1. **Problem statement:** Given that lines KL ∥ MN and LM ∥ NO, find the value of $x$ given the angles $3x^\circ$, $96^\circ$, and $2y^\circ$ in the two adjacent triangles KLM and MNO. 2. **Key properties:** Since KL ∥ MN and LM ∥ NO, alternate interior angles and corresponding angles are equal. 3. **Analyze angles at point M:** The angle between LM and MN is $96^\circ$. 4. **Use the fact that the sum of angles around point M is $360^\circ$:** The angles around M are $3x^\circ$, $96^\circ$, and $2y^\circ$ plus the remaining angle. 5. **Use parallel lines to relate angles:** Since LM ∥ NO, angle $3x^\circ$ corresponds to angle $2y^\circ$, so $3x = 2y$. 6. **Sum of angles in triangle KLM:** Angles are $3x^\circ$, $96^\circ$, and the third angle. Sum is $180^\circ$. 7. **Sum of angles in triangle MNO:** Angles are $2y^\circ$, $96^\circ$, and the third angle. Sum is $180^\circ$. 8. **From step 5, substitute $2y = 3x$ into the triangle MNO sum:** $$ 2y + 96 + \text{third angle} = 180 \implies 3x + 96 + \text{third angle} = 180 $$ 9. **Calculate the third angle in triangle MNO:** $$ \text{third angle} = 180 - 96 - 3x = 84 - 3x $$ 10. **Similarly, in triangle KLM, the third angle is:** $$ 180 - 96 - 3x = 84 - 3x $$ 11. **Since the third angles are equal, the triangles are similar, confirming the relationships.** 12. **Use the straight line at M:** The sum of angles around point M on a straight line is $180^\circ$, so $$ 3x + 96 + 2y = 180 $$ 13. **Substitute $2y = 3x$ into the above:** $$ 3x + 96 + 3x = 180 $$ $$ 6x + 96 = 180 $$ 14. **Solve for $x$:** $$ 6x = 180 - 96 $$ $$ 6x = 84 $$ $$ x = \frac{84}{6} $$ $$ x = 14 $$ **Final answer:** $$ \boxed{14} $$