This image shows a triangle where the angles $\angle ABC = 60^\circ$ and $\angle ACD = 58^\circ$, with the unknown angles labeled as $x$ and $y$ in the triangle.

To find $x$ and $y$, we can use the properties of triangles, specifically that the sum of the interior angles in any triangle equals $180^\circ$.

Step 1: Find $y$

The sum of the angles in triangle $ABC$ is 180°: $\angle ABC + \angle BAC + \angle BCA = 180^\circ$ Given that $\angle ABC = 60^\circ$ and $\angle BAC = y$, we can calculate $y$ by knowing that the angle $\angle BCA$ is $58^\circ$. So, $60^\circ + y + 58^\circ = 180^\circ$ $y + 118^\circ = 180^\circ$ $y = 180^\circ - 118^\circ = 62^\circ$

Step 2: Find $x$

In the triangle, the angle $\angle ACD = 58^\circ$, and the angle $\angle ABC = 60^\circ$. The remaining angle $x$ can be found by using the triangle angle sum again: $\angle ACD + x + 60^\circ = 180^\circ$ $58^\circ + x + 60^\circ = 180^\circ$ $x + 118^\circ = 180^\circ$ $x = 180^\circ - 118^\circ = 62^\circ$

So, both $x = 62^\circ$ and $y = 62^\circ$.

Would you like further clarification or have any questions? Here are some related questions to consider:

1. What is the difference between exterior and interior angles in a triangle?
2. How can the sum of angles in polygons help in geometry problems?
3. What other methods can we use to solve for unknown angles in polygons?
4. How does the triangle inequality theorem relate to finding angles?
5. How would we solve for missing sides in a triangle using trigonometry?

Tip: Always check the sum of the angles in a triangle. If the sum doesn't add up to 180°, something may be wrong with your calculations or assumptions.