem l0 32 HQ 10 180 93 4939 m HQ a 70 Q Practice Problem 000 w lt3 W 1950 mo 660 1 60 FL AOO 4656 Application Find the third angle of a triangle if the other two angles are known 0 Find the missing angle The Clock Example How many degrees does the minute hand of a clock turn through a in 18 minutes b in 28 miutes OiDOwgsww mmma 99 lg X W Example How many degrees does the hour hand of a clock turn through in 1 hour quot 39 7100M Wmmwr Example How many degrees does the hour hand of a clock tum through a in 30 minutes b in 28 minutes Q hours m MOE mm T mm Du mm m rm mud 0 mm W mm 6quot W m b Q m Nquot Example Find the angle formed by the minute hand and the hour hand of a standard 12 hour clock at 1100 hOUiS X 00 iW 53500 Example Find the angle formed by the minute hand and the hour hand of a standard 12 hour clock at 1041 0 mii iiiit i iOi d 400 QUU rx x ya a g 7 r W I G b i mu i IOX at 00 ii om mm 1532 mm h a xx 20960 HIHII n a s a if 2 571 y 0 f a gt 39w39 r is J t 2 W5 r775 cm x i The Hiker O 3Q quotquot I W 9 A hiker started heading due north then turned to the right 43quot then turned to the left 35 and next t med to the right 49 To resume heading due north what turn must the hiker ma e3 5

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