a Let's mark the congruent segments on the diagram and focus on quadrilateral BCDE. It is given that two sides of this quadrilateral are congruent. Let's show that the other two sides are also congruent.
There is a congruence given in the question which is not indicated by markers on the diagram.
AC≅ CFLet's use that congruent segments have the same length and apply the Segment Addition Postulate for the two segments in this congruence.
AC= CF
⇓
AB+BC=CD+DF
We can use the additional congruences indicated on the diagram to identify segments of equal length and rewrite this equality.
Segments BC and DE have the same length, so they are congruent. We now know that both pairs of opposite sides of quadrilateral BCDE are congruent. According to Theorem 6.9, this means that BCDE is a parallelogram. By definition, this means that opposite sides are parallel.
BE∥CD
Let's sumarize the steps above in a paragraph proof.
Congruent segments have the same length, so AC=CF. According to the Segment Addition Postulate, we can write both sides as a sum, AB+BC=CD+DF. Segments AB and CD are congruent, so we can subtract AB=CD to get BC=DF. Segments DF and DE are also congruent, so we can replace DF with DE to get BC=DE. This means that segments BC and DE are congruent. Since it is given that segments CD and BE are also congruent, Theorem 6.9 guarantees that BCDE is a parallelogram. Opposite sides of a parallelogram are parallel, so BE∥CD.
b Let's mark the given AB=12 inches and DF=8 inches on the diagram. Let's also put the same measurements on the congruent segments.
The width of the copy is the width of the original object multiplied by the scale.
5.5* CF/BE
Using the Segment Addition Postulate, we can rewrite the numerator of the scale as CF=CD+DF. Let's substitute the measures to find the width.
