Given: `AB"||"DE. AB=DE,AC"||"DFandAC"||"=DF "To prove:"BC"||"EF and BC=EF` Proof:
Prove: `AC"||"DF" "("given")`
`and" "AC=DF" "("given")`
`thereforeACFD` is a parallelogram.
`implies" "AD"||"CF" "(because"opposite sides of a parallelogram are parallel")...(1)`
`and" "AD"||"CF" "(because"opposite sides of a parallelogram are equal")...(2)`
`Now," "AB"||"DE" "("given")`
`and" "AB"||"DE" "("given")`
`therefore` ABED is a parallelogram.
`implies" "AD"||"BE" "(because"opposite sides of a parallelogram are parallel")...(3)`
`implies" "AD=BE" "(because"opposite sides of a parallelogram are equal")...(4)`
From (1) and (3), we get
`" "CF"||"BE`
And, from (2) and (4), we get
`" "CF=BE`
`therefore` BCEF is a parallelogram.
`implies" "BC"||"EF" "(because"opposite sides of a parallelogram are parallel")`
`implies" "BC=EF" "(because"opposite sides of a parallelogram are equal")`
