Download Proving lines parallel - Benjamin N. Cardozo High School

LINES CUT BY A TRANSVERSAL
Geometry Lesson: Proving Lines are
Parallel
3
Aim: When are lines parallel?
Do Now:
1) Name 4 pairs of corresponding angles.
1, 5 3, 7 2, 6 4, 8
2) Name 2 pairs of alternate interior angles.
3, 6
4, 5
l
3) Name 2 pairs of alternate exterior angles.
k
12
34
56
7 8
1, 8
2, 7
4) If lines l and k are extended and they never intersect,
what can we say about l and k ?
l || k
5) If lines l and k are extended and they do intersect,
what can we say about l and k ?
l is not || k
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Def.
Def: Parallel:
Parallel lines have no points in common or have
all points in common.
B
AB || CD
EF || EF
F
D
A
E
C
Def:
Def: Transversal
A transversal is a line that intersects two other lines in
two different points.
l
k
Line m is “transverse” to lines l and k.
m
Geometry Lesson: Proving Lines are
Parallel
8
Transversals/ angle pairs:
48
3 7
2 6
1 5
Corresponding angle pairs:
1,3 2,4 5,7
6,8
Alternate interior angles:
2,7 3,6
Alternate exterior angles:
1,8 4,5
Geometry Lesson: Proving Lines are
Parallel
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Proving lines parallel
l
48
3 7
m
2 6
Theorem :
1 5
k
Two lines cut by a transversal are parallel if a pair of
corresponding angles are congruent. Ex: If 8  6, then m || k .
Theorem :
Two lines cut by a transversal are parallel if a pair of alternate
interior angles are congruent. Ex: If 2  7, then m || k .
Theorem :
Two lines cut by a transversal are parallel if a pair of interior
angles on the same side of the transversal are supplementary.
Ex: If m2  m3  180, then m || k .
Theorem :
Two lines perpendicular to the same line are parallel.
Ex: If m  l and k  l , then m || k .
Ex: Proving lines parallel
In each case, what reason can be given to prove that AB || CD ?
E
1)
C
D
2)
Alt. interior 's,
D
C
A
B
Corresp. 's,
A
EDC  DBA
3) A
B
B
4)
E
48
C
132
C
D
Int. 's, same side are suppl.
C  B.
Or
CD  CB and BA  CB
Alt. interior 's,
D
CDA  BAD.
A
B
mB +mD  180
Geometry Lesson: Proving Lines are
Parallel
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Ex: Proving lines parallel
D
C
If mA  100  3x and mB  80  3x,
show that AD || BC.
A
B
A and B are interior angles
on the same side of transversal.
?
mA  mB  180
mA  mB  100  3x  80  3x
mA  mB  180 
Since A and B are supplementary, AD || BC.
Geometry Lesson: Proving Lines are
Parallel
12
C
Ex Proving lines parallel:
Given: BD bisects ABC
BC  CD
Prove: CD || BA
B 12
3
D
A
Statements
Reasons
1) BD bisects ABC
2) BC  CD
1) Given
2) Given
4) 1  3
5) 2  3
3) Def. angle bisector
4) Base 's of isosceles 's are  .
5) Transitive Postulate
6) CD || BA
6) Two lines are || if
3) 1  2
alt. interior 's are  .
Geometry Lesson: Proving Lines are
Parallel
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