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Lines in a Plane

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Title: Lines in a Plane


1
Chapter 3
  • Lines in a Plane

2
Line Relationships
Intersecting lines always coplanar
Perpendicular
Oblique (not -)
Non-intersecting lines maybe coplanar
Parallel (coplanar)
Skew (non-coplanar)
3
Theorem. - Transitivity of Parallel Lines
If two lines are parallel to the same line, then
they are parallel to each other.
l 3
l 3
l 1
l 2
l 2
l 1
l 1 l 2
l 2 l 3
l 1 l 3
4
Theorem Property of Perpendicular Lines
If two coplanar lines are perpendicular to the
same line, then they are parallel to each other.
l 1
l 2
l 3
l 1 - l 3
l 2 l 3
l 1 - l 2
5
Parallel Postulate
If there is a line and a point not on the line,
then there is exactly one line through the point
parallel to the given line.
This an enabling postulate, allows us to draw the
parallel line.
Any other coplanar line through the point will
eventually intersect the line.
6
Perpendicular Postulate
If there is a line and a point not on the line,
then there is exactly one line through the point
perpendicular to the given line.
This an enabling postulate, allows us to draw the
perpendicular line.
Any other coplanar line through the point will
not be perpendicular to the line.
7
Coordinate Geometry Solutions to systems
y m1x b1 y m2x b2
line 1 line 2
m1 m2, b1 ? b2
m1 ? m2
m1 m2, b1 b2
no intersection no solution
one point intersection unique solution
coincident (same) lines infinitely many solutions
8
Using Laws of Logic
  • Negation
  • Contrapositive
  • Structure of an argument
  • Law of syllogism
  • Law of detachment

9
Review conditional statements
If a bird is a pelican, then it eats fish.
p ? q
Conditional
hypothesis
conclusion
q ? p
If a bird eats fish, then it is a pelican
Converse
Two angles are complementary iff the sum of their
measures is 90.
Bi-conditional
p ? q
Negation formed by denying the hypothesis and
conclusion
If a bird is not a pelican, then it does not eat
fish.
p ? q
negation doesnt tell us what it does eat, simply
that it is not fish
10
Contrapositive formed by writing the negation
of the converse of a statement

If a bird is not a pelican, then it does not eat
fish.
p ? q
If a bird does not eat fish, then it is not a
pelican.
q ? p
The truth value of a conditional statement and
its contrapositive are always the same.
They are either both true or both false.
11
Practice
If a fruit is a banana, then it is high in
potassium.
conditional
p ? q
write each of the following
negation
If a fruit is not a banana, then it is not high
in potassium.
p ? q
converse
q ? p
If a fruit is high in potassium, then it is a
banana.
contrapositive
q ? p
If a fruit is not high in potassium, then it is
not a banana.
12
Structure of a Logical Argument
1. Theorem Hypothesis, Conclusion
2. Argument body Series of logical statements,
beginning with the Hypothesis and ending with the
Conclusion.
3. Restatement of the Theorem. (I told you so)
13
If you are careless with fire, Then a fish will
die.
  • If you are careless with fire, Then there will
    be a forest fire.
  • If there is a forest fire, Then there will be
    nothing to trap the rain.
  • If there is nothing to trap the rain,
  • Then the mud will run into the river.
  • If the mud runs into the river, Then the gills of
    the fish will get clogged with silt
  • If the gills of the fish get clogged with silt,
  • Then the fish cant breathe.
  • If a fish cant breath, Then a fish will die

?If you are careless with fire, Then a fish will
die.
14
If Hans goes to Insbruck, then the town will be
smothered
  1. If Hans goes to Insbruck, then he will go skiing.
  2. If Hans begins to yodel, then he will start an
    avalanche.
  3. If Hans is inspired, then he will begin to yodel.
  4. If he goes skiing, then he will be inspired.
  5. If there is an avalanche, then the town will be
    smothered.

15
If Hans goes to Insbruck, then the town will be
smothered
  • If Hans goes to Insbruck, then he will go skiing.
  • If he goes skiing, then he will be inspired.
  • If Hans is inspired, then he will begin to yodel.
  • If Hans begins to yodel, then he will start an
    avalanche.
  • If there is an avalanche, then the town will be
    smothered.

16
Laws of logic
  • Law of syllogism
  • p ? q
  • q ? r
  • ? p ? r
  • domino theory is true
  • Law of detachment
  • p ? q
  • p is true
  • ? q is true
  • must tip first domino

17
example assume each statement is true
  • If Mike visits Norfolk, then he will go to Busch
    Gardens.
  • If Mike goes to Busch Gardens, then he will ride
    the Drachen Fire.
  • Can you conclude Mike rode the Drachen Fire?
  • p Mike visits Norfolk
  • q Mike goes to Busch Gardens
  • r Mike rides the Drachen Fire.
  • Since p ? q, q ? r, are true law of syllogism
    infers p ? r.
  • Did Mike ride the Drachen Fire?
  • Need law of detachment.
  • Mike went to Norfolk last July. ? Mike rode the
    Drachen Fire.

18
example
If the cat does not run out of the room, then
Chad will not scream.
contrapositive
If Chad screams then the cat runs out of the room
If a mouse jumps out of the dresser, then Chad
will scream
If the dresser drawer is opened, then a mouse
will jump out
Chad opened the dresser drawer
  • If the dresser drawer is opened, then a mouse
    will jump out
  • If a mouse jumps out of the dresser, then Chad
    will scream
  • If Chad screams

?
, then the cat runs out of the room
The cat runs out of the room
123/1 5, 7 13 odd, 15, 20 24
19
Lesson 3.4 No Name Theorems
Th. 3.3 - If two lines are perpendicular, then
they intersect to form 4 right angles.
Def. of perpendicular lines If two lines meet
to form a right angle then they are perpendicular.
l 1
1
4
l 2
3
2
statement
reason
2. l 1 - l 2
3. ?1, ?2, ?3, ?4 are right angles
3. - lines meet form 4 rt. ?s
20
Lesson 3.4 No Name Theorems
Th. 3.4 All right angles are congruent.
use flow proof
If two angles are both right angles, then they
are congruent
Given ?1 ?2 are right angles
Prove ?1 ? ?2
2
1
?1 is a rt. ?
m ?1 90
Given
Def. rt. angle
m ?1 m ?2
?1 ? ?2
Transitive prop
Def ? angles
?2 is a rt. ?
m ?2 90
Given
Def. rt. angle
21
Lesson 3.4 No Name Theorems
Th. 3.4 All right angles are congruent.
2 column proof
If two angles are both right angles, then they
are congruent
Given ?1 ?2 are right angles
Prove ?1 ? ?2
2
1
reason
statement
1. ?1 is a rt. ?
1. Given
2. ?2 is a rt. ?
2. Given
3. m ?1 90
3. Def. rt. angle
4. m ?2 90
4. Def. rt. angle
5. m ?1 m ?2
5. Transitive prop
6. ?1 ? ?2
6. Def ? angles
22
Lesson 3.4 No Name Theorems
Th. 3.5 If two lines intersect to form a pair
of adjacent congruent angles, then the lines are
perpendicular.
l 1
Given ?1 ? ? 2
1
2
l 2
Prove l 1 ? l 2
reason
statement
1. ?1 ? ? 2
1. Given
2. ?1 ?2 are a linear pair
2. Def. of a linear pair
3. m ?1 m ?2 180
3. Linear pair postulate
4. m ?1 m ?2
4. Definition of congruent angles
5. m ?1 m ?1 180
5. Substitution
6. 2(m ?1) 180
6. Distributive property
7. m ?1 90
7. Division prop
8. ?1 is a right angle
8. Def. of a right angle
9. l 1 ? l 2
9. Def. of perpendicular lines
23
Assignment 129/9, 10 133/1 4, 7 11 write
a two column proof for both problems 10 11.
24
10.
Given ?5 ? ?6
5
7
6
Prove ?5 ? ?7
reason
statement
25
11.
Given m?A m?B 90 m?C m?B 90
no diagram
Prove ?A ? ?C
reason
statement
26
Angles formed by a Transversal
A transversal is a line that intersects two or
more coplanar lines at different points
exterior
1
2
?1 ? 3, linear pair
?5 ? 8 vertical angles
3
4
?1 ? 5 corresponding angles
?6 ? 3 alternate interior angles
?2 ? 7 alternate exterior angles
interior
?4 ? 6 consecutive interior angles
6
5
exterior
8
7
alternating opposite sides of transversal
consecutive same side of transversal
27
Corresponding Angles Postulate
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles are
congruent.
1
m
Given m n
Conclusion ?1 ? ? 2
n
2
1. m n
2. ?1 ? ? 2
2. Corresponding Angles Post.
28
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are
congruent.
m
1
Given m n
Conclusion ?1 ? ? 2
n
2
3
create a loop on diagram, use transitive property
1. m n
1. Given
2. ?1 ? ? 3
2. Corresponding Angles Post.
3. ?2 ? ? 3
3. Vertical Angles Theorem
4. ?1 ? ? 2
4. Transitive Property
29
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are
supplementary.
3
m
1
Given m n
Conclusion ?1 ? 2 are supplementary
n
2
1. m n
1. Given
2. m?1 m? 3 180
2. Linear Pair Post.
3. ?2 ? ? 3
3. Corresponding Angles Post
4. m?2 m? 3
4. Def. of Congruent Angles
5. Substitution
5. m?1 m? 2 180
6. ?1 ? 2 are supplementary
6. Def of Supplementary Angles
30
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate exterior angles are
congruent.
1
m
Given m n
Conclusion ?1 ? ? 2
3
n
2
create a loop on diagram, use transitive property
1. m n
1. Given
2. ?1 ? ? 3
2. Corresponding Angles Post.
3. ?2 ? ? 3
3. Vertical Angles Theorem
4. ?1 ? ? 2
4. Transitive Property
31
Perpendicular Transversal Theorem
r
If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the
second line.
1
m
m n
Given r ? m
Conclusion r ? n
2
n
1. r ? m, m n
1. Given
2. ?1 is a right angle
2. Def. of Perpendicular Lines
3. m ?1 90
3. Def of Right Angle
4. ?1 ? ? 2
4. Corresponding Angles Post.
5. m ?1 m ?2
5. Def. of Congruent Angles
6. m ?2 90
6. Substitution
7. ?2 is a right angle
7. Def of Right Angle
8. r ? n
8. Def. of Perpendicular Lines
32
Proving Lines are Parallel
  • Corresponding Angles Converse
  • Alternate Interior Angles Converse
  • Consecutive Interior Angles Converse
  • Alternate Exterior Angles Converse

33
Corresponding Angles Converse
If two lines are cut by a transversal so that
corresponding angles are congruent, then the
lines are parallel.
1
m
Given ?1 ? ?2
n
2
Conclusion m n
1. ?1 ? ? 2
2. m n
2. Corresponding Angles Converse
34
Alternate Interior Angles Converse
If two lines are cut by a transversal so that
alternate interior angles are congruent, then the
lines are parallel.
m
1
Given ?1 ? ?2
n
2
Conclusion m n
1. ?1 ? ? 2
2. m n
2. Alternate Interior Angles Converse
35
Consecutive Interior Angles Converse
If two lines are cut by a transversal so that
consecutive interior angles are supplementary,
then the lines are parallel.
3
m
Given ?1 ? 2 are supplementary
1
Conclusion m n
n
2
1. ?1 ? 2 are supplementary
2. m n
2. Consecutive Interior Angles Converse
36
Alternate Exterior Angles Converse
If two lines are cut by a transversal so that
alternate exterior angles are congruent, then the
lines are parallel.
1
m
Given ?1 ? ?2
n
Conclusion m n
2
1. ?1 ? ? 2
2. m n
2. Alternate Exterior Angles Converse
37
1
m
What do we know about this diagram?
n
2
144/1 16 Test Moved to Wednesday 11/2
38
1
m
What do we know about this diagram?
n
2
Parallel Lines?
Circular Reasoning
Law of Detachment!!
Congruent Angles?
39
1
m
What do we know about this diagram?
n
2
Parallel Lines?
Circular Reasoning
Law of Detachment!!
Congruent Angles?
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