Answer
1. If the trapezoid is isosceles, then its base angles are equal to each other.
2. When the midpoints are joined in order, they create four triangles.
3. Since these midpoint lines are parallel to the midpoint line opposite to them, this means that it is a parallelogram.
Thus, we now just need to prove that consecutive sides are congruent.
4. By the definition of midpoints, the two sides next to the base angles are congruent.
5. From (4) and (2), the triangles on the bottom of the trapezoid are congruent by ASA.
6. Likewise, by ASA, the triangles on the upper part of the trapezoid are congruent.
7. Because CPCTC, this means that the bottom sides are the same length, and the top sides are the same length.
8. Because it is a parallelogram with consecutive sides of the same lengths, it is a rhombus.
