Geometry

Journal 3Michelle 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.Skew Lines: Lines that do not intersect and arenot coplanar and will never touch each other.Examples: Side-walk for a

handicap

Mirror:

Transversal:

Line that intersects two parallel lines at two different points. The transversal “t” and other two lines “r” and “s” forming special pairs of angles such as: corresponding, alternate exterior & interior and same side interior. (eight angles.)

t

r

s

http://2.bp.blogspot.com/_nmM9aWNBB2E/TNGckkc0kzI/AAAAAAAAAB4/LlocG1kq_F4/s1600/beatles_abbyroad.jpg

1 2

53 4

6

7 8

Angles:Corresponding: Lie on the same side of thetransversal. One inside and one outside the parallel

lines. Alternate Exterior: Lie on opposite sides ofthe transversal outside the paralleles. Alternate Interior: Not adjacent angles that lie on the

opposite sides of the transversal inside the parallel lines.

Same-side Interior: Are the angle pair that are on the inside of the two parallel lines forming a pair of supplementary angles.

Alternate Exterior:Alternate Interior:

Corresponding:

Same- Side Interior:

Corresponding Angles Postulate & 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 23 4

5 67 8

1 & 52 & 63 & 74 & 8

Trains Transversal via

http://www.westechme.com/rides%20library/trains/images/CARTOON%20TRAIN_jpg.jpg

Alternate Interior Angles Theorem & 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:

Parking Lot

Electric stairs in mall

http://thumbs.dreamstime.com/thumblarge_136/1175613399z4N9YZ.jpg

Same- Side Interior Angles Theorem & 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:

Neighbors

Boxing Ring:

Alternate Exterior Angles Theorem & 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:Supermarket halls (baskets):

http://www.miss-thrifty.co.uk/wp-content/uploads/2008/08/shopping-basket.jpghttp://www.clipartguide.com/_named_clipart_images/0511-0810-1419-0773_Cartoon_Airplane_clipart_image.jpg

Airport:

Perpendicular Transversal Theorems:

In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other parallel line.

Examples:If a is perpendicular to c and b is perpendicular to c then a is perpendicular to c.

a b d

c a

cbal

m n o

If room a has lines m and n perpendicular to l then, o and q are also perpendicular to l.

q

s

t

p q r

If p is perpendicular to s and q is perpendicular to s then p and q are parallel.

Transitive Property:

Parallel Lines: If one line is parallel to the second one and that second one is parallel to a third one then, the three lines are parallel to each other.Perpendicular Lines: If p is perpendicular to q and r is perpendicular to q also, then p and r must be parallel.Perpendicular Lines Theorem: If two intersecting lines form a linear pair of congruent angles then the lines must be perpendicular.

Examples:l m n

If line l is parallel to line m and line m is parallel to line n then, l and n are also parallel.

a

b

If a is perpendicular to c and a is parallel to b then b is perpendicular to c.

Slope of a Line:

How to find the slope of a line?You need to have atleast 2 points or a graph of a line to be able to find the vertical change over the horizontal change.(rise/run)

Formula: m=y2-y1/ x2- x1Parallel Lines: Two parallel lines must have

same slope.Perpendicular Lines: Have opposites and

reciprocal slopes.

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

4x+2y=62y= 4x+6Y=-2x+3M=-2Y-y1=m (x-x1)Y-(-1)=-2 (x-2)Y+1=-2x+4Y=-2x+3

Y-5= -2/5x=6/5Y=-2/5x=6/5+5Y=-2/5x+31/5

Examples of Parallel:

(6,-1) mll=1/2Y-y1=m (x-x1)Y-(-1)=1/2 (x-6)Y+1=1/2 x-3Y+1-1=1/2 x -3 -1Y=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+ 8/33y+15 = -4x+84x+3y=-7

Equations:

Slope Intercept Form: To write an equation in this form you must know the slope and atleast one point in order to find the y- intercept.

Formula: y=mx+bPoint Slope Form: It is used to write an equation when knowing the slope and a point that crosses the line.

Formula: (y-y1) =m (x-x1)When to use each form:Slope Intercept Form: is useful when you need to graph the

line.Point Slope Form: is useful when ever you have to write an

equation. Real Life Situations are:BusinessSells, constructionGrowth of Population

Examples:Slope- Intercep Form:

Point- Slope Form: