given trapezoid abcd with ad || bc, ab=ad+bc, and e is a point on CD such that ae is the angle bisector of angle dab. prove e is the midpoint of cd

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 \(CD\)**.

---

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.

Given trapezoid ABCD with AD || BC, AB = AD + BC, and E is a point on CD such that AE is the angle bisector of angle DAB. Prove E is the midpoint of CD.

This geometry problem involves proving that point E is the midpoint of side CD in trapezoid ABCD. Using the Angle Bisector Theorem and the condition AB = AD + BC, the solution demonstrates how the bisector of angle DAB divides side CD proportionally, showing E as the midpoint. Suitable for students in Grades 9-11.

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