Here is an example where angles with a common arm and vertex overlap. Name all the obtuse angles adjacent to

Transcription

1 djcent tht do not overlp shre n rm from the sme vertex point re clled djcent ngles. me the djcent cute ngles in this digrm rm is shred y + + me vertex point for is djcent to + djcent simply mens 'next to' ere is the officil wy to sy it: The rm is common to oth ngles. The vertex is common to oth ngles. ere is n exmple where ngles with common rm vertex overlp. me ll the otuse ngles djcent to + rm is lso common to these otuse ngles + + The vertex is common to ll ngles rm is lso common to these otuse ngles: +, is djcent to the otuse ngles +, +, + +. e creful: g: + + shre common rm vertex, ut they re not djcent ecuse they overlp. rm vertex common to oth RI TI 13

2 T * T * here does it work? our Turn djcent 1 me pir of djcent cute ngles in ech of these digrms: K.../.../ me one reflex ngle ll the cute ngles djcent to these ngles mrked with dot. Rememer to write the word reflex infront of the reflex ngles T (iii) (iv) R I T I I R * 3 rw n otuse ngle lel it + R. rw n cute ngle + djcent to it. 4 ch of these ngles shre n rm. xplin why they re not djcent to ech other re not djcent ecuse: + + re not djcent ecuse: 14 RI TI

3 omplementry supplementry ngles These specil nmes re given to pirs of ngles tht dd together to totl of 90c or 180c. omplementry ngles re pir of ngles tht mke right-ngle (90c) when put together. lculte the size of + if it is the complement of + Rememer like this: omplementry ngles mke orner. 50c ize of = 90c ` c = 90c ` + = 40c If you drw complementry ngles djcent to ech other, you will mke right-ngle! + + re complementry ngles 40c = 50c ere is nother exmple. me the pir of complementry ngles in this digrm 42c 47c 45c 43c ook for pir of ngles tht dd to 90c = 43c + 47c = 90c ` + + re complementry ngles RI TI 15

4 upplementry ngles re pir of ngles tht mke stright-ngle (180c) when put together. lculte the supplement of 132c The supplement of 132c is: 180c - 132c = 48c upplementry ngles dd to 180c lculte the size of n ngle supplementry to + Rememer like this: upplementry ngles mke tright ngle. 44c 65c + = = 65c + 44c = 9c + is formed y two djcent ngles + + ngle = 180c ` 9c + ngle = 180c ` ngle = 180c - 9c ` ngle = 71c ` the size of the ngle supplementry to + is 71c uestions with mny ngles need closer investigtion. rite down the pir of djcent, supplementry ngles from this digrm igrm not drwn to scle K 29c 5c 19c 46c + = = 46c + 5c + = 151c The totl size of other djcent ngles is sometimes needed K = 151c + 29c = 180c + + K re supplementry ngles 16 RI TI

5 TR TR * here does it work? our Turn omplementry supplementry ngles 1 lculte the complement (the ngle tht mkes it 90c) of these ngles: 30c 80c c 46c.../.../20... d 11c e 23. 5c f 18. 3c 2 lculte the supplementry (the ngle tht mkes it 180c) of these ngles: 0c 90c c 165c d 9c e c f c 3 lculte the size of the missing complementry ngles elow: 11. 5c R T 71c + = + T = 4 lculte the size of the missing supplementry ngles elow: I 7c + = + I = RI TI 17

6 our Turn omplementry supplementry ngles 5 me the pir of supplementry ngles in this digrm: int: wht is the size of + 64c 44c 46c re supplementry ngles 6 me the two pirs of complementry ngles in this digrm: 24c 35c 23c 30c irst pir: 37c + econd pir: + 31c It's como time! 7 me the pir of djcent complementry ngles in this digrm: R.../.../ c 25c 65c 35c T TI * TI * TI * 25c If + is drwn djcent to + T s shown, wht size must it e to mke + stright ngle? int: the ngles must ll dd to 180c + = 18 RI TI

7 erticlly opposite ngles hen two stright lines cross ech other, four ngles re creted If you mesured ech of these ngles with your protrctor, you will discover tht: ngle 1 = ngle 3 ngle 2 = ngle 4 In themtics we cll these equl ngles, verticlly opposite ngles. me the pirs of verticlly opposite ngles in this digrm: re two stright lines crossing ech other t ` 1 st pir of verticlly opposite ngles re: + + ` 2 nd pir of verticlly opposite ngles re: + + djcent ngles formed y the intersection of two stright lines re supplementry. lculte the size of these ngles: 140c + + = + erticlly opposite ngles re the sme size ` + = 140c + + is djcent to + ` = 180c djcent ngles of intersecting lines re supplementry ` c = 180c ` + = 40c RI TI 19

8 RTI IT * RTI IT * here does it work? our Turn erticlly opposite ngles 1 me shde ll the pirs of verticlly opposite ngles elow: R.../.../20... T K + TR + T irst pir: econd pir: 2 lculte the size of these ngles: I + I c + 49c 56c re stright lines 3 This digrm is mde up of four stright lines,, intersecting t the sme point. me ten different pirs of verticlly opposite ngles. (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) 20 RI TI

9 rllel lines rllel lines never cross ech other, so on their own they never form n ngle. sed to show tht the lines re prllel to ech other rllel lines re nmed like this: ;; This symol mens 'is prllel to' e get ngles formed when nother line tht is not prllel crosses them. line tht crosses prllel lines is clled trnsversl The opposite of prllel is perpendiculr. The symol for this is =. It mens the lines cross t 90cto ech other ngles tht re on lternte sides of the trnsversl inside pir of prllel lines re the sme size. + = + + = + These re clled lternte ngles, they form zigzg shpe when highlighted. et s cll them ngles. ind the size of + 124c K + = 124c + = + ` + = 124c lternte ngles in prllel lines, K ;; lternte ngles in prllel lines re the sme size RI TI 21

10 ngles tht re in corresponding (mtching) positions on pir of prllel lines re the sme size. + = + + = + These re clled corresponding ngles they form n shpe when highlighted. et s cll them ngles. ind the size of + 124c + = 124c + = + ` + = 124c K orresponding ngles in prllel lines, K ;; orresponding ngles in prllel lines re the sme size ngles on the sme side of the trnsversl inside pir of prllel lines re supplementry = 180c = 180c These re clled cointerior ngles they form shpe when highlighted. et s cll these ngles. ind the size of + 124c + = 124c = 180c ` + = 56c K ointerior ngles in prllel lines, K ;; ointerior ngles in prllel lines re supplementry 22 RI TI

11 our Turn rllel lines 1 or ech of these digrms: me the trnsversl. me the pir of prllel lines using the correct symol. R I * R I *.../.../ me ll the pirs of ngles, ngles ngles in these digrms: T ngles (lternte ngles) ngles (lternte ngles) ngles (corresponding ngles) There re four pirs of this type ngles (corresponding ngles) There re four pirs of this type (iii) (iii) (iv) (iv) ngles (cointerior ngles) ngles (cointerior ngles) RI TI 23

12 our Turn rllel lines 3 stright cle, runs underneth rilwy trck s shown. se the ngles ngles properties to complete the tle with ll the other ngles tht re the sme size s the two given. 63c 117c 63c Rememer you + + cn look for 117c verticlly opposite ngles too s they re lso equl. K 4 ind the size of ech of these ngles include one of the properties elow you used to find them: roperties: lternte, corresponding, cointerior, verticlly opposite, stright 46c + = roperty used: orresponding ngles + = roperty used: ngles + = roperty used: ngles + = roperty used: ngles Try this one with 3 prllel lines! (psst! ou will need to use more thn one property) + = roperty used: 128c orresponding ngle to + T, ( + ) = 128c + is verticlly opposite to + = 128c + = roperty used: T 128c + = roperty used: 24 RI TI

13 our Turn rllel lines ince the rules for ngles, ngles ngles only work when lines re prllel, you cn use them to find out whether pir of lines re prllel or not! 5 or ech of these: circle prllel or not prllel for the lines drwn write reson why you circled the one you did! The line re: 153c Reson: R T R 37c The cointerior ngles do not dd up to 180c or The cointerior ngles re not supplementry 81c T The line T re: Reson: R T R 81c c R The line R re: 1c 0c Reson: R T R d K The line re: Reson: R T R (write ll the properties used here) 158c I 22c RI TI 25