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To solve for the missing angles \( b \) and \( c \), we can apply the property that vertically opposite angles are equal and that angles on a straight line sum up to 180°.

### Step-by-Step:
1. **Vertically opposite angles**: Since angle \( b \) is vertically opposite to the given angle \( 142^\circ \), we know that:
   \[
   b = 142^\circ
   \]

2. **Supplementary angles**: Angles \( b \) and \( c \) are adjacent and form a straight line, so they must sum to 180°:
   \[
   b + c = 180^\circ
   \]
   Substituting \( b = 142^\circ \) into the equation:
   \[
   142^\circ + c = 180^\circ
   \]
   Solving for \( c \):
   \[
   c = 180^\circ - 142^\circ = 38^\circ
   \]

### Final Answer:
- \( b = 142^\circ \)
- \( c = 38^\circ \)

Let me know if you need further details or if you have any questions!

Here are 5 related questions to expand your understanding:
1. How would the solution change if angle \( b \) were \( 100^\circ \)?
2. What is the definition of supplementary angles?
3. Can you explain why vertically opposite angles are equal?
4. What is the sum of all angles around a point?
5. How would you calculate the missing angles if both \( b \) and \( c \) were unknown?

**Tip**: Always remember that angles around a point total to 360°, and supplementary angles always sum to 180°.

Find the measure of the missing angles. Given angle 142°, find angles b and c.

Learn how to find the missing angles using the properties of vertically opposite angles and supplementary angles. This geometry problem helps students apply the Vertically Opposite Angle Theorem and the Supplementary Angle Theorem to find angles b and c when one angle is given as 142°. Suitable for students in Grades 6-8.

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