Question 840921: ABCD is a parallelogram. M is a point in the middle of CD. A straight line is drawn from A to M and extended to arrive the extension of BC. Prove that BC = CM
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Some things were lost in translation here.
BC = CM may happen in some cases, but will not always be that way, so it cannot be proven.
There is no point (pun intended) in
drawing a line passing through A and M,
extending BC, and
seeing the lines AM and BC intersect,
if you are not going to name and use the intersection point.
If ABCD is a parallelogram,
M is the midpoint of segment CD, and
P is the point where line AM intersects line BC,
then BC = CP, and you can prove that.
Lines AD and BP are parallel because opposite sides of parallelogram ABCD are part of those lines, and opposite sides of a parallelogram are parallel.
Angles ADM and PCM are congruent because they are alternate interior angles, on either side of line DC, between parallel lines AD and BP.
(You could instead use the fact that angles MAD and MPC are congruent because they are alternate interior angles, on either side of line AM, between parallel lines AD and BP).
Line segments DM and MC are congruent because M is the midpoint of DC and that's the definition of midpoint.
Angles AMD and PMC are congruent because they are vertical angles.
Triangles ADM and PCM are congruent by ASA congruency (because they have a pair of congruent sides between pairs of congruent angles, as shown above.
Sides CP and AD are congruent by CPCTC (corresponding parts of congruent triangles are congruent).
AD and BC are congruent because they are opposite sides of parallelogram ABCD, and opposite sides of a parallelogram are congruent.
Since CP and AD are congruent, and AD and BC are congruent,
then CP and BC are congruent. (Maybe you have to say by the transitive property of congruency or something like that.)
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