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G e o m e t r y J o u r n a l 3 Michelle Habie

G e o m e t r y J o u r n a l 3 Michelle Habie. Parallel Lines & Planes * Skew Lines :. Parallel Lines : are coplanar and do not intersect , always keeping the same distance from each other . Parallel Planes: two planes that do not intersect .

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G e o m e t r y J o u r n a l 3 Michelle Habie

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  1. GeometryJournal 3Michelle Habie

  2. ParallelLines & Planes*SkewLines: ParallelLines: are coplanarand do notintersect, always keepingthesamedistancefromeachother. Parallel Planes: two planes that do notintersect. SkewLines: Linesthat do notintersectand are notcoplanarandwillnevertoucheachother. Examples: Side-walkfor a handicap Mirror:

  3. Transversal: Linethatintersectstwoparallellines at twodifferentpoints. The transversal “t” andothertwolines “r” and “s” formingspecialpairsofanglessuch as: corresponding, alternate exterior & interior andsameside interior. (eightangles.) t s 1 2 3 4 r 5 6 7 8 http://2.bp.blogspot.com/_nmM9aWNBB2E/TNGckkc0kzI/AAAAAAAAAB4/LlocG1kq_F4/s1600/beatles_abbyroad.jpg

  4. Angles: Corresponding: Lie onthesamesideofthe transversal. Oneinsideandoneoutsidetheparallellines. Alternate Exterior: Lie onoppositesidesof the transversal outsidetheparalleles. Alternate Interior: Notadjacentanglesthatlieontheoppositesidesofthe transversal insidetheparallellines. Same-side Interior: Are theanglepairthat are ontheinsideofthetwoparallellinesforming a pairofsupplementaryangles. Alternate Exterior: Alternate Interior: Corresponding: Same- Side Interior:

  5. CorrespondingAnglesPostulate & converse: If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. Examples: 1 & 5 2 & 6 3 & 7 4 & 8 1 2 3 4 5 6 8 Trains Transversal via 7 http://www.westechme.com/rides%20library/trains/images/CARTOON%20TRAIN_jpg.jpg

  6. Alternate Interior AnglesTheorem& converse: If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. Examples: Electric stairs in mall  http://thumbs.dreamstime.com/thumblarge_136/1175613399z4N9YZ.jpg Parking Lot

  7. Same- Side Interior AnglesTheorem & Converse: If two coplanar lines are cut by a transversal so that a pair of same- side interior angles are supplementary, then the two lines are parallel. Examples: Boxing Ring: Neighbors

  8. Alternate Exterior AnglesTheorem & Converse: If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. Examples: Airport: Supermarket halls (baskets): http://www.miss-thrifty.co.uk/wp-content/uploads/2008/08/shopping-basket.jpg http://www.clipartguide.com/_named_clipart_images/0511-0810-1419-0773_Cartoon_Airplane_clipart_image.jpg

  9. Perpendicular Transversal Theorems: In a plane, if a transversal is perpendicular tooneoftwoparallellines, thenitis perpendicular totheotherparallelline. Examples: Ifpis perpendicular tosandqis perpendicular tosthenpandq are parallel. Ifroom a has linesmandn perpendicular tolthen, o andq are also perpendicular tol. a l b d q If a is perpendicular tocandbis perpendicular tocthen a is perpendicular toc. b c a c a s t m o n p q r

  10. TransitiveProperty: ParallelLines: Ifonelineisparalleltothesecondoneandthatsecondoneisparallelto a thirdonethen, thethreelines are paralleltoeachother. Perpendicular Lines: Ifpis perpendicular toqandris perpendicular toqalso, thenpandrmust be parallel. Perpendicular LinesTheorem: Iftwointersectinglinesform a linear pairofcongruentanglesthenthelinesmust be perpendicular. Examples: Iflinelisparalleltolinemandlinemisparalleltolinenthen, landn are alsoparallel. m l n a If a is perpendicular tocand a isparalleltobthenbis perpendicular toc. b

  11. Slopeof a Line: How tofindtheslopeof a line? Youneedtohaveatleast 2 pointsor a graphof a lineto be abletofindthe vertical changeoverthe horizontal change.(rise/run) Formula: m=y2-y1/ x2- x1 ParallelLines: Twoparallellinesmusthavesameslope. Perpendicular Lines: Haveoppositesandreciprocalslopes.

  12. examples perpendicular: M=y2-y1/x2-x1 = (3)- (-1) / (-1)- (1)= 4/2= -2 Y-y1= m (x-x1) Y-(-1)=-2 (x-1) y+1= -2x+2 Y=-2x+1 4x+2y=6 2y= 4x+6 Y=-2x+3 M=-2 Y-y1=m (x-x1) Y-(-1)=-2 (x-2) Y+1=-2x+4 Y=-2x+3 Y-5= -2/5x=6/5 Y=-2/5x=6/5+5 Y=-2/5x+31/5

  13. ExamplesofParallel: (6,-1) mll=1/2 Y-y1=m (x-x1) Y-(-1)=1/2 (x-6) Y+1=1/2 x-3 Y+1-1=1/2 x -3 -1 Y=1/2x-4 M=y2-y1/ x2-x1= (-1)-3/ 2-(-2)= -1-3/2+2 =-4/4 =-1 Y=(-5)= -4/3 (x-2) Y+5= -4/3x+ 8/3 3y+15 = -4x+8 4x+3y=-7

  14. Equations: SlopeInterceptForm: Towriteanequation in thisformyoumust know theslopeandatleastonepoint in ordertofindthe y- intercept. Formula: y=mx+b PointSlopeForm: Itisusedtowriteanequationwhenknowingtheslopeand a pointthatcrossestheline. Formula: (y-y1) =m (x-x1) Whento use eachform: SlopeInterceptForm: isusefulwhenyouneedtographtheline. PointSlopeForm: isusefulwheneveryouhavetowriteanequation. Real LifeSituations are: Business Sells, construction GrowthofPopulation

  15. Examples: Slope- IntercepForm:

  16. Point- SlopeForm:

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