Geometry proofs (parallelograms) 1) given: < reu=< tfs, ue=sf, re=ft   prove rstu is a parallelogram 2) given: quadrilateral abcd, ab = cd, ab is parallel to cd prove abcd is a parallelogram images for the two:

Since measure REU equal measure SFT, RE=FT and SF=EU then the two triangles REU et SFT are similar. 
Then we deduce that the two sides RU and ST are equal, RU=ST. 
Also, since the two triangles above are similar, then the two angles FST and RUE are equal. We deduce that the two lines RU and ST are parallel (interior opposite angles principles.)
We have two facts now:
RU = ST and RU parallel to ST, we deduce that the quadrilateral is a parallelogram. 

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