Geo journal 3
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Transcript of Geo journal 3
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3rd Geometry Journal
By: Ignacio Rodríguez
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Parallel lines/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
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TransversalIt is a line that intersects two other lines.
EX:
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Angles Formedby the TransversalCorresponding: angles that lie in the same side of the transversal.EX: <1and<5, <4and<8, etc. 1 2Alternate exterior: angles in the 3 4opposite side of the transversal but in the outside. Ex: <1and<8 and <2and<7 5 6 7 8Alternate interior: angles in the oppositeside 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
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Corresponding AnglesPostulate: 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
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Alternate Exterior
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
Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Exterior angles are congruent.
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Alternate Interior
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
Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Interior angles are congruent.
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Same-Side Interior
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
Postulate: If two parallel lines are cut by a transversal, then the pairs of Same-Side Interior angles are supplementary.
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PerpendicularTransversal TheoremTheorem: 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 JC E
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How does theTransative propertyApply in Parallel andPerpendicular 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
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SlopeSlope 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
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Slope´s relationWith Parallel and
Perpendicular linesParallel: 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 itscomplementing pair.
slopes: line1= -1/3 line1=1/6 line2= 3/1 line2= -6/1
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Slope/InterceptForm
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
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Point/InterceptForm
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)