Let's analyze the problem and find the measure of angle $\angle ECA$.

Given Information:

• $\angle D = 134^\circ$
• $\angle B = 50^\circ$
• We are looking for $\angle ECA$.

Solution Approach:

1. Observe the Circle Properties: Since the points $D$, $E$, $C$, and $A$ are on a circle, we can apply properties of cyclic quadrilaterals and inscribed angles.

2. Inscribed Angle Theorem: The angle subtended by the same arc at the circumference is half of the angle subtended at the opposite side. Also, in a cyclic quadrilateral, opposite angles sum up to $180^\circ$.

3. Finding $\angle DCA$: Since $\angle B = 50^\circ$, we can deduce that $\angle DCA = 50^\circ$ as both angles $\angle B$ and $\angle DCA$ subtend the same arc $DA$.

4. Using Triangle $\triangle DCA$:

   • $\angle D + \angle DCA + \angle ECA = 180^\circ$ (since it’s a triangle)
   • Substituting the values: $134^\circ + 50^\circ + \angle ECA = 180^\circ$
   • $184^\circ + \angle ECA = 180^\circ$
   • $\angle ECA = 180^\circ - 184^\circ = 16^\circ$

Conclusion:

The measure of $\angle ECA$ is $16^\circ$.

Answer: D. $16^\circ$

Would you like further details on this solution or have any other questions?

Related Questions:

1. What are the properties of cyclic quadrilaterals?
2. How do we use inscribed angles to find unknown angles in a circle?
3. What other methods can we use to solve angle problems in a cyclic quadrilateral?
4. How do opposite angles in a cyclic quadrilateral relate to each other?
5. How can we apply triangle angle-sum properties to find unknown angles?

Tip:

In cyclic quadrilaterals, always remember that the sum of opposite angles is $180^\circ$. This property can simplify many angle-finding problems.