The Midpoint-Connector Theorem for Trapezoids states:
"If a line bisects one leg of a trapezoid and is parallel to the base, then it contains the median, whose length is 1/2 the sum of the lengths of the 2 bases."
"Conversely, the median of a trapezoid is parallel to each of the 2 bases, and its length equals 1/2 the sum of the lengths of the bases."
We proved the theorem in class, and now for homework, we are to prove the converse.
The converse is: The median of a trapezoid is parallel to either base. (By transitivity of parallelism, I only need to show the median parallel to one of the bases.)
My work so far:
Proof: Given trapezoid ABCD with median LN. AB parallel to DC. LN = 1/2 (AB + DC). L midpoint of AD. N midpoint of BC.
Construct diagonal AC.
ALLD.
BNNC.
∠BAC∠DCA.
ACAC.
AMMC.
∠AML∠CMN.
My approach is to use the Z-Property of parallel lines to show LN parallel to DC. So I want to show the alternate interior angles
∠NMC∠MCD.
Then by Transitivity of Parallelism if LN is parallel to DC and DC is parallel to AB, then LN is also parallel to AB.
Now I am stuck as to how I can show the alternate interior angles congruent. Or is it even a legitimate approach?
A hint was given in the text: Suppose the median is not parallel to the base; construct a line passing through the midpoint of one leg parallel to the base, and intersecting the other leg. What must happen?)
The hint just further confused me rather than helped because it seems to contradict the theorem that we previously proved. I am VERY lost in this class (Modern Geometry). The professor is very disorganized, and I always leave the class more confused than before.