Download Geometry - 6.2-6.3

Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
• Ch 5 Review Problems pp. 206­209 #15­50 due FRIDAY 01/06
• TEST #2 ­ Monday 01/09??
• Ch 6 Review Problems pp. 250­254 #9­19, 33­53
6.2 – Proving Lines Parallel
Def: Two lines are parallel iff they lie in the same plane and do not intersect.
A transversal is a line that intersects two or more lines in different points.
l1
l2
t
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Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
When a transversal intersects two lines that lie in the same plane, it forms pairs of angles that are given special names:
Alternate interior angles
Corresponding angles
Interior angles on the same side of the transversal
Theorem 17: Equal corresponding angles mean that lines are parallel.
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Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
Corollary 1: Equal alternate interior angles mean that lines are parallel.
Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel.
Corollary 3: In a plane, two lines perpendicular to a third line are parallel.
Corollary 1: Equal alternate interior angles mean that lines are parallel.
Given: 1= 2
Prove: a||b
1
2
Proof:
Statements
3
Reasons
a
b
c
1. Given
2.
Vertical angles are equal
3. Substitution
4.
Equal corresponding angles mean that lines are parallel.
3
Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel.
Given: 1 and 2 are supplementary
Prove: a||b
a
1
2
Proof:
Statements
Reasons
1. Given
2.
The angles in a linear pair are supplementary
3. Supplements of the same angle are equal
4.
Equal corresponding angles mean that lines are parallel.
b
3
c
Corollary 3: In a plane, two lines perpendicular to a third line are parallel.
Given: a c and b c
1
Prove: a||b
2
Proof:
Statements
Reasons
1. Given
2.
Perpendicular lines form right angles
3. All right angles are equal
4.
Equal corresponding angles mean that lines are parallel.
a
b
c
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Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
Given: ABD and BDE are right angles
A
B
Prove:AB||DE
29. Proof:
Statements Reasons
1. ABD and BDE are right angles
Given
C
D
2. AB BD and BD DE
E
3. AB||DE
30. Proof:
Statements
Reasons
1. ABD and BDE are right angles
Given
2. ABD= BDE
3. AB||DE
39. E
Given: AE=AD and E= BCE
D
C
Prove: AD||BC
A
B
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Geometry ­ 6.2­6.3 ­ Parallel Lines
January 05, 2017
6.3 – The Parallel Postulate
Construction 7: To construct a line parallel to a given line through a given point.
183
122°
Theorem 17: Equal corresponding angles mean that lines are parallel.
Corollary 1: Equal alternate interior angles mean that lines are parallel.
Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel.
Corollary 3: In a plane, two lines perpendicular to a third line are parallel.
The Parallel Postulate – Through a point not on a line, there is exactly one line parallel to the given line.
Theorem 18: In a plane, two lines parallel to a third line are parallel to each other.
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