From the image, we are given the following setup:

• $\angle ABD = 60^\circ$
• The task is to find the measure of $\angle DBC$.

Step-by-step Solution:

1. Understanding the Diagram:

   • The angles $\angle ABD$ and $\angle DBC$ are adjacent angles formed by the rays $AB$, $BD$, and $BC$ meeting at point $B$.
   • Together, $\angle ABD$ and $\angle DBC$ form a straight line. This means the sum of these two angles is $180^\circ$ because they are supplementary angles.

2. Relationship Between the Angles: $m\angle ABD + m\angle DBC = 180^\circ$

3. Substitute the Given Value: $60^\circ + m\angle DBC = 180^\circ$

4. Solve for $m\angle DBC$: $m\angle DBC = 180^\circ - 60^\circ$ $m\angle DBC = 120^\circ$

Final Answer:

$m\angle DBC = 120^\circ$

Would you like me to elaborate further on any part of this solution? Here are some related questions for practice:

1. What are supplementary angles, and how are they used in geometry?
2. If $m\angle DBC$ were $110^\circ$, what would $m\angle ABD$ be?
3. How would the diagram change if $AB$ and $BC$ formed a right angle at $B$?
4. How do adjacent angles relate to linear pairs in geometry?
5. What is the difference between complementary and supplementary angles?

Tip:

When solving problems involving angles, always look for relationships like straight lines ($180^\circ$), right angles ($90^\circ$), or full circles ($360^\circ$). These help simplify your calculations!