1. **State the problem:** We need to find the measures of angles $x$ and $y$ adjacent to a triangle with known angles $62^\circ$ and $51^\circ$. 2. **Recall the angle sum rule:** The sum of angles in a triangle is always $180^\circ$. 3. **Calculate the third angle of the triangle:** $$ \text{Third angle} = 180^\circ - 62^\circ - 51^\circ = 67^\circ $$ 4. **Use the linear pair rule:** Angles on a straight line sum to $180^\circ$. 5. **Find angle $y$:** Since $y$ is adjacent to the $67^\circ$ angle on the straight line, $$ y + 67^\circ = 180^\circ $$ $$ y = 180^\circ - 67^\circ = 113^\circ $$ 6. **Find angle $x$:** Angles $x$ and $y$ are adjacent on a straight line, so $$ x + y = 180^\circ $$ Substitute $y = 113^\circ$: $$ x + 113^\circ = 180^\circ $$ $$ x = 180^\circ - 113^\circ = 67^\circ $$ **Final answer:** $$ x = 67^\circ, \quad y = 113^\circ$$