LINES CUT BY A TRANSVERSAL

3Geometry Lesson: Proving Lines are Parallel

7

Aim: When are lines 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 || k4) If lines l and k are extended and they never intersect, what can we say about l and k ?

l is not || k5) If lines l and k are extended and they do intersect, what can we say about l and k ?

8Geometry 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.

9Geometry 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

Proving lines parallel

Two lines cut by a transversal are parallel if a pair of alternate interior angles are congruent.

Theorem :

1234 87

65

m

k

Ex: If 2 7, then || .m k

Theorem : 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 :

Ex: If 2 3 180, then || .m m m k

Two lines perpendicular to the same line are parallel.Theorem :

l

Ex: If and , then || .m l k l m k

11Geometry 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

12Geometry 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 angleson the same side of transversal.A B

13Geometry Lesson: Proving Lines are Parallel

Ex Proving lines parallel: C

D

A

B 12

3

Statements Reasons1) 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 .