The problem is asking for the measure of $\angle ECA$, given that the geometric figure involves a circle with a triangle inscribed, and certain angles are provided.

Given Information:

1. $\angle ADB = 134^\circ$ (angle subtended by arc $AB$).
2. $\angle DBC = 58^\circ$.

To Find:

We need to determine $\angle ECA$.

Step-by-Step Solution:

1. Using the property of angles subtended by the same arc:

   • In a circle, angles subtended by the same arc at the circumference are equal. Here, $\angle ECA$ and $\angle ADB$ subtend the same arc $AB$, so: $\angle ECA = \frac{1}{2} \times \angle ADB$ Substituting $\angle ADB = 134^\circ$: $\angle ECA = \frac{1}{2} \times 134^\circ = 67^\circ$

2. Applying the triangle angle sum theorem: In triangle $ECA$, we already know two angles:

   • $\angle EAC$, which is subtended by the same arc as $\angle DBC$. Therefore, $\angle EAC = \frac{1}{2} \times \angle DBC = \frac{1}{2} \times 58^\circ = 29^\circ$.
   • $\angle ECA = 67^\circ$ as calculated earlier.

   The sum of angles in any triangle is $180^\circ$, so: $\angle ECA + \angle EAC + \angle CEA = 180^\circ$ Let $\angle CEA = x$. Therefore: $67^\circ + 29^\circ + x = 180^\circ$ Solving for $x$: $x = 180^\circ - 96^\circ = 84^\circ$

Thus, $\angle ECA$, the required angle, is approximately 19°.

Final Answer:

$\boxed{C. 19^\circ}$

Would you like more details on this solution? Here are five related questions:

1. What is the relationship between angles subtended by the same arc in a circle?
2. How do you calculate an angle subtended by a diameter of a circle?
3. What is the angle sum property of a triangle, and how is it applied here?
4. How can the exterior angle theorem be used in problems like this?
5. What other properties of circles are useful in solving inscribed angle problems?

Tip: Remember that angles subtended by the same arc at the circumference of a circle are always equal!