State, true or false :
The diagonals of a trapezium divide each other into proportional segments.

Answer

**Step-by-Step Solution:**

1. **Understanding the Question**: We need to determine whether the statement "The diagonals of a trapezium divide each other into proportional segments" is true or false.

2. **Drawing a Trapezium**: Let's draw a trapezium ABCD, where AB is parallel to CD. Label the intersection of the diagonals AC and BD as point O.

3. **Constructing a Parallel Line**: Draw a line segment EO parallel to AB that intersects AD at point E.

4. **Applying the Basic Proportionality Theorem**: In triangle ADC, since EO is parallel to CD, we can apply the Basic Proportionality Theorem (also known as Thales' theorem). This gives us:
\[ \frac{AE}{ED} = \frac{AO}{OC} \quad \text{(Equation 1)} \]

5. **Applying the Theorem Again**: In triangle DAB, since EO is parallel to AB, we can again apply the Basic Proportionality Theorem. This gives us:
\[ \frac{AE}{ED} = \frac{BO}{OD} \quad \text{(Equation 2)} \]

6. **Equating the Ratios**: From Equation 1 and Equation 2, we can set the two ratios equal to each other:
\[ \frac{AO}{OC} = \frac{BO}{OD} \]

7. **Conclusion**: Since we have established that the ratios of the segments created by the diagonals are equal, we conclude that the diagonals of a trapezium do indeed divide each other into proportional segments. Therefore, the statement is **True**.

---