The given image depicts a quadrilateral, and you're asked to find the values of the angles $x$ and $y$.

Here are the key observations from the figure:

• There is a quadrilateral labeled as $ABCD$.
• $\angle A = 120^\circ$ is marked at point $A$.
• The angles $x$ and $y$ are internal angles of the quadrilateral at points $D$ and $B$ respectively.
• Two sides of the quadrilateral, $AB$ and $BC$, are marked as equal (with double lines indicating congruent lengths).

Since this is a cyclic quadrilateral (as it appears from the geometry and marked angles), we can use the following property:

• The sum of the opposite angles of a cyclic quadrilateral is $180^\circ$.

Solution:

• Using the property for angles $A$ and $C$: $\angle A + \angle C = 180^\circ$ $120^\circ + x = 180^\circ \implies x = 60^\circ$

• Similarly, using the property for angles $B$ and $D$: $\angle B + \angle D = 180^\circ$ Since $\angle D = x = 60^\circ$: $y + 60^\circ = 180^\circ \implies y = 120^\circ$

Final Answer:

• $x = 60^\circ$
• $y = 120^\circ$

Let me know if you want further clarifications or details! Here are five related questions to expand on this concept:

1. How would the solution change if the quadrilateral were not cyclic?
2. What is the property of opposite angles in any quadrilateral?
3. How can you determine if a quadrilateral is cyclic based on side lengths or diagonals?
4. Can this method apply to a quadrilateral with different internal angles?
5. How does the symmetry of the quadrilateral help in solving for the unknown angles?

Tip: Always check if a quadrilateral is cyclic to apply the sum of opposite angles theorem effectively.