UNIT 31 Angles nd Symmetry Dt Sheets Dt Sheets 31.1 Line nd Rottionl Symmetry 31.2 Angle Properties 31.3 Angles in Tringles 31.4 Angles nd Prllel Lines: Results 31.5 Angles nd Prllel Lines: Exmple 31.6 Angle Symmetry in Regulr Polygons
Dt Sheet 31.1 Line nd Rottionl Symmetry An ojet hs rottionl symmetry if it n e rotted out point so tht it fits on top of itself without ompleting full turn. The numer of times this n e done is the order of rottionl symmetry. Shpes hve line symmetry if mirror ould e pled so tht one side of the shpe is n ext refletion of the other. Exmple Rottionl symmetry of order 2 2 lines of symmetry (shown with dotted lines) Rottionl symmetry of order 3 3 lines of symmetry (shown with dotted lines) Exerises Wht is () the order of rottionl symmetry, () the numer of lines of symmetry of eh of these shpes? Show the lines of symmetry. () () () () () () () ()
Dt Sheet 31.2 Angle Properties 1. Angles t Point The ngles t point will lwys dd up to 360. It does not mtter how mny ngles re formed t the point their totl will lwys e 360. d + + + d = 360 2. Angles on Line Any ngles tht form stright line dd up to 180. 3. Angles in Tringle The ngles in ny tringle dd up to 180. + + = 180 + + = 180 4. Angles in n Equilterl Tringle In n equilterl tringle ll the ngles re 60 nd ll the sides re the sme length. 60 o 60 o 60 o 5. Angles in n Isoseles Tringle In n isoseles tringle two sides re the sme length nd the two ngles re the sme size. 6. Angles in Qudrilterl The ngles in ny qudrilterl dd up to 360. equl ngles d + + + d = 360
Dt Sheet 31.3 Angles in Tringles Note tht the ngles in ny tringle sum to 180. Exmple In this figure, ABC is n isoseles tringle with CAB = p nd ABC = ( p + 3 ). C () Write n expression in terms of p for the vlue of the ngle t C. A p o () Determine the size of EACH ngle in the tringle. () As ABC is n isoseles tringles, Not drwn to sle (p+3) o B ACB ˆ = () For tringle ABC, + + = 180 p + = 180 p = p =
Dt Sheet 31.4 Angles nd Prllel Lines: Results d g f h e Results Corresponding ngles re equl. e.g. d = f, = e Alternte ngles re equl. e.g. = f, = e Supplementry ngles sum to 180. e.g. + f = 180 Thus If orresponding ngles re equl, then the two lines re prllel. If lternte ngles re equl, then the two lines re prllel. If supplementry ngles sum to 180, then the two lines re prllel.
Dt Sheet 31.5 Angles nd Prllel Lines: Exmple Exmple E F In this digrm, AB is prllel to CD, EG is prllel to FH, ngle IJL = 50 nd ngle KIJ = 95. Clulte the vlues of x, y nd z, showing lerly the steps in your lultions. A C y o J N G K I 95 o z o L 50 o x o M H B Not drwn to sle D x Angles BIG nd END re ngles, so But ngles END 95 +END ˆ = END ˆ = nd FMD re ngles, so x = y Angles BCD nd ABC re ngles, so In tringle BIJ, y = y + + = 180 So y = 180 y = z Angles AKH nd FMD re ngles, so z =
Dt Sheet 31.6 Angle Symmetry in Regulr Polygons Exmple 1 Find the interior ngle of regulr dodegon. A regulr dodegon hs sides. interior ngle The ngle, mrked x, is given y x = 360 = x The other ngle in eh of the isoseles ( tringles is 180 ) = 2 The interior ngle is 2 = Exmple 2 Find the sum of the interior ngles of regulr heptgon. You n split regulr heptgon into isoseles tringles. Eh tringle ontins three ngles tht sum to. Totl of ll ngles= 7 =. We need to exlude the ngles round the entre tht sum to. Hene sum of interior ngles = = Note: Is the result the sme for n irregulr heptgon?