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Geo journal 3
 

Geo journal 3

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M2 3rd Geometry Journal!

M2 3rd Geometry Journal!

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    Geo journal 3 Geo journal 3 Presentation Transcript

    • 3rd Geometry
      Journal
      By: IgnacioRodríguez
    • Parallellines/planes
      and skew lines
      Parallel lines: two coplanar lines that never intersect.
      Ex: AB and CD, AC and EG, FH and EG
      Parallel planes: two planes that never intersect.
      Ex:[] ABDC and [] EFHG, [] EACG and [] FBDH, [] ABFE and [] CDHG
      Skew lines: two lines that have no relationship whatsoever.
      Ex: AC and EF, GH and AE, BD and CG
      A B
      E F
      ([] means plane)
      C D
      G H
    • Transversal
      It is a line that intersects two other lines.
      EX:
    • AnglesFormed
      by the Transversal
      Corresponding: angles that lie in
      the same side of the transversal.
      EX: <1and<5, <4and<8, etc.
      1 2
      Alternate exterior: angles in the 3 4
      opposite side of the transversal
      but in the outside.
      Ex: <1and<8 and <2and<7 5 6
      7 8
      Alternate interior: angles in the opposite
      side of the transversal but in the interior.
      Ex: <3and<6, <4and <5
      Same-side interior: same side of the transversal in the interior.
      Ex: <3and<5, <4and<6
    • Corresponding Angles
      Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
      Converse: if the pairs of corresponding angles are congruent, then two parallel lines have to be cut by a transversal.
      Corresponding angles:
      <1and<5 1 2
      <2and<6 3 4
      <3and<7
      <4and<8
      5 6
      7 8
    • Alternate Exterior
      Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Exterior angles are congruent.
      Converse: If the pairs of Alternate Exterior angles are congruent, then two parallel lines were cut by a transversal.
      Alternate exterior angles:
      <1and<8 1 2
      <2and<7 3 4
      5 6
      7 8
    • Alternate Interior
      Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Interior angles are congruent.
      Converse: If the pairs of Alternate Interior angles are congruent, then two parallel lines were cut by a transversal.
      Alternate exterior angles:
      <3and<6 1 2
      <4and<5 3 4
      5 6
      7 8
    • Same-Side Interior
      Postulate: If two parallel lines are cut by a transversal, then the pairs of Same-Side Interior angles are supplementary.
      Converse: If the pairs of Same-Side Interior angles are Supplementary, then two parallel lines were cut by a transversal.
      Alternate exterior angles:
      <3and<5 1 2
      <4and<6 3 4
      5 6
      7 8
    • Perpendicular
      Transversal Theorem
      Theorem: If a line is perpendicular to one of the parallel lines, then it must be perpendicular to the other line too.
      Ex:
      A _|_ B  A _|_ C I _|_ G  I _|_ Y M _|_ J  M _|_ E
      A G Y I
      B
      I J
      C
      E
    • Howdoes the
      Transative property
      Apply in Parallel and
      Perpendicular lines?
      We know that parallel lines never touch so if line A is parallel to line B and line B is parallal to line C, then line A is parallel to line C.
      In perpendicular lines this is not possible because if line A is perpendicular to line B and B is perpendicular to line C then line A and line C mudt be parallel.
      Ex: B B A B C
      C
      A C A
      B
      C A
    • Slope
      Slope is the rise of a line over the run of that same line (rise/run)
      In many equations slope is represented by the lower-case letter m.}
      Formula: Y¹ –Y² (X,Y) (X,Y)
      X¹- X²
      1 no -1/3
      slope 0
    • Slope´srelation
      With Parallel and
      Perpendicular lines
      Parallel: All parallel lines have the
      same slope as its complementing pair.
      slopes: line1=1 line1= -1/3
      line2=1 line2= -1/3
      Perpendicular: All perpendicular lines
      have the negative reciprocal slope of its
      complementing pair.
      slopes: line1= -1/3 line1=1/6
      line2= 3/1 line2= -6/1
    • Slope/Intercept
      Form
      Formula: Y=mX+b
      You would use it when the slope and interceps are given.
      Ex:
      Y=1X+2 Y=1/2+1 Y=-2/3-2
    • Point/Intercept
      Form
      Formula: Y-Y¹= m(X-X¹)
      You would use it when points are given.
      Ex:
      Y-3=1(X+2) Y-0=1/2(X+3) Y+1=(X+0)