https://file.aiquickdraw.com/mathbot/file/1730322048169b6oxmi0z.jpg

To solve this problem, let's break down the information provided:

1. **Angle BPD**:
   Since \( \angle BPD \) is an inscribed angle that intercepts arc \( BD \) of \( 70^\circ \), we can use the inscribed angle theorem. This theorem states that an inscribed angle is half of the measure of the intercepted arc. Therefore:
   \[
   \angle BPD = \frac{1}{2} \times m \overset{\frown}{BD} = \frac{1}{2} \times 70^\circ = 35^\circ
   \]

2. **Sum of \( m \overset{\frown}{BC} + m \overset{\frown}{AD} \)**:
   The circle has a total arc measure of \( 360^\circ \). Given that \( m \overset{\frown}{BD} = 70^\circ \) and \( m \overset{\frown}{CA} = 170^\circ \), we can find the remaining arcs \( m \overset{\frown}{BC} \) and \( m \overset{\frown}{AD} \) by subtracting from \( 360^\circ \):
   \[
   m \overset{\frown}{BC} + m \overset{\frown}{AD} = 360^\circ - (m \overset{\frown}{BD} + m \overset{\frown}{CA}) = 360^\circ - (70^\circ + 170^\circ) = 120^\circ
   \]

**Final Answers**:
- \( \angle BPD = 35^\circ \)
- \( m \overset{\frown}{BC} + m \overset{\frown}{AD} = 120^\circ \)

Would you like more details on any of these steps? 

Here are 5 related questions to deepen understanding:
1. What is the inscribed angle theorem, and why does it apply here?
2. How do we find other arc measures in a circle if only a few arcs are given?
3. Can you explain how central angles differ from inscribed angles?
4. What if \( m \overset{\frown}{CA} \) was different; how would it change \( m \overset{\frown}{BC} + m \overset{\frown}{AD} \)?
5. How can we use symmetry in circles to simplify problems with arcs and angles?

**Tip**: In circle problems, identifying intercepted arcs can simplify finding angle measures.

Consider circle O, where m∠BD = 70° and m∠CA = 170°. Find the measure of angle BPD and the sum of arcs BC and AD.

Learn how to find the measure of angle BPD and the sum of arcs BC and AD in circle O, where m∠BD = 70° and m∠CA = 170°. This solution involves applying the Inscribed Angle Theorem and basic properties of arcs in a circle, suitable for students in Grades 9-10.

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