In a circle the line BD intersect the diameter the circle AC at X, BCA= 26°, CAD=47°

Let's solve the problem step by step.

### Given:
- \( \overline{BD} \) is a line intersecting the diameter \( \overline{AC} \) of the circle at point \( X \).
- \( \angle BCA = 26^\circ \)
- \( \angle CAD = 47^\circ \)

We are asked to find the angle \( \angle BXD \).

### Approach:
1. **Use the property of angles in a circle**: The angle subtended by the diameter on the circumference is always \(90^\circ\). Therefore, \( \angle BAC = 90^\circ \).

2. **Apply the sum of angles in triangle \(ABC\)**:
   In triangle \(ABC\),
   \[
   \angle BAC + \angle BCA + \angle ABC = 180^\circ
   \]
   Substitute the known angles:
   \[
   90^\circ + 26^\circ + \angle ABC = 180^\circ
   \]
   So,
   \[
   \angle ABC = 180^\circ - 116^\circ = 64^\circ
   \]

3. **Use the given angle \( \angle CAD \)**: 
   We know \( \angle CAD = 47^\circ \). We can use this along with \( \angle ABC \) to find \( \angle BXD \).

4. **Calculate \( \angle BXD \)**:
   Since \( \overline{BD} \) is a chord intersecting the diameter at \( X \), the angles subtended by \( \overline{BD} \) on the opposite sides of \( \overline{AC} \) are equal. Hence, \( \angle BXD = \angle ABC + \angle CAD \).

   Thus,
   \[
   \angle BXD = 64^\circ + 47^\circ = 111^\circ
   \]

So, the angle \( \angle BXD = 111^\circ \).

Would you like to go over the steps in more detail or have any questions?

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### Related Questions:
1. What is the significance of the angle subtended by the diameter on the circumference?
2. How would the solution change if \( \overline{BD} \) did not intersect at the diameter?
3. How do you find angles in cyclic quadrilaterals?
4. What are the properties of angles formed by intersecting chords in a circle?
5. Can the angle \( \angle BXD \) be more than \(180^\circ\)? Why or why not?
6. How can you determine if \( \overline{BD} \) is a tangent to the circle?
7. What other methods could be used to solve for \( \angle BXD \)?
8. What if \( \angle BCA \) were not given? How would you approach the problem?

### Tip:
When working with angles in circles, always consider properties like the angle subtended by the diameter, angles in the same segment, and cyclic quadrilaterals to simplify your calculations.

Explore how to solve a circle geometry problem involving angle calculations with specific intersections. Given angles and a diameter intersecting a circle, find the angle BXD step-by-step. Suitable for high school students studying circle geometry.

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