Let's break down the problem step by step:

We are given a triangle $ABC$ with the following information:

• A bisector $AD$ is drawn from vertex $A$ to side $BC$, making $D$ the point on $BC$.
• The length of the bisector $AD = 10 \, \text{cm}$.
• The sum of the lengths of sides $AC + DC = 25 \, \text{cm}$.
• A line $DE$ is drawn parallel to $AB$, and $ED = BD$.

Key Insights:

1. Angle Bisector Theorem: This theorem tells us that the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. Specifically: $\frac{AC}{DC} = \frac{AB}{BD}$

2. Parallel Line Implication: Since $DE$ is parallel to $AB$, triangles $ADE$ and $ABD$ are similar by AA (Angle-Angle) similarity.

Step-by-Step Approach:

1. Use the Angle Bisector Theorem to relate the sides of the triangle: From the theorem: $\frac{AC}{DC} = \frac{AB}{BD}$ Let $AB = x$ and $BD = y$. Then, from the theorem: $\frac{AC}{DC} = \frac{x}{y}$

2. Express the sides in terms of known quantities: From the sum $AC + DC = 25$, we can express $AC$ as $AC = 25 - DC$, where $DC = y$ (since $D$ is on side $BC$). Thus: $AC = 25 - y$ The equation becomes: $\frac{25 - y}{y} = \frac{x}{y}$ Simplifying, we get: $25 - y = x$ So, $x = 25 - y$, meaning $AB = 25 - BD$.

3. Use the similarity of triangles: Since $DE \parallel AB$, triangles $ADE$ and $ABD$ are similar. Therefore, the corresponding sides are proportional: $\frac{AE}{AB} = \frac{AD}{AB}$ This simplifies to: $\frac{AE}{AB} = \frac{AD}{AB}$ Therefore: $AE = AD = 10 \, \text{cm}.$

Final Answer:

The length of $AE$ is 10 cm.

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Would you like further clarification on any of these steps? Here are some related questions:

1. How do you prove the Angle Bisector Theorem in a triangle?
2. What does the similarity of triangles tell us about proportionality?
3. Can the Angle Bisector Theorem be used in non-triangular shapes?
4. What is the significance of parallel lines in triangle geometry?
5. How does similarity in geometry relate to real-world applications?

Tip: When working with similar triangles, always look for proportional relationships between corresponding sides. They help simplify the problem!