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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
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines:
2. Parallel Planes:
3. Skew Lines:
4. Transversal:
5. Interior Angles:
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes:
3. Skew Lines:
4. Transversal:
5. Interior Angles:
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes: Planes that do not intersect
3. Skew Lines:
4. Transversal:
5. Interior Angles:
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes: Planes that do not intersect
3. Skew Lines: Noncoplanar lines that do not intersect and are not parallel
4. Transversal:
5. Interior Angles:
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes: Planes that do not intersect
3. Skew Lines: Noncoplanar lines that do not intersect and are not parallel
4. Transversal: A line that intersects two other coplanar lines at different points
5. Interior Angles:
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes: Planes that do not intersect
3. Skew Lines: Noncoplanar lines that do not intersect and are not parallel
4. Transversal: A line that intersects two other coplanar lines at different points
5. Interior Angles: Angles formed in between two coplanar lines when intersected by a transversal
6. Exterior Angles:
Tuesday, January 25, 2011
VOCABULARY1. Parallel Lines: Coplanar lines that do not intersect
2. Parallel Planes: Planes that do not intersect
3. Skew Lines: Noncoplanar lines that do not intersect and are not parallel
4. Transversal: A line that intersects two other coplanar lines at different points
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
Tuesday, January 25, 2011
VOCABULARY7. Alternate Interior Angles:
8. Same-side Interior Angles:
9. Alternate Exterior Angles:
10. Corresponding Angles:
Tuesday, January 25, 2011
VOCABULARY7. Alternate Interior Angles: Interior angles on opposite sides of a
transversal; these angles are congruent (parallel lines)
8. Same-side Interior Angles:
9. Alternate Exterior Angles:
10. Corresponding Angles:
Tuesday, January 25, 2011
VOCABULARY7. Alternate Interior Angles: Interior angles on opposite sides of a
transversal; these angles are congruent (parallel lines)
8. Same-side Interior Angles: Interior angles on the same side of a transversal; these angles are supplementary (parallel lines)
9. Alternate Exterior Angles:
10. Corresponding Angles:
Tuesday, January 25, 2011
VOCABULARY7. Alternate Interior Angles: Interior angles on opposite sides of a
transversal; these angles are congruent (parallel lines)
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)
10. Corresponding Angles:
Tuesday, January 25, 2011
VOCABULARY7. Alternate Interior Angles: Interior angles on opposite sides of a
transversal; these angles are congruent (parallel lines)
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)
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)
Tuesday, January 25, 2011
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Interior Angles:
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Interior Angles: 3, 4, 5, 6
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Interior Angles: 3, 4, 5, 6
Exterior Angles:
Tuesday, January 25, 2011
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Tuesday, January 25, 2011
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles:
Tuesday, January 25, 2011
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Tuesday, January 25, 2011
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles:
Tuesday, January 25, 2011
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b
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles: 3 & 5, 4 & 6
Tuesday, January 25, 2011
1 23 4
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles: 3 & 5, 4 & 6
Alternate Exterior Angles:
Tuesday, January 25, 2011
1 23 4
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a
b
m
Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles: 3 & 5, 4 & 6
Alternate Exterior Angles: 1 & 8, 2 & 7
Tuesday, January 25, 2011
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b
m
Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles: 3 & 5, 4 & 6
Alternate Exterior Angles: 1 & 8, 2 & 7Corresponding Angles:
Tuesday, January 25, 2011
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Interior Angles: 3, 4, 5, 6
Exterior Angles: 1, 2, 7, 8
Alternate Interior Angles: 3 & 6, 4 & 5
Same-side Interior Angles: 3 & 5, 4 & 6
Alternate Exterior Angles: 1 & 8, 2 & 7Corresponding Angles: 1 & 5, 2 & 6, 3 & 7, 4 & 8
Tuesday, January 25, 2011
Parallel Line Postulates
Tuesday, January 25, 2011
Parallel Line Postulates
If two parallel lines are intersected by a transversal, then corresponding angles are congruent
Tuesday, January 25, 2011
Parallel Line Postulates
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
Tuesday, January 25, 2011
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
Tuesday, January 25, 2011
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
*
Tuesday, January 25, 2011
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
*
*
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
2x =10
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
2x =10 2 2
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
2x =10 2 2 x = 5
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
2x =10 2 2 x = 5
m∠AEF = (4(5) +10)°
Tuesday, January 25, 2011
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
*
*
4x +10 = 2x + 20 −2x −2x −10 −10
2x =10 2 2 x = 5
m∠AEF = (4(5) +10)°
m∠AEF = 30°
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20 −2x −2x
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20 −2x −2x −40 −40
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20 −2x −2x −40 −40
5x = −20
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20 −2x −2x −40 −40
5x = −20 5 5
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
A B
F
E
C D
H
G
*
*
7x + 40 = 2x + 20 −2x −2x −40 −40
5x = −20 5 5 x = −4
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
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)°
Tuesday, January 25, 2011
EXAMPLE 2
In the figure, AB CD
. If m∠FEB = (7x + 40)° and
m∠HFD = (2x + 20)°, find m∠BEF .
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°
Tuesday, January 25, 2011
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.
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Complementary:
Supplementary:
Tuesday, January 25, 2011
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.
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Complementary:None
Supplementary:
Tuesday, January 25, 2011
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.
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Complementary:None
Supplementary:
∠2 and ∠4
Tuesday, January 25, 2011
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.
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Vertical:
Adjacent:
Tuesday, January 25, 2011
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.
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Vertical:
Adjacent:
∠2 and ∠4
Tuesday, January 25, 2011
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.
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Vertical:
Adjacent:
∠2 and ∠4
∠1 and ∠4
Tuesday, January 25, 2011
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.
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Interior:
Alternate Interior:
Tuesday, January 25, 2011
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.
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Interior:
Alternate Interior:
∠3 and ∠6
Tuesday, January 25, 2011
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.
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Interior:
Alternate Interior:
∠3 and ∠6
∠3 and ∠4
Tuesday, January 25, 2011
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.
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Same-side Interior:
Exterior:
Tuesday, January 25, 2011
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.
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Same-side Interior:
Exterior:
∠1 and ∠7
Tuesday, January 25, 2011
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.
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Same-side Interior:
Exterior:
∠1 and ∠7
∠3 and ∠5
Tuesday, January 25, 2011
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.
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Alternate Exterior:
Corresponding:
Tuesday, January 25, 2011
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.
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Alternate Exterior:
Corresponding:
∠1 and ∠5
Tuesday, January 25, 2011
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.
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Alternate Exterior:
Corresponding:
∠1 and ∠5
∠1 and ∠8
Tuesday, January 25, 2011
PROBLEM SET
Tuesday, January 25, 2011
PROBLEM SET
p. 204 #1-35 odd
“The talent of success is nothing more than doing what you can do, well.” - Henry W. Longfellow
Tuesday, January 25, 2011