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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
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