Midpoint Parallel 82249F

1. **Problem Statement:** Given a quadrilateral $ABCD$ where $E$ is the midpoint of both $AD$ and $BC$, prove that the line segments $AB$ and $CD$ are parallel. 2. **Key Concept:** If a line segment joining the midpoints of two sides of a quadrilateral is parallel to the other two sides, then those sides are parallel. 3. **Step 1: Define Midpoints** Since $E$ is the midpoint of $AD$, we have: $$E = \frac{A + D}{2}$$ Similarly, since $E$ is the midpoint of $BC$, we have: $$E = \frac{B + C}{2}$$ 4. **Step 2: Equate Midpoints** From the above, we get: $$\frac{A + D}{2} = \frac{B + C}{2}$$ Multiply both sides by 2: $$A + D = B + C$$ 5. **Step 3: Rearrange to find vector relation** Rearranging: $$A - B = C - D$$ This means the vector from $B$ to $A$ is equal to the vector from $D$ to $C$: $$\overrightarrow{AB} = \overrightarrow{DC}$$ 6. **Step 4: Conclusion** Since $\overrightarrow{AB}$ and $\overrightarrow{DC}$ are equal vectors, they have the same direction and magnitude. Therefore, the line segments $AB$ and $CD$ are parallel and equal in length. **Final answer:** $$AB \parallel CD$$