To solve this problem, we will analyze the angles in the given figure. Here's the breakdown of the key information:

• Quadrilateral $ABCD$ is given.
• $\angle A = y$, $\angle C = x$.
• $AB$ is extended to point $E$ such that $\angle EBC = a$.
• $CD$ is extended to point $F$ such that $\angle ADF = b$.
• We are asked to find $x + y$.

Step-by-step solution:

1. Understanding the external angles:

   • $\angle EBC$ is the external angle at vertex $B$, and is given as $a$.
   • $\angle ADF$ is the external angle at vertex $D$, and is given as $b$.

2. Using the Exterior Angle Theorem:

   • The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
   • For triangle $ABC$, the exterior angle $\angle EBC$ is the sum of the two interior angles $\angle A$ and $\angle C$. Thus: $a = x + y$
   • For triangle $ADF$, the exterior angle $\angle ADF$ is the sum of the two opposite interior angles $\angle D$ and $\angle A$. Since the quadrilateral $ABCD$ is a trapezoid, the internal structure can be ignored as the calculation was completed with triangle $ABC$.

Thus, the result is: $x + y = a$

Conclusion:

The value of $x + y$ is $a$. If you have any further questions or would like more details, feel free to ask!

Related Questions:

1. What is the Exterior Angle Theorem and how is it used in geometry?
2. How would you calculate unknown angles if $a$, $b$, or other angles were missing?
3. How can we solve for $x$ or $y$ individually, given another relationship?
4. Can this principle be extended to other polygons like pentagons or hexagons?
5. What are the properties of trapezoids, and how do external angles behave in such quadrilaterals?

Tip:

Always look for relationships between interior and exterior angles when working with extended lines and polygons!