Prove that a line that divides two sides of a triangle proportionally is parallel to the third side. Be sure to create and name the appropriate geometric figures.

I've gotten as far as saying this:

If we said we had triangle ABC, like drawn here: http://prntscr.com/94jhd5
If we drew the line that would proportionally divide the triangle, like this: http://prntscr.com/94ji00
We can name that line DE, like I did in the example figure.

I've read up on this a lot and I figured it would be the converse of the Triangle Proportionality Theorem, but I'm lost as to how to show that.

Question

Prove that a line that divides two sides of a triangle proportionally is parallel to the third side. Be sure to create and name the appropriate geometric figures.

I've gotten as far as saying this: 

If we said we had triangle ABC, like drawn here: http://prntscr.com/94jhd5 
If we drew the line that would proportionally divide the triangle, like this: http://prntscr.com/94ji00
We can name that line DE, like I did in the example figure. 

I've read up on this a lot and I figured it would be the converse of the Triangle Proportionality Theorem, but I'm lost as to how to show that.

leahhhmorgannn · Answer

@phi

phi · Answer

If the corresponding legs of two triangles are "in proportion", then the two triangles are similar.  I think there is a theorem that says that , or we have to prove that.

once we know that, then we have "corresponding angles" are congruent
and then we use that to show corresponding angles of a transversal are equal
which means the lines crossed by the transversal are parallel

leahhhmorgannn · Answer

I saw someone proving this by saying angles B and D are corresponding, but I wasn't sure how that proved proportionality because I only know that to prove similarity.

phi · Answer

You are given
 a line that divides two sides of a triangle proportionally

so we know the sides are divided proportionally
we use that fact to show the triangles are similar.

leahhhmorgannn · Answer

And how do we do that?

phi · Answer

I think it is a definition of similar:
if corresponding sides of two shapes are "in proportion", then the shapes are similar.

leahhhmorgannn · Answer

How would I write a proof for all of this though? That's what I need help with.

phi · Answer

Given  triangle ABC and line DE that divides the sides AB and BC in proportion.
1) triangle ABC is similar to triangle ADE;  reason: sides are proportion (given)
2) angle D is congruent to angle B;  reason:  corresponding angles of similar triangles
3) corresponding angles of transversal AB crossing lines AC and DE are congruent; reason: step 2
4) lines AC and DE are parallel;  reason:  corresponding angles are congruent by step 3

phi · Answer

here is the picture I am assuming
|dw:1447949309778:dw|