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

Given the problem, where the line \( \overline{BD} \) intersects and crosses the diameter \( \overline{AC} \) of the circle at point \( X \), and the angles \( \angle BCA = 26^\circ \) and \( \angle CAD = 47^\circ \), let's find the angle \( \angle BXD \).

### Approach:
1. **Understand the Problem Geometry**:
   - \( \overline{BD} \) is a line intersecting the diameter \( \overline{AC} \) at point \( X \) inside the circle.
   - \( \angle BCA \) is given as \( 26^\circ \).
   - \( \angle CAD \) is given as \( 47^\circ \).

2. **Use the Inscribed Angle Theorem**:
   - The angle subtended by the diameter \( \overline{AC} \) on the circumference \( \angle BAC = 90^\circ \).

3. **Calculate \( \angle BAC \)**:
   \[
   \angle BAC = 180^\circ - (\angle BCA + \angle CAD)
   \]
   Substituting the values:
   \[
   \angle BAC = 180^\circ - (26^\circ + 47^\circ) = 180^\circ - 73^\circ = 107^\circ
   \]

4. **Use the Fact that Opposite Angles in Intersecting Chords are Equal**:
   The angles \( \angle BXD \) is related to the opposite angles \( \angle BCA \) and \( \angle CAD \).
   
   Since \( \overline{BD} \) and \( \overline{AC} \) intersect at \( X \), the angle \( \angle BXD \) formed at the intersection can be found by:
   \[
   \angle BXD = \angle BCA + \angle CAD
   \]
   
   Substituting the given values:
   \[
   \angle BXD = 26^\circ + 47^\circ = 73^\circ
   \]

Thus, the angle \( \angle BXD \) is \( 73^\circ \).

Would you like to go over any of these steps in more detail, or do you have any questions?

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### Related Questions:
1. How do angles in a circle relate to the diameter?
2. What is the significance of the intersecting chord theorem?
3. How do you find angles between intersecting lines in a circle?
4. Can \( \angle BXD \) ever be obtuse in this configuration?
5. How would the solution change if \( \overline{BD} \) were a tangent?
6. What is the relationship between \( \angle BXD \) and the angles of the triangle \( \triangle BXC \)?
7. How does the inscribed angle theorem apply to different cases of \( \overline{BD} \)?
8. What is the role of the angle subtended by the diameter in circle geometry?

### Tip:
When dealing with angles in circles, consider how the angles relate to the arcs and chords of the circle, especially when they intersect at a point on the circle. This can often simplify the problem.

Learn how to calculate angle BXD in a circle geometry problem where line BD intersects diameter AC. Given angles BCA = 26° and CAD = 47°, find angle BXD using the Inscribed Angle Theorem and Intersecting Chords Theorem. Suitable for high school students.

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