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The problem provides a diagram where line \( BD \) bisects \( \angle ABC \), which means it divides the angle into two equal parts. We are given different angle measures and are tasked with finding the missing angles based on this information.

Let’s break down the solutions for each question:

### Problem 15:
- Given: \( m\angle ABD = 57^\circ \)
- Since \( BD \) bisects \( \angle ABC \), we know that:
  \[
  m\angle ABD = m\angle DBC
  \]
  So, \( m\angle DBC = 57^\circ \).

- To find \( m\angle ABC \), we add the two angles:
  \[
  m\angle ABC = m\angle ABD + m\angle DBC = 57^\circ + 57^\circ = 114^\circ
  \]

Thus:
- \( m\angle DBC = 57^\circ \)
- \( m\angle ABC = 114^\circ \)

### Problem 16:
- Given: \( m\angle ABD = 70^\circ \)
- Again, since \( BD \) bisects \( \angle ABC \):
  \[
  m\angle DBC = 70^\circ
  \]

- Now, we calculate \( m\angle ABC \):
  \[
  m\angle ABC = m\angle ABD + m\angle DBC = 70^\circ + 70^\circ = 140^\circ
  \]

Thus:
- \( m\angle DBC = 70^\circ \)
- \( m\angle ABC = 140^\circ \)

### Problem 17:
- Given: \( m\angle ABC = 110^\circ \)
- Since \( BD \) bisects \( \angle ABC \), the two parts are equal:
  \[
  m\angle ABD = m\angle DBC = \frac{m\angle ABC}{2} = \frac{110^\circ}{2} = 55^\circ
  \]

Thus:
- \( m\angle ABD = 55^\circ \)
- \( m\angle DBC = 55^\circ \)

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Do you have any questions or would you like further details?

Here are 5 related questions to consider:
1. How does the concept of angle bisectors help in solving geometric problems?
2. If \( BD \) were not an angle bisector, how would the solution change?
3. What is the sum of interior angles in a triangle, and how does it relate to angle bisectors?
4. Can a line segment bisect more than one angle in a triangle? Why or why not?
5. How does the Angle Addition Postulate apply to the problems given here?

**Tip:** Always remember that an angle bisector divides an angle into two equal parts, which is key to solving many geometric problems.

BD bisects \(\angle ABC\). Given the angle measures in the diagram, find the indicated angle measures. Specifically, find m\(\angle DBC\) and m\(\angle ABC\) when m\(\angle ABD\) = 57°, 70°, or when m\(\angle ABC\) = 110°.

This problem involves the bisector of an angle in a triangle, focusing on finding missing angle measures using key theorems. The exercises provided test your knowledge of angle bisectors and the Angle Addition Postulate. Suitable for students in Grades 8-10.

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