Int Math 2 Section 5-3 1011

SECTION 5-3Parallel Lines and Transversals

Tuesday, January 25, 2011

ESSENTIAL QUESTIONS

How do you identify angles formed by parallel lines and transversals?

How do you identify and use properties of parallel lines?

Where you’ll see this:

Construction, safety, navigation, music


VOCABULARY1. Parallel Lines:

2. Parallel Planes:

3. Skew Lines:

4. Transversal:

5. Interior Angles:

6. Exterior Angles:


VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect

2. Parallel Planes:

3. Skew Lines:

4. Transversal:

5. Interior Angles:

6. Exterior Angles:



2. Parallel Planes: Planes that do not intersect

3. Skew Lines:

4. Transversal:

5. Interior Angles:

6. Exterior Angles:




3. Skew Lines: Noncoplanar lines that do not intersect and are not parallel

4. Transversal:

5. Interior Angles:

6. Exterior Angles:





4. Transversal: A line that intersects two other coplanar lines at different points

5. Interior Angles:

6. Exterior Angles:






5. Interior Angles: Angles formed in between two coplanar lines when intersected by a transversal

6. Exterior Angles:






5. Interior Angles: Angles formed in between two coplanar lines when intersected by a transversal

6. Exterior Angles: Angles formed outside two coplanar lines when intersected by a transversal


VOCABULARY7. Alternate Interior Angles:

8. Same-side Interior Angles:

9. Alternate Exterior Angles:

10. Corresponding Angles:


VOCABULARY7. Alternate Interior Angles: Interior angles on opposite sides of a

transversal; these angles are congruent (parallel lines)

8. Same-side Interior Angles:






8. Same-side Interior Angles: Interior angles on the same side of a transversal; these angles are supplementary (parallel lines)







9. Alternate Exterior Angles: Exterior angles on opposite sides of a transversal; these angles are congruent (parallel lines)






9. Alternate Exterior Angles: Exterior angles on opposite sides of a transversal; these angles are congruent (parallel lines)

10. Corresponding Angles: These angles will have the same position around a transversal and the lines it intersects with; these angles are congruent (parallel lines)


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a

b

m


1 23 4

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a

b

m

Interior Angles:


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a

b

m

Interior Angles: 3, 4, 5, 6


1 23 4

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a

b

m


Exterior Angles:


1 23 4

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a

b

m


Exterior Angles: 1, 2, 7, 8


1 23 4

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a

b

m



Alternate Interior Angles:


1 23 4

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a

b

m



Alternate Interior Angles: 3 & 6, 4 & 5


1 23 4

8765

a

b

m




Same-side Interior Angles:


1 23 4

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a

b

m




Same-side Interior Angles: 3 & 5, 4 & 6


1 23 4

8765

a

b

m





Alternate Exterior Angles:


1 23 4

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a

b

m





Alternate Exterior Angles: 1 & 8, 2 & 7


1 23 4

8765

a

b

m





Alternate Exterior Angles: 1 & 8, 2 & 7Corresponding Angles:


1 23 4

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a

b

m





Alternate Exterior Angles: 1 & 8, 2 & 7Corresponding Angles: 1 & 5, 2 & 6, 3 & 7, 4 & 8


Parallel Line Postulates



If two parallel lines are intersected by a transversal, then corresponding angles are congruent



If two parallel lines are intersected by a transversal, then corresponding angles are congruent

If two lines are intersected by a transversal so that corresponding angles are congruent, then the lines are parallel


EXAMPLE 1

In the figure, AB CD

. If m∠AEF = (4x +10)° and

m∠EFD = (2x + 20)°, find m∠AEF .

A B

F

E

C D

H

G


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10

2x =10


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10

2x =10 2 2


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10

2x =10 2 2 x = 5


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10

2x =10 2 2 x = 5

m∠AEF = (4(5) +10)°


EXAMPLE 1


. If m∠AEF = (4x +10)° and


A B

F

E

C D

H

G

*

*

4x +10 = 2x + 20 −2x −2x −10 −10

2x =10 2 2 x = 5

m∠AEF = (4(5) +10)°

m∠AEF = 30°


EXAMPLE 2


. If m∠FEB = (7x + 40)° and

m∠HFD = (2x + 20)°, find m∠BEF .

A B

F

E

C D

H

G


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40

5x = −20


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40

5x = −20 5 5


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40

5x = −20 5 5 x = −4


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40

5x = −20 5 5 x = −4

m∠BEF = (7(−4) + 40)°


EXAMPLE 2


. If m∠FEB = (7x + 40)° and


A B

F

E

C D

H

G

*

*

7x + 40 = 2x + 20 −2x −2x −40 −40

5x = −20 5 5 x = −4

m∠BEF = (7(−4) + 40)°

m∠BEF =12°


EXAMPLE 3Refer to the figure. For each type of angle relationship mentioned, list a pair of angles that would satisfy the

relationship.

For example, ∠1 and ∠2 are supplementary.

1 23 4

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Complementary:

Supplementary:



relationship.


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Complementary:None

Supplementary:



relationship.


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Complementary:None

Supplementary:

∠2 and ∠4



relationship.


1 23 4

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Vertical:

Adjacent:



relationship.


1 23 4

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Vertical:

Adjacent:

∠2 and ∠4



relationship.


1 23 4

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Vertical:

Adjacent:

∠2 and ∠4

∠1 and ∠4



relationship.


1 23 4

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Interior:

Alternate Interior:



relationship.


1 23 4

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Interior:

Alternate Interior:

∠3 and ∠6



relationship.


1 23 4

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Interior:

Alternate Interior:

∠3 and ∠6

∠3 and ∠4



relationship.


1 23 4

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Same-side Interior:

Exterior:



relationship.


1 23 4

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Same-side Interior:

Exterior:

∠1 and ∠7



relationship.


1 23 4

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Same-side Interior:

Exterior:

∠1 and ∠7

∠3 and ∠5



relationship.


1 23 4

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Alternate Exterior:

Corresponding:



relationship.


1 23 4

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Alternate Exterior:

Corresponding:

∠1 and ∠5



relationship.


1 23 4

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Alternate Exterior:

Corresponding:

∠1 and ∠5

∠1 and ∠8


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PROBLEM SET

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Int Math 2 Section 5-3 1011

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