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Parallel Lines Congruent Segments Theorem

Interact with the following applet for a few minutes.  Be sure to change the location of each BIG point.  Be sure to move the slider at some point as well.  Then complete the activity that follows.  
The applet above suggests a very powerful, yet often unused theorem in all of geometry. See if you can rearrange the list of words below to form a conditional ("If-then") statement that states what is shown in the applet above!   segments        one           parallel             then every      cut        transversal    segments      lines or       congruent      they       transversal     cut If      congruent     off       on       more 3   on    off
Write a formal proof of the conditional statement (whose words you've arranged) above.