This problem involves a parallelogram, where we will apply geometric angle properties to solve for $x$ and $y$.

Key Observations:

1. Opposite Angles in a Parallelogram: Opposite angles are equal.
2. Supplementary Angles: Adjacent angles in a parallelogram are supplementary, i.e., their sum is $180^\circ$.
3. The diagonal divides the parallelogram into two congruent triangles, giving additional relationships.

Step 1: Use the angles provided

• One angle is $31^\circ$, and its adjacent angle is $98^\circ$. Their sum confirms $31^\circ + 98^\circ = 129^\circ$, so they satisfy the geometry rules.

Step 2: Solve for $6x + 3$

• Angle $6x + 3$ is opposite $98^\circ$, so: $6x + 3 = 98$ $6x = 95 \implies x = \frac{95}{6} \approx 15.83$

Step 3: Solve for $4y + 2$

• Angle $4y + 2$ is opposite $31^\circ$, so: $4y + 2 = 31$ $4y = 29 \implies y = \frac{29}{4} = 7.25$

Final Answer:

$x = \frac{95}{6} \text{ or approximately } 15.83, \quad y = \frac{29}{4} \text{ or } 7.25$

Would you like me to clarify any step or provide additional details?

Follow-Up Questions:

1. What are the properties of diagonals in a parallelogram, and how do they help solve angle problems?
2. Can supplementary angles exist in other geometric shapes besides parallelograms?
3. Why are opposite angles in parallelograms always equal?
4. How would the solution change if additional angles were labeled?
5. What are the formulas to calculate interior and exterior angles of polygons?

Tip:

When solving geometry problems, always verify relationships like supplementary, opposite, or complementary angles for consistency.