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Let's analyze the geometry problem step by step based on the provided diagram and the description.

### Given Information:
- The circle has a center \( O \) and passes through points \( A \), \( B \), \( C \), and \( D \).
- \( BOD \) is a straight line.
- \( \angle BAC = 38^\circ \).
- The line \( BQ \) is a tangent to the circle at point \( B \).

### Required:
1. \( \angle DAC \)
2. \( \angle DBC \)
3. \( \angle CBQ \)

---

### (a) \( \angle DAC \)

We know that \( \angle BAC = 38^\circ \), and from the property of a circle, angles subtended by the same arc are equal. Since both \( \angle DAC \) and \( \angle BAC \) are subtended by the same arc \( AC \), we can conclude:

\[
\angle DAC = \angle BAC = 38^\circ
\]

### (b) \( \angle DBC \)

To find \( \angle DBC \), we need to use the fact that \( \angle BOD \) is a straight line. In a circle, the angle at the center is twice the angle subtended on the circumference. Therefore, using this property for arc \( BD \), we have:

\[
\angle BOD = 2 \times \angle BCD
\]

Given that \( BOD \) is a straight line, \( \angle BOD = 180^\circ \). Thus:

\[
180^\circ = 2 \times \angle BCD
\]

Solving for \( \angle BCD \):

\[
\angle BCD = \frac{180^\circ}{2} = 90^\circ
\]

Therefore, \( \angle DBC \) is:

\[
\angle DBC = 90^\circ
\]

### (c) \( \angle CBQ \)

Finally, \( BQ \) is a tangent to the circle at \( B \). By the tangent-secant theorem, the angle between the tangent and a chord is equal to the angle subtended by the chord on the opposite side of the circle. In this case, the chord is \( BC \), and the angle subtended by \( BC \) at \( A \) is \( \angle BAC = 38^\circ \). Hence:

\[
\angle CBQ = \angle BAC = 38^\circ
\]

---

### Final Answers:
(a) \( \angle DAC = 38^\circ \)  
(b) \( \angle DBC = 90^\circ \)  
(c) \( \angle CBQ = 38^\circ \)

Would you like further explanation or have any questions about this solution?

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**Related Questions:**
1. What is the relationship between the tangent and the radius at the point of contact in a circle?
2. How do we calculate the angles subtended by the same arc in a circle?
3. What is the cyclic quadrilateral property, and how does it apply to this problem?
4. How would the solution change if \( \angle BAC \) were different?
5. Can you explain the tangent-secant theorem in more detail?

**Tip:** In a circle, the angle subtended by an arc at the center is always double the angle subtended at any point on the circumference.

The diagram shows a circle, center O, passing through points A, B, C, and D. BOD is a straight line, and angle BAC = 38°. The line BQ is a tangent to the circle at B. Find (a) angle DAC, (b) angle DBC, (c) angle CBQ.

Learn how to solve a circle geometry problem involving angles subtended by the same arc, tangent-secant theorem, and angle properties in a cyclic quadrilateral. Given angle BAC = 38°, this problem guides you through finding angles DAC, DBC, and CBQ, ideal for Grades 9-10.

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