To solve for $x$ and $y$ in the triangle, let's break it down step-by-step.

1. Key Observations:

   • The triangle is isosceles, as indicated by the equal marks on the two sides $\overline{AB}$ and $\overline{AC}$.
   • In isosceles triangles, the base angles are equal. Therefore, $\angle B = x^\circ$ and $\angle C = x^\circ$.
   • The vertex angle is divided into two equal parts, and each half is $y^\circ$.
   • The sum of all interior angles of a triangle is $180^\circ$.

2. Equation Setup:

   • Using the sum of angles in $\triangle ABC$: $\angle A + \angle B + \angle C = 180^\circ$ Substituting $\angle A = 2y$, $\angle B = x$, and $\angle C = x$: $2y + x + x = 180$ Simplifies to: $2y + 2x = 180$ Dividing through by 2: $y + x = 90$

3. Use of Given Information:

   • It is given that one of the base angles, $\angle C$, is $47^\circ$. This means $x = 47^\circ$.

4. Find $y$:

   • Substitute $x = 47$ into $y + x = 90$: $y + 47 = 90$ $y = 90 - 47$ $y = 43$

5. Final Answer:

   • $x = 47^\circ$ and $y = 43^\circ$.

The correct choice is A: $x = 47$ and $y = 43$.

Would you like further clarification or more details? Here are some related questions you might find helpful:

1. How do you identify isosceles triangles and their properties?
2. Why does the sum of angles in a triangle always equal $180^\circ$?
3. How do you divide a vertex angle into equal parts geometrically?
4. What are some common mistakes when solving for unknown angles in triangles?
5. Can this problem be extended to non-isosceles triangles?

Tip: Always verify the sum of angles when working with triangles to ensure your calculations are consistent.