Loading...

PPT – Lines in a Plane PowerPoint presentation | free to download - id: 6a2805-MjhkZ

The Adobe Flash plugin is needed to view this content

Chapter 3

- Lines in a Plane

Line Relationships

Intersecting lines always coplanar

Perpendicular

Oblique (not -)

Non-intersecting lines maybe coplanar

Parallel (coplanar)

Skew (non-coplanar)

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

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

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.

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.

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

Using Laws of Logic

- Negation
- Contrapositive
- Structure of an argument
- Law of syllogism
- Law of detachment

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

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.

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.

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)

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.

If Hans goes to Insbruck, then the town will be

smothered

- If Hans goes to Insbruck, then he will go skiing.
- If Hans begins to yodel, then he will start an

avalanche. - If Hans is inspired, then he will begin to yodel.
- If he goes skiing, then he will be inspired.
- If there is an avalanche, then the town will be

smothered.

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.

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

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.

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

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

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

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

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

Assignment 129/9, 10 133/1 4, 7 11 write

a two column proof for both problems 10 11.

10.

Given ?5 ? ?6

5

7

6

Prove ?5 ? ?7

reason

statement

11.

Given m?A m?B 90 m?C m?B 90

no diagram

Prove ?A ? ?C

reason

statement

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

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.

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

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

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

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

Proving Lines are Parallel

- Corresponding Angles Converse
- Alternate Interior Angles Converse
- Consecutive Interior Angles Converse
- Alternate Exterior Angles Converse

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

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

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

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

1

m

What do we know about this diagram?

n

2

144/1 16 Test Moved to Wednesday 11/2

1

m

What do we know about this diagram?

n

2

Parallel Lines?

Circular Reasoning

Law of Detachment!!

Congruent Angles?

1

m

What do we know about this diagram?

n

2

Parallel Lines?

Circular Reasoning

Law of Detachment!!

Congruent Angles?