UNIT 5 Angles Activities

Transcription

1 UNIT 5 Angles Ativities Ativities 5.1 Compss Rose Constrution 5.2 Sm Loyd's Dissetion 5.3 Overlpping Squres 5.4 Construting Tringles 5.5 Angles in Tringles 5.6 Angles in Qudrilterls 5.7 Interior Angles in Polygons

2 ACTIVITY 5.1 Compss Rose Constrution Here is ompss rose, symmetri out oth the NS nd the EW lines, so you n onstrut it using instrutions for ny one qurter N 1. Introduing oordinte xes, e.g. x in the E diretion, nd y in the N diretion, s shown, wht re the oordintes of the verties of the rose in the positive qudrnt? 2. Give omplete set of instrutions for drwing ll lines in the positive qudrnt. 3. Give diretions to now omplete the digrm, using refletions. Drw your own ompss rose, giving omplete set of instrutions for ompleting the drwing.

3 ACTIVITY 5.2 Sm Loyd's fmous dissetion prolem ws designed in the 1920s. Drw 5 m squre s shown on the right. Find the midpoints (A, B, C nd D in digrm) on eh side nd join them up s indited. Using the digrm s guide, ut your squre into 5 piees long the old lines. Do not ut long the dotted lines. Use the 5 piees to mke ll the shpes elow. D Sm Loyd's Dissetion A B C Qudrilterl Cross Retngle Tringle Prllelogrm

4 ACTIVITY 5.3 Overlpping Squres Tke two squres nd put them down on surfe so tht they overlp. The squres n e of ny size, not neessrily the sme. 1. Whih of the following shpes n e formed y the overlp: () retngle () squre () kite (d) rhomus? 2. Cn two squres interset so tht tringle is formed y the overlp? 3. Cn two squres interset so tht the overlp forms polygon of n sides for vlues of n equl to () 5 () 6 () 7 (d) 8 (e) 9 (f) 10? 4. Wht hppens when two tringles overlp? s 1. Wht hppens when two pentgons overlp? 2. Wht hppens when two different shpes, e.g. squre nd tringle, overlp?

5 ACTIVITY 5.4 Construting Tringles Construting tringles, using ruler nd protrtor, is strightforwrd when suffiient informtion is given. Sometimes not enough informtion is ville: t other times you my e given too muh informtion, some of whih my e redundnt (i.e. not needed). 1. Using the informtion on the skethes, drw urtely the following tringles. () () () 4 m 3 m 5 m m 60 4 m 60 6 m In eh se, mesure ll other sides nd ngles. Eh of these tringles is extly defined with suffiient informtion (ut not too muh) to enle you to drw the tringle. We refer to these ses s 'SSS' (three side lengths given), 'ASA' (ngle, side, ngle) nd 'SAS' (side, ngle, side). 2. () Drw urtely the following tringle nd then mesure ll other sides nd ngles. () Compre this tringle with tringle () in question 1. Wht do you notie? 60 5 m So, if you re given the tringle opposite to drw you tully hve more informtion thn you need! You will hve to deide whih informtion to use m 5 m Drw urtely the tringle skethed opposite. Be reful s there re two distint possiilities! 40 6 m 5 m This is the 'ASS' se, nd it does not neessrily hve unique (only one) solution. As you sw in question 3, there were two distint tringles tht greed with the informtion given, i.e. there ws insuffiient informtion given for unique solution. Cn you onstrut either of the tringles elow? () () 3 m 3.85 m 40 6 m 40 6 m

6 ACTIVITY 5.5 Angles in Tringles In eh of the three tringles, mesure ll the ngles s urtely s possile, nd dd up the vlues. Tringle Angles + + Wht do you notie? Repet the exerise with your own tringles.

7 ACTIVITY 5.6 Angles in Qudrilterls In eh of the three qudrilterls, mesure ll the ngles s urtely s possile, nd dd up the vlues. Qudrilterl Angles d d d d d Wht do you notie? Repet the exerise with your own qudrilterls. Given tht the interior ngles of tringle sum to 180, show tht the interior ngles of qudrilterl sum to 360.

8 ACTIVITY 5.7 Interior Angles in Polygons You n find the sum of the interior ngles in ny polygon, y dividing it up into tringles with lines onneting the verties. For exmple, the hexgon shown opposite hs een divided into 4 internl tringles. The sum of ll the interior ngles of the hexgon is equl to the sum of ll the ngles in eh tringle, so: sum of interior ngles = = Repet the sme nlysis for the following shpes: () qudrilterl () pentgon () heptgon (d) otgon (e) nongon (f) dodegon. 2. Copy nd omplete the tle. Nme of Numer Numer of Sum of Polygon of Sides Tringles Interior Angles Tringle Qudrilterl Pentgon Hexgon Heptgon Otgon Nongon Dodegon Wht is the formul for the sum of the interior ngles of polygon with n sides?

9 ACTIVITIES Notes nd Solutions Notes nd solutions re given only where pproprite (7, 0), (3, 1), (4, 4), (1, 3), (0, 7) 2. Join (7, 0) to (3, 1) to (4, 4) to (1, 3) to (0, 7) nd (0, 0) to (7, 0) ; (0, 0) to (3, 1) ; (0, 0) to (4, 4) ; (0, 0) to (1, 3) : nd (0, 0) to (0, 7). 3. Reflet shpe in x -xis, nd then reflet new shpe in y-xis (or the other wy round) () Yes () Yes () No (d) No 2. Yes : Yes : 9-10 : No 4. Overlp forms polygon of n sides with n () 90, 37, 53 () 90, 5 m, 8.7 m () 5.3 m, 79, () 10 m, 8.7 m, 30 () identil tringles (ongruent) 3. There re two distint tringles s shown elow m 5 m m 5 m m 89.5 () not possile () unique nswer (with ngle of 90 t vertex)

10 ACTIVITIES Notes nd Solutions Notes nd solutions re given only where pproprite. 5.5 ngles dd to 180 (ut note tht this is not proof!) 5.6 ngles dd to 360 (ut, gin, note tht this is not proof!) Any qudrilterl n e divided into two tringles. 5.7 For polygon with n sides, the sum of the interior ngles is 180 ( n 2); hene the vlues in the tle should e 180, 360, 540, 720, 900, 1080, 1160, 1340.