In `DeltaABC`, two points D and E are taken on the lines AB and BC respectively in such a way that AC is parallel to DE. Then `DeltaABC and DeltaDBE `are

Options

A. similar only if D lies outside the line segment AB
B. congruent only if D lies out side the line segment AB
C. always similar
D. always congruent

Answer

To solve the problem, we need to establish the relationship between triangles ABC and DBE given that AC is parallel to DE.

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Start by sketching triangle ABC. Mark points D on line AB and E on line BC such that AC is parallel to DE.

2. **Identify Corresponding Angles**:
- Since AC is parallel to DE, we can use the properties of parallel lines to identify corresponding angles.
- Angle BAC (∠BAC) is equal to angle BDE (∠BDE) because they are corresponding angles formed by the transversal line AB.

3. **Identify Another Pair of Corresponding Angles**:
- Similarly, angle ACB (∠ACB) is equal to angle DEB (∠DEB) for the same reason (corresponding angles).

4. **Identify the Common Angle**:
- Angle ABC (∠ABC) is equal to angle DBE (∠DBE) because angle B is common to both triangles ABC and DBE.

5. **Establish Similarity**:
- We have now established that:
- ∠BAC = ∠BDE
- ∠ACB = ∠DEB
- ∠ABC = ∠DBE
- Since all three pairs of angles are equal, we can conclude by the Angle-Angle (AA) similarity criterion that triangle ABC is similar to triangle DBE.

6. **Conclusion**:
- Therefore, triangles ABC and DBE are always similar regardless of the position of point D on line AB.

### Final Answer:
Triangles ABC and DBE are **always similar**.