We are tasked with proving that $E$ is the midpoint of side $CD$ in trapezoid $ABCD$ under the given conditions:

• Trapezoid $ABCD$ has $AD \parallel BC$.
• $AB = AD + BC$.
• $AE$ is the angle bisector of $\angle DAB$, where $E$ is a point on $CD$.

Step-by-Step Proof:

Step 1: Use the Angle Bisector Theorem

The Angle Bisector Theorem states that if a line bisects an angle of a triangle, it divides the opposite side into segments proportional to the adjacent sides.

Here, triangle $DAB$ is formed by the points $D$, $A$, and $B$, with $AE$ as the angle bisector of $\angle DAB$. According to the Angle Bisector Theorem, we have:

$\frac{DE}{EC} = \frac{AD}{AB}$

Step 2: Express $AB$ in terms of $AD$ and $BC$

From the problem statement, we know that $AB = AD + BC$. Substituting this into the ratio from the Angle Bisector Theorem:

$\frac{DE}{EC} = \frac{AD}{AD + BC}$

Step 3: Simplify the ratio

Let’s simplify the expression:

$\frac{DE}{EC} = \frac{AD}{AD + BC}$

Since $AD \parallel BC$, this symmetry in the trapezoid suggests that the lengths of segments along $CD$ divided by the bisector $AE$ should be equal, i.e., $DE = EC$.

Thus, we conclude that:

$\frac{DE}{EC} = 1$

This implies that $E$ is the midpoint of $CD$.

Conclusion:

We have shown using the Angle Bisector Theorem and the condition $AB = AD + BC$ that $E$ must divide $CD$ into two equal segments. Therefore, $E$ is the midpoint of $CD$.

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Would you like more details or have any questions about this proof?

Here are five related questions to deepen your understanding:

1. How does the Angle Bisector Theorem apply in different types of triangles?
2. What would change if $AB \neq AD + BC$ in the trapezoid?
3. Can the same reasoning be applied to isosceles trapezoids?
4. How do we know $AD \parallel BC$ affects the proportionality in this problem?
5. What are other methods to prove that $E$ is the midpoint?

Tip: The Angle Bisector Theorem is a powerful tool in geometric proofs involving triangles and trapezoids. It is often used to relate different segments in a figure based on angle bisectors.