To prove that two lines cut by a transveral are parallel on must show that?

Answer 1

You can show that two lines cut by a transversal are parallel in a number of ways.

(1) Show that the consecutive interior angles are supplementary. Let's say your lines are arranged like this (ignore the periods, they're just there so the spacing is right):

......................1 | 2

--------------------|-----------------------

.......................8| 3

.........................|

......................7 | 4

--------------------|-----------------------

......................6 | 5

If the lines are parallel, the measures of all the consecutive interior angles should be supplementary. The following should be true:

Angle 8 + Angle 3 = 180 degrees

Angle 3 + Angle 4 = 180 degrees

Angle 4 + Angle 7 = 180 degrees

and

Angle 7 + Angle 8 = 180 degrees

(2) You can also prove that the lines are parallel by showing that the corresponding angles are congruent. Using the line arrangement above, prove any of the following to be true:

Angle 1 = Angle 7

Angle 2 = Angle 4

Angle 3 = Angle 5

or

Angle 8 = Angle 6

(3) Finally, you can use alternate angles (either interior or exterior).

To use alternate interior angles, prove that:

Angle 3 = Angle 7

or

Angle 4 = Angle 8

To use alternate exterior angles, prove that:

Angle 1 = Angle 5

or

Angle 2 = Angle 6

Well, there you have it! Best of luck!