USING PARALLEL AND USING PARALLEL AND PERDENDICULAR LINESPERDENDICULAR LINES

--Solving problems by using a diagramSolving problems by using a diagram

-Use properties of parallel lines-Use properties of parallel lines

-Use slope to identify parallel and perpendicular lines-Use slope to identify parallel and perpendicular lines

-Prove lines parallel-Prove lines parallel

-Apply distance relationships among points, lines, -Apply distance relationships among points, lines, and planesand planes

OBJECTIVESOBJECTIVES

ProofsProofs• Proofs are built on ‘if—then’ statements• ALWAYS make a diagram of the data & mark the

‘givens’• The givens are ‘ifs’• Start with ‘if’ and find a postulate, theorem or

definition that will take you to the next step. • The last step is what was to be proven.• Transitivity & substitution are often used for

justification • An example follows at the end of the power point.

Types of Lines

• Parallel lines—same plane, never cross

• Skew lines—different planes, never cross

• Traversal lines –same plane, crosses

• Perpendicular lines—same plane, crosses to form a right angle

Angles from lines cut by transversals• Exterior angles

• Interior angles

• Consecutive interior angles

• Alternate interior angles

• Alternate exterior

• Corresponding angles

12

34

56

7

8

1, 2, 7, 8

3, 4, 5, 6

3& 5, 4& 6

3& 6, 4& 5 1& 8, 2 & 7

1& 5, 2 & 6,

3& 7, 4 & 8

Theorems: IF TWO PARALLEL LINES CUT BY A TRANSVERSAL-- • Each pair of corresponding angles is congruent

(corr /_‘s ~ ) = • Each pair of alternate interior angles is congruent

(AIA)• Each pair of consecutive interior angles is

supplementary (CIA)• Each pair of alternate exterior angles is congruent

(AEA)

Slopes: Parallel & Perpendicular Lines-Given (x1, y1) & (x2, y2): m = , x1≠ x2

-Two nonvertical lines have the same slope if and only if they are parallel.

m1 = m2 iff ℓ1 // ℓ2

-Two nonvertical lines are perpendicular if and only if the products of their slopes is –1

ℓ1┴ ℓ2 iff m1·m2 = –1

2 1

2 1

y y

x x

** iff means if and only if: if m1 = m2 ℓ1 // ℓ2 AND

if ℓ1 // ℓ2 m1 = m2

… then the lines are parallel

If 2 co-planar lines are cut by a transversal that makes:

-the corresponding angles congruent…-or alt.exterior angles congruent…-or a pair of consec.interior angles congruent…-or a pair of alt.interior angles are congruent…

or If 2 co-planar lines are perpendicular to the same line…

If you want to prove lines are parallel match the IF

If there is a line & a point not one the line, only one line can go through the point & be parallel to the line

DistanceDefinition: The distance between a line and a point

not on the line = the length of the perpendicular segment that joins the point to the line

Definition: the distance between two parallel lines = the length of the perpendicular segment that joins the two line (90°angles)

If a line is perpendicular to one of two parallel line, then it is perpendicular to both.

2

2 21 2 1( ) ( )d x x y y

• distance

EXAMPLE: Proof

1 Given

2 Corresponding

3 Transitivity

4 Transitivity

5

Given: , ,

Prove:

1 2 3 4

1 2 3 4

KO||AN

JO||KN

J K A

O N

JO||KN1 ,

2

3

4

5

1 2

3 4

1 3

3 2

2 4

KO||AN||Corr lines

Web Resourceshttp://math.about.com/gi/dynamic/offsite.htm?zi=1/XJ/Ya&sdn=math&cdn=education&tm=29&gps=87_9_796_428&f=00&tt=14&bt=1&bts=0&zu=http%3A//library.thinkquest.org/C0110248/geometry/analytic.

htm

http://hotmath.com/?referrer=goo-cg-geo&gclid=COOiwIXV7IwCFRE4OAodJVev6g

http://www.math.psu.edu/geom/koltsova/index.html