geo journal 3

3rd Geometry Journal

By: Ignacio Rodríguez

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

TransversalIt is a line that intersects two other lines.

EX:

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

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

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.

Alternate Interior

Converse: If the pairs of Alternate Interior angles are congruent, then two parallel lines were cut by a transversal.


<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.

Same-Side Interior

Converse: If the pairs of Same-Side Interior angles are Supplementary, then two parallel lines were cut by a transversal.


<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.

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

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

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

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

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

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)

geo journal 3

Education