Parallel and Perpendicular Lines

Question

1.) Prove analytically that the line joining the midpoints of the nonparallel sides of a trapezoid is parallel to the bases of a trapezoid.

Answer 1

Assume a trapezoid ABCD starting left down clockwise.  Let M be the midpoint of AB and N the midpoint of DC.  The mid-segmant will be MN. The question is why MN is parallel to the bases AD and BC.

To prove that connect B and N with line BN and extend this line until it intersects the extension of AD to the right at point X.  Compare triangles BCN and NDX.  They are congruent by two angles and included side.  How?

<BNC = <DNX and

<BCN = <NDX and finally

CN = ND

Since triangles BCN and NDX are congruent, the sides BN and NX are equal.

In the triangle ABX, the line MN intersect with AB and BX at midpoints M and N. Therefore:

MN is parallel to the side AX.  This means that the line MN is parallel to the base AD of the trapezoid since AD is part of AX.