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We are given a circle with two secants: PBA and PDC. AD is the diameter of the circle, and we are tasked with finding the measure of ∠DBC, knowing the following:

- ∠A = 40° (angle subtended by arc AB at point A),
- ∠P = 20° (angle subtended by arc PD at point P),
- AD is the diameter.

### Step-by-Step Solution

1. **Angle at the Center (Inscribed Angle Theorem):**
   Since AD is the diameter of the circle, ∠AOD is a straight line and measures 180°. By the **Inscribed Angle Theorem**, the angle subtended by an arc at the center is twice the angle subtended at any point on the circumference. So:
   - ∠AOB = 2 × ∠A = 2 × 40° = 80°.

2. **Relation Between ∠DBC and Arc AB:**
   ∠DBC is an angle formed by secants intersecting outside the circle (point B), and the angle it subtends is related to the difference between the arcs it intercepts. The intercepted arcs are:
   - Arc AB (which subtends angle ∠AOB = 80°),
   - Arc PD (which subtends angle ∠P = 20°).

   The formula for the angle formed by two secants intersecting outside a circle is:
   \[
   \angle DBC = \frac{1}{2} (\text{arc AB} - \text{arc PD})
   \]
   Substituting the values:
   \[
   \angle DBC = \frac{1}{2} (80° - 20°) = \frac{1}{2} \times 60° = 30°
   \]

### Final Answer:
The measure of ∠DBC is **30°**.

Would you like any additional details or further explanation on any of the steps?

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Here are some related questions you might find interesting:

1. How do you find the angle formed by two intersecting chords inside a circle?
2. What is the relationship between angles subtended by the same arc at the circumference?
3. How is the angle at the center of a circle related to the angle at the circumference subtended by the same arc?
4. How does the angle subtended by a diameter relate to a right angle?
5. How would the problem change if the angle at point P were increased?

**Tip:** The angle formed by secants outside a circle is always related to the intercepted arcs through subtraction!

PBA and PDC are two secants. AD is the diameter of the circle with center at O. ∠A = 40°, ∠P = 20°. Find the measure of ∠DBC.

In this problem, we are given two secants, PBA and PDC, and asked to find the measure of ∠DBC. Using the inscribed angle theorem and the formula for angles formed by secants, the solution involves subtracting the arcs subtended by the secants. This problem is suitable for Grades 9-10.

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