https://file.aiquickdraw.com/mathbot/file/1727052271874n042k8oj.jpg

Let's work through the questions in the image. Here, **BD bisects \(\angle ABC\)**, which means that \(\angle ABD = \angle DBC\).

### 15. Given:
- \(\angle ABD = 57^\circ\)

Since **BD** bisects the angle, \(\angle DBC = \angle ABD = 57^\circ\).

Now, \(\angle ABC = \angle ABD + \angle DBC = 57^\circ + 57^\circ = 114^\circ\).

### Answer for 15:
- \(\angle DBC = 57^\circ\)
- \(\angle ABC = 114^\circ\)

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### 16. Given:
- \(\angle ABD = 70^\circ\)

Since **BD** bisects the angle, \(\angle DBC = \angle ABD = 70^\circ\).

Now, \(\angle ABC = \angle ABD + \angle DBC = 70^\circ + 70^\circ = 140^\circ\).

### Answer for 16:
- \(\angle DBC = 70^\circ\)
- \(\angle ABC = 140^\circ\)

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### 17. Given:
- \(\angle ABC = 110^\circ\)

Since **BD** bisects the angle, we know that \(\angle ABD = \angle DBC\).

Thus, \(\angle ABD = \angle DBC = \frac{\angle ABC}{2} = \frac{110^\circ}{2} = 55^\circ\).

### Answer for 17:
- \(\angle ABD = 55^\circ\)
- \(\angle DBC = 55^\circ\)

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Would you like further details or have any other questions? 

Here are some related questions to consider:
1. What happens to \(\angle ABD\) if \(\angle ABC\) increases?
2. How do you prove that **BD** is an angle bisector using geometry?
3. How would the calculation change if **BD** did not bisect \(\angle ABC\)?
4. What is the sum of the interior angles of a triangle, and how does it apply to this diagram?
5. If \(\angle ABC = 180^\circ\), what would \(\angle ABD\) and \(\angle DBC\) be?

**Tip:** When you know an angle is bisected, you can always divide the larger angle in half to get the two equal smaller angles!

BD bisects ∠ABC. Use the diagram and the given angle measure to find the indicated angle measures. 15. m∠ABD = 57°. Find m∠DBC and m∠ABC. 16. m∠ABD = 70°. Find m∠DBC and m∠ABC. 17. m∠ABC = 110°. Find m∠ABD and m∠DBC.

Learn how to find the unknown angles in a geometric figure where BD bisects ∠ABC. This problem explores angle bisectors and involves the angle bisector theorem, suitable for Grades 7-10.

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