Let's analyze the given problem:

We are asked to find the missing angles $b$ and $c$ in the diagram. The provided angle is $21^\circ$, and the lines are intersecting, forming several angles.

Step 1: Use Vertical Angles Theorem

The angle opposite $21^\circ$ is equal to it because vertical angles are congruent. So, $b = 21^\circ$

Step 2: Use Linear Pair Theorem

Angles $b$ and $c$ form a linear pair, meaning they are supplementary (i.e., their sum is $180^\circ$). Therefore, we can find $c$ by subtracting $b$ from $180^\circ$: $c = 180^\circ - b = 180^\circ - 21^\circ = 159^\circ$

Final Answers:

• $b = 21^\circ$
• $c = 159^\circ$

Would you like further explanation on any of these steps? Here are a few related questions:

1. What is the Vertical Angles Theorem and when is it used?
2. How do you know if two angles form a linear pair?
3. Can two angles be both vertical and form a linear pair at the same time?
4. What are complementary and supplementary angles, and how do they differ?
5. How do intersecting lines help in finding unknown angles?

Tip: Always check for relationships between angles, such as vertical or supplementary, when working with intersecting lines!