To show that $ABCD$ is a trapezium, we need to verify that one pair of opposite sides of the quadrilateral is parallel. This can be done by checking if the corresponding angles along these sides add up to $180^\circ$, as this is a property of parallel lines.

Step-by-Step Solution:

1. Identify the given angles:

   • At $A$: $2x + 15^\circ$
   • At $B$: $4x + 15^\circ$
   • At $C$: $4x + 8^\circ$
   • At $D$: $3x^\circ$

2. Trapezium Property: A quadrilateral is a trapezium if one pair of opposite sides is parallel. To verify this, we need to check if the angles on the same side of one pair of opposite sides (for example, $AB$ and $CD$) are supplementary ($180^\circ$).

3. Check $AB \parallel CD$:

   • Angles at $A$ and $D$: Check if $(2x + 15) + (3x) = 180$: $2x + 15 + 3x = 180$ $5x + 15 = 180$ $5x = 165 \quad \Rightarrow \quad x = 33$

4. Substitute $x = 33$ into all angles:

   • At $A$: $2(33) + 15 = 66 + 15 = 81^\circ$
   • At $B$: $4(33) + 15 = 132 + 15 = 147^\circ$
   • At $C$: $4(33) + 8 = 132 + 8 = 140^\circ$
   • At $D$: $3(33) = 99^\circ$

5. Verify parallelism:

   • For $AB \parallel CD$, check if $\text{Angle at } A + \text{Angle at } D = 180^\circ$: $81 + 99 = 180^\circ$ This is true, so $AB \parallel CD$.

6. Conclusion: Since one pair of opposite sides is parallel, $ABCD$ is a trapezium.

Do you have any questions or would you like more details?
Here are 5 related questions for practice:

1. How can you verify if a quadrilateral is a parallelogram using angles?
2. What are the properties of a trapezium that distinguish it from other quadrilaterals?
3. How would the solution change if the angles at $A$ and $D$ did not sum to $180^\circ$?
4. What is the sum of all interior angles of any quadrilateral, and how can you use it here?
5. If $x = 30$, can $ABCD$ still be a trapezium? Why or why not?

Tip: Always double-check angle sums and substituting values to avoid small calculation errors!