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Find the measure of a and b in the following figure 

We are given a geometric figure with angles labeled \(a\) and \(b\). Here's the strategy to find the values of \(a\) and \(b\) using the properties of angles:

### Step-by-step Solution:

1. **Focus on \( \angle Q \) (labeled as 110°):**
   - Since \( \angle DQR = 110^\circ \) is an exterior angle to the triangle \( PQR \), we can use the **exterior angle theorem**, which states:
     \[
     \text{Exterior angle} = \text{sum of opposite interior angles}
     \]
   - Thus, \( 110^\circ = b + 30^\circ \).

2. **Solve for \(b\):**
   \[
   b = 110^\circ - 30^\circ = 80^\circ
   \]

3. **Now focus on finding \(a\):**
   - \(a\) is on a straight line with \( \angle DQR \), and they form a linear pair. A linear pair of angles sums up to 180°. Therefore:
     \[
     a + 110^\circ = 180^\circ
     \]
   - Solve for \(a\):
     \[
     a = 180^\circ - 110^\circ = 70^\circ
     \]

### Final Answer:
- \( a = 70^\circ \)
- \( b = 80^\circ \)

Would you like a further breakdown of any part of the solution or additional details?

Here are 5 relative questions:
1. What is the exterior angle theorem and how is it applied in various cases?
2. How do linear pairs of angles help in solving geometric problems?
3. What are some key properties of angles formed by a transversal and parallel lines?
4. Can the exterior angle theorem be used in quadrilaterals? Why or why not?
5. How does the sum of interior angles in polygons differ from triangles?

**Tip:** Always remember that the sum of angles in a triangle is 180°, which is useful for many geometric problems.

Find the measure of a and b in the given geometric figure.

Learn how to find the measures of angles a and b in a geometric figure using the exterior angle theorem and linear pairs. This problem covers key concepts in geometry and is suitable for students in Grades 8-10.

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