To solve for the angle $x$ in the given triangle configuration, we'll analyze the geometry in the diagram.

Step 1: Identify the known angles and their relationships

• The diagram shows a large triangle $ABC$ and a smaller triangle $BCD$, with lines intersecting and forming various angles.
• The angles given are:
  • $\angle BAC = 3x$
  • $\angle BCA = x$
  • $\angle ABC = 2x$
  • $\angle ACD = x$
  • $\angle BDA = x$

Step 2: Apply the sum of angles in a triangle

For any triangle, the sum of the interior angles is always $180^\circ$. We can apply this to the relevant triangles in the diagram:

1. In Triangle $ABC$: $\angle BAC + \angle ABC + \angle BCA = 180^\circ$ Substituting the known angles: $3x + 2x + x = 180^\circ$ Simplifying, we get: $6x = 180^\circ$ $x = 30^\circ$

Conclusion:

The angle $x$ is $30^\circ$.

Would you like further details or have any questions?

Related Questions:

1. What are the angles of triangle $ABC$ if $x = 30^\circ$?
2. How would the solution change if one of the angles was different?
3. Can the triangle be classified as isosceles or equilateral with the given angles?
4. What is the significance of each angle in the context of the triangle’s geometry?
5. How would the geometry change if $\angle BAC$ were $4x$ instead of $3x$?
6. How can trigonometric identities be used to verify the value of $x$?
7. What are the properties of the triangle formed by the points $B$, $D$, and $C$?
8. How does this problem relate to the Law of Sines or Law of Cosines?

Tip:

When solving geometry problems involving angles, always verify by summing the angles in a triangle to ensure they add up to $180^\circ$.