Medians and Mid-Segments

For this problem, you might want to review similar triangles and parallel lines. Below are steps to show why $\overline{DE}$ is parallel to $\overline{BC}$ in the diagram below.

Given that DE is a mid-segment of $\triangle ABC$, show that $\overline{DE}//\overline{BC}$

1. Since DE is the mid-segment, D and E are the mid-points of $\overline{AB}$ and $\overline{AC}$
2. Since D and E are mid-points, $\frac{AD}{AB}=\frac{1}{2}=\frac{AE}{AC}$
3. We also know $\angle DAE = \angle BAC$
4. Because of 2 and 3, we can conclude $\triangle DAE \sim \triangle BAC$
5. Since the two triangles are similar, $\angle AED = \angle ACB$
6. $\overline{DE}//\overline{BC}$