Aline goes through the point of intersection of the diagonals of parallelogram abcd so that it intersects sides bc and ad at points e and f respectively. find the length of the sides bc and ad if line be = 2 in and line a f = 2.8 in.

Given Parallelogram ABCD in which diagonals meet at G.
A segment is constructed through G intersecting BC at E and AD at F.
mBE = 2.0 in.
mAF = 2.8 in.
Need to find measures of BC and AD.

In geometry, the first step is to draw a diagram according to given information, if one is not already provided.  (see attachment).

Next, based on basic theorems, deduce as much information as possible, and take note.

Properties of a parallelogram include (but not limited to):
- opposite sides are parallel .................(1)
- opposite sides are congruent. ............(2)
- diagonals bisect each other.................(3)

Strategy: 
prove congruence of triangles BEG and DFG
=> BE=FD
=> mAD=2.8+2.0=4.8 = mBC

Congruence of BEG and DFG
Consider transversal BD, we conclude that
angle EBG is congruent to angle FDG   (alternate interior angles)...(A)
Consider diagonal BD,
mBG = mGD    (property (3) above)..................................................(S)
Consider transversal EF, we conclude that 
angle BEG is congruent to angle FGD  (alternate interior angles)....(A)

=> triangle BEG is congruent to triangle DFG    (ASA).
=> FD=BE=2.0      ............(corresponding sides of congruent triangles)
=> AD=2.8+2.0=4.8
=> BC=AD=4.8........................(property (2) above).

BC=AD=4.8 inches.

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