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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!

Find the measure of the missing angles in the diagram, where one angle is given as 21°.

Learn how to find the missing angles in a diagram with intersecting lines. Using the Vertical Angles Theorem and Linear Pair Theorem, this step-by-step guide helps solve for the unknown angles, b = 21° and c = 159°. Ideal for students in Grades 6-8.

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