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.