How do you prove this theorem on trapezoids and its median? The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

Question

Claire Akin · Accepted Answer

The proof of this theorem about mid-segment of a trapezoid is below.

Let #ABCD# be a trapezoid with lower base #AD# and upper base #BC#. #M# is a midpoint of left leg #AB# and #N# is a midpoint of right leg #CD#. Connect vertex #B# with midpoint #N# of opposite leg #CD# and extend it beyond point #N# to intersect with continuation of lower base #AD# at point #X#. Consider two triangles #Delta BCN# and #Delta NDX#. They are congruent by angle-side-angle theorem because (a) angles ∠#BNC# and ∠#DNX# are vertical, (b) segments #CN# and #ND# are congruent (since point #N# is a midpoint of segment #CD#), (c) angles ∠#BCN# and ∠#NDX# are alternate interior angles with parallel lines #BC# and #DX# and transversal #CD#. Therefore, segments #BC# and #DX# are congruent, as well as segments #BN# and #NX#, which implies that #N# is a midpoint of segment #BX#. Now consider triangle #Delta ABX#. Since #M# is a midpoint of leg #AB# by a premise of this theorem and #N# is a midpoint of segment #BX#, as was just proven, segment #MN# is a mid-segment of triangle #Delta ABX# and, therefore, is parallel to its base #AX# and equal to its half. But #AX# is a sum of lower base #AD# and segment #DX#, which is congruent to upper base #BC#. Therefore, #MN# is equal to half of sum of two bases #AD# and #BC#. End of proof. The lecture dedicated to this and other properties of quadrilaterals as well as many other topics are addressed by a course of advanced math for high school students at Unizor.

Avery Allen · Answer

undefined To prove the theorem on trapezoids and their medians:

1. Draw a trapezoid ABCD where AB is parallel to DC.
2. Draw the median EF, where E is the midpoint of AD and F is the midpoint of BC.
3. Use the properties of triangles and parallel lines to show that EF is parallel to AB and DC.
4. Show that EF is one half the sum of the lengths of the bases AB and DC.
   - Prove that AE is congruent to ED and BF is congruent to FC.
   - Use the segment addition postulate to show that EF = AE + BF = ED + FC = 1/2(AD + BC).
5. Conclude that the median EF of the trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.