1geometry lesson: aim: how do we prove lines are parallel? do now: 1) name 4 pairs of corresponding...
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1Geometry Lesson: Proving Lines are Parallel
Aim: How do we prove lines are parallel?Do Now:
1) Name 4 pairs of corresponding angles.
2) Name 2 pairs of alternate interior angles.
3) Name 2 pairs of alternate exterior angles.
1, 5 3, 7 2, 6 4, 8
3, 6 4, 5
1, 8 2, 7
k2
3 4
5 67 8
l
1
l || k
4) If lines l and k are extended and they never intersect, what can we say about l and k ?
l is not || k
5) If lines l and k are extended and they do intersect, what can we say about l and k ?
2Geometry Lesson: Proving Lines are Parallel
Def. Parallel:Def: Parallel lines have no points in common or have all points in common.
A
B
D
C
||AB CD���������������������������������������� ���
F
E
||EF EF���������������������������������������� ���
l km
Line m is “transverse” to lines l and k.
Def: TransversalDef: A transversal is a line that intersects two other lines in two different points.
3Geometry Lesson: Proving Lines are Parallel
Transversals/ angle pairs:
1234 87
65
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
4Geometry Lesson: Proving Lines are Parallel
Ex: Transversals/ angle pairs
State the type of each angle pair:
123
5 467 1) 2, 5
2) 1, 6 3) 3, 4 4) 1, 7
1 243
5) 1, 4
6) 2, 3
alt. interiorcorrespondingalt. interioralt. exterior
alt. interior
alt. interior
5Geometry Lesson: Proving Lines are Parallel
Proving lines parallel
Two lines cut by a transversal are parallel if a pair of alternate interior angles are congruent.
Theorem #12:
1234 87
65
m
k
Ex: If 2 7, then || .m k
Theorem #11: Two lines cut by a transversal are parallel if a pair of corresponding angles are congruent. Ex: If 8 6, then || .m k
Two lines cut by a transversal are parallel if a pair of interior angles on the same side of the transversal are supplementary.
Theorem #13:
Ex: If 2 3 180, then || .m m m k
Two lines perpendicular to the same line are parallel.Theorem #14:
l
Ex: If and , then || .m l k l m k
6Geometry Lesson: Proving Lines are Parallel
Ex: Proving lines parallelIn each case, what reason can be given to prove that || ?AB CD
A B
DC
E
(
(
1)
A B
DC
E
)(
4)3) A B
DC132
48
A B
C D2)Alt. interior 's,
C . B
and CD CB BA CB Or
Corresp. 's,
EDC DBA
Int. 's, same side are suppl.
+ 180m B m D
Alt. interior 's,
. CDA BAD
7Geometry Lesson: Proving Lines are Parallel
Ex: Proving lines parallel
A B
CDIf 100 3 and 80 3 ,
show that || .
m A x m B x
AD BC
180m A m B ?
100 3 80 3m A m B x x 180m A m B
Since and are supplementary, || .A B AD BC
and are interior angles
on the same side of transversal.
A B
8Geometry Lesson: Proving Lines are Parallel
Ex Proving lines parallel: C
D
A
B 12
3
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
6) 6)
Given bisects BD ABC
BC CD Given
Given:
Prove:
bisects BD ABC
||CD BABC CD
1 2 Def. angle bisector1 3 2 3
Base 's of isosceles 's are . Transitive Postulate
||CD BA Two lines are || if
alt. interior 's are .
9Geometry Lesson: Proving Lines are Parallel
Proving lines parallel: R
A D
S
L1) Given:
Prove: and bisect each other at RS AD L || RA DS
2) Given:
Prove:P D
M
S
Q, PQS QP QD
MQD QPD
||QM PD
1 2 3 4M B R C
FE3) Given: , 1 2, 3 4MBRC �������������� �
Prove: ||BE RF����������������������������