In the image, we have intersecting lines that form several angles. One of the angles is labeled as $30^\circ$, and the other two missing angles are labeled as $b$ and $c$. Here's how we can solve for these angles:

1. Vertical Angles: Vertical angles are equal. Since the $30^\circ$ angle is vertically opposite to angle $b$, we know that: $b = 30^\circ$

2. Linear Pair: Adjacent angles that form a straight line sum up to $180^\circ$. The angles $b$ and $c$ form a straight line, so: $b + c = 180^\circ$ Substituting $b = 30^\circ$ into this equation: $30^\circ + c = 180^\circ$ Solving for $c$: $c = 180^\circ - 30^\circ = 150^\circ$

Thus, the values of the missing angles are:

• $b = 30^\circ$
• $c = 150^\circ$

Would you like further details or explanations on any of these steps?

Here are 5 related questions to extend this problem:

1. What are vertical angles, and why are they always equal?
2. How do we apply the concept of supplementary angles to find missing angles?
3. Can we extend this problem to three-dimensional space with intersecting planes?
4. How do alternate interior angles work in parallel lines cut by a transversal?
5. How would the calculation change if the given angle was $45^\circ$ instead of $30^\circ$?

Tip: Always remember that angles formed by intersecting lines follow simple rules: vertical angles are equal, and adjacent angles sum to $180^\circ$.