We are tasked with finding $\angle DBC$ in the trapezium $ABCD$ with the given conditions:

1. $AB \parallel DC$, so $AB$ and $DC$ are parallel.
2. $AB = AD$, so $\triangle ABD$ is isosceles.
3. $\angle ADC = 64^\circ$ and $\angle BCD = 54^\circ$.

Step 1: Properties of a Trapezium

In $\triangle ADC$:

• The interior angles sum to $180^\circ$, so: $\angle CAD + \angle ADC + \angle ACD = 180^\circ.$ Since $\angle ADC = 64^\circ$ and $\angle ACD = \angle BCD = 54^\circ$ (corresponding angles due to $AB \parallel DC$): $\angle CAD = 180^\circ - 64^\circ - 54^\circ = 62^\circ.$

Step 2: Analyze $\triangle ABD$

• $AB = AD$, so $\triangle ABD$ is isosceles with $\angle CAD = 62^\circ$.
• Therefore, $\angle DAB = \angle DBA$.

Using the angle sum property in $\triangle ABD$: $\angle DAB + \angle DBA + \angle ADB = 180^\circ.$ Here, $\angle DAB = \angle DBA$ and $\angle ADB = 64^\circ$: $2\angle DAB + 64^\circ = 180^\circ.$ $2\angle DAB = 116^\circ \quad \Rightarrow \quad \angle DAB = 58^\circ.$

Thus, $\angle DBA = 58^\circ$.

Step 3: Find $\angle DBC$

Finally, in $\triangle DBC$:

• $\angle BCD = 54^\circ$, and $\angle DBC$ is the required angle.
• Using the angle sum property in $\triangle DBC$: $\angle DBC + \angle BCD + \angle DCB = 180^\circ.$ Here, $\angle DCB = \angle DBA = 58^\circ$ (as $\triangle ABD$ and $\triangle DBC$ share diagonal $DB$): $\angle DBC + 54^\circ + 58^\circ = 180^\circ.$ $\angle DBC = 180^\circ - 112^\circ = 72^\circ.$

Final Answer:

$\boxed{(b) \; 72^\circ.}$

Let me know if you need further clarifications!

5 Related Questions:

1. How do corresponding angles work in parallel lines within a trapezium?
2. Can we determine the length of $AB$ or $DC$ with this data?
3. How would the problem change if $AB \neq AD$?
4. Can you find $\angle ADB$ directly from other properties of trapezium?
5. How does the sum of angles in a triangle help in such problems?

Tip:

Always double-check relationships like parallel lines or equal sides when working with trapeziums; these often simplify complex angle calculations.