Sum of Exterior Angles of Polygons TEACHER NOTES


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1 Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary). Studets will determie that if oe exterior agle is draw at each vertex of a covex polygo, the the sum of the measures of those exterior agles is 360. Studets will determie a formula for the measure of oe exterior agle of a regular polygo ad use this to discover a alterative form for the formula that is typically used to calculate the measure of the iterior agle of a regular polygo. Vocabulary exterior agle of a polygo regular polygo iterior agle of a polygo irregular polygo About the Lesso This lesso ivolves movig a arrow alog the side of regular ad irregular polygos to form a exterior agle with the adjacet side of the polygo. As a result studets will: Discover that a iterior ad exterior agle of a polygo form a liear pair. Determie that the sum of the measures of the exterior agles of ay covex polygo is 360. Determie that a exterior agle of a regular go must measure 360 ad that the measure of the iterior agle of 360 a regular polygo ca be foud by the formula TINspire Navigator Live Preseter Quick Poll Class Capture Activity Materials Compatible TI Techologies: TINspire Apps for ipad, TINspire CX Hadhelds, TINspire Software Tech Tips: This activity icludes scree captures from the TINspire CX hadheld. It is also appropriate for use with the TINspire family of products icludig TINspire software ad TINspire App. Slight variatios to these directios may be required if usig other techologies besides the hadheld. Watch for additioal Tech Tips throughout the activity for the specific techology you are usig. Access free tutorials at ators/pd/us/olie Learig/Tutorials Lesso Materials: Studet Activity Sum_of_Exterior_Agles_of_ Polygos_Studet.PDF Sum_of_Exterior_Agles_of_ Polygos_Studet.DOC TINspire documet Sum_of_Exterior_Agles_of_ Polygos.ts Visit for lesso updates ad tech tip videos Texas Istrumets Icorporated 1 educatio.ti.com
2 Sum of Exterior Agles of Polygos TEACHER NOTES Discussio Poits ad Possible Aswers Tech Tip: If studets experiece difficulty draggig the poit, check to make sure that they have moved the arrow util it becomes a had ( ) gettig ready to grab the poit. Press / x to grab the poit ad close the had ({). Move to page The figure o page 1.2 is a regular petago. Teacher Tip: You may wat to demostrate ad have studets follow you o the first rotatio. a. Move the arrow by draggig poit T alog the side of the petago. What appears whe the arrow reaches the vertex? Aswer: A dotted ray appears. This ray forms a exterior agle with the ext side of the polygo. The measure of the iterior agle appears. The measure of the agle betwee the arrow ad the dotted ray appears. Sice the arrow is lied up with the dotted ray, this measure is 0. b. Make a cojecture about the umber of degrees eeded to rotate the arrow for it to lie up with the ext side of the petago. What mathematical relatioship betwee the exterior ad iterior agles could help you determie the umber of degrees eeded to rotate the arrow? Aswer: Aswers may vary. If studets cojecture correctly, they will determie that they should rotate the vector 72. Studets should ote that sice the iterior agle ad the exterior agle at that vertex form a liear pair, they are supplemetary ad the sum of their measures is 180. Therefore, you must rotate the arrow Texas Istrumets Icorporated 2 educatio.ti.com
3 Sum of Exterior Agles of Polygos TEACHER NOTES c. Press d. Test your cojecture by grabbig poit T ad rotatig the arrow so that it "saps" to the ext side. What do you otice about the sum of the exterior agle ad the adjacet iterior agle? Aswer: Their sum is 180. They are supplemetary agles. Teacher Tip: After pressig d, grab the arrow ad rotate to the side. It should sap to the side, but this may take a little practice. If studets have trouble, have them move the poit a little closer to the ext vertex ad try agai. The poit will jump slightly whe it saps ito place. Tech Tip: To release poit T, tap the white space outside of the polygo. TINspire Navigator Opportuity: Live Preseter See Note 1 at the ed of this lesso. d. Press d. Move poit T to the ext vertex ad complete the eeded rotatio for the arrow to lie up with the ext side of the polygo. Explai the ew exterior agle sum. Aswer: The ew sum is 144 because of aother rotatio of 72. e. Repeat steps a d to cotiue movig ad rotatig the arrow util You re doe! appears o the scree. How does the Exterior Agle Sum relate to the arrow s movemet aroud the petago? Aswer: As the arrow moved aroud the petago, the Exterior Agle Sum icreased by the amout that the arrow was rotated at each vertex. The sum vector rotated the same amout as the arrow o the petago, but the Exterior Agle Sum shows the total amout the arrow was rotated as it moved aroud the petago. The exterior agle sum is 360. Teacher Tip: Studets may wish to udo their moves at some poit. If so, have them use / d to udo their work, or close ad reope the documet Texas Istrumets Icorporated 3 educatio.ti.com
4 Sum of Exterior Agles of Polygos TEACHER NOTES Move to page The figure o page 2.1 is a irregular hexago. a. Fid the fial exterior agle sum for the irregular hexago. Before you begi, do you thik the fial exterior agle sum for the irregular hexago will be the same as it was for the regular petago? Why or why ot? Aswer: Aswers may vary. Studets may assume correctly that the total will agai be 360, but may may icorrectly assume that the total will ot be 360 because the shape is irregular. b. At each vertex, explai how you ca determie the umber of degrees eeded to rotate the arrow for it to lie up with the ext side. Aswer: The amout eeded to rotate the arrow to the ext side will always be 180 mius the measure of the iterior agle at that vertex. TINspire Navigator Opportuity: Quick Poll See Note 2 at the ed of this lesso. c. Cotiue movig the arrow util the message You re doe! appears o the scree. Observe the fial exterior agle sum. How do the results compare to your expectatios i part 2a? Aswer: Aswers may vary, but studets should see that the total is ideed 360. Depedig o their aswers to part 2a, they will make differet comparisos. Teacher Tip: Studets will eed to drop T ad grab it agai alog each side util they have made it all the way aroud the polygo. d. Based o your fidigs, what do you thik is true for the exterior agle sum of ay regular or irregular polygo? Aswer: The sum of the exterior agles of ay regular or irregular polygo is Texas Istrumets Icorporated 4 educatio.ti.com
5 Sum of Exterior Agles of Polygos TEACHER NOTES TINspire Navigator Opportuity: Quick Poll See Note 3 at the ed of this lesso. 3. O page 3.1, there is a polygo with a exterior agle at each vertex. a. Drag poit P. What do you observe about the exterior agles as the polygo chages size? Aswer: The exterior agles do ot chage. b. Drag poit P so that the polygo shriks to a poit. What do you observe about the exterior agles? Aswer: They seem to meet at a poit like radii of a circle (or spokes of a wheel). c. Do some more experimets by movig poit P. Click the up ad dow arrows at the top of the scree to chage the umber of sides of the polygo. Drag the ope circle at the bottom of the scree to chage the polygo from regular to irregular. What do you observe about the exterior agles? Aswer: Whe you drag poit P so that the polygo shriks to a poit, it does ot matter how may sides the polygo has, or whether it is regular or irregular. The exterior agles always meet at a poit like radii of a circle, ad thus their sum is always 360. TINspire Navigator Opportuity: Class Capture See Note 4 at the ed of this lesso Texas Istrumets Icorporated 5 educatio.ti.com
6 Sum of Exterior Agles of Polygos TEACHER NOTES d. Based o your fidigs, what is the sum of the measures of the exterior agles of ay polygo? Aswer: 360 e. What is the sum of the exterior agles of a dodecago (12sided polygo)? Aswer: a. If you kow the sum of the exterior agles of ay regular go, what formula could you use to determie the measure of oe of its exterior agles? Aswer: The polygo is regular, so each iterior agle must have the same measure. Therefore, each exterior agle would have the same measure. Sice the sum of the exterior agles of the go is 360, the measure of oe of the exterior agles is 360. b. If you kow the measure of a exterior agle of a regular go, what formula could you use to determie the measure of oe of its iterior agles? 360 Aswer: The iterior agle would have to measure ( ) c. How does the formula i part 4b relate to a commo formula  measure of the iterior agle of a regular go? that is give for the Aswer: Studets will eed to use their algebra skills to show that the formulas are equivalet. ( ) = =  = Teacher Tip: Possible extesios for studets could iclude the exploratio of the sum of the exterior agles of cocave polygos Texas Istrumets Icorporated 6 educatio.ti.com
7 Wrap Up Sum of Exterior Agles of Polygos TEACHER NOTES Upo completio of the discussio, the teacher should esure that studets uderstad: At ay vertex of a polygo, the exterior agle ad the iterior agle are supplemetary. If oe exterior agle is draw at each vertex of ay covex polygo, the sum of the measures of these exterior agles is 360. The measure of a exterior agle of a regular go is The measure of the iterior agle of a regular go is TINspire Navigator Note 1 Questio 1c, Live Preseter: Make a studet the Live Preseter ad have the studet demostrate how to pull poit T out alog the side of the polygo, release it, ad the grab it agai to rotate it i towards the ext side. Note 2 Questio 2b, Quick Poll: Oe of the iterior agles of a petago measures 118. What is the measure of the exterior agle at the same vertex? Aswer: 62 Note 3 Questio 2d, Quick Poll: Four of the exterior agles of a petago measure 95, 100, 70, ad 65. What is the measure of the fifth exterior agle? Aswer: 30 Note 4 Questio 3c, Class Capture: As studets chage the value of ad switch from regular to irregular o page 3.1, use Class Capture so that they ca see results for a variety of gos Texas Istrumets Icorporated 7 educatio.ti.com
Figure 40.1. Figure 40.2
40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called
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