1. The problem asks to find the value of each variable $x$ and $y$ given some angles and relationships. 2. From the first part, we have a triangle with angles $60^\circ$, $x$, and $y$ such that $x + y + 60^\circ = 180^\circ$ because the sum of angles in a triangle is always $180^\circ$. 3. Therefore, the formula is: $$x + y + 60 = 180$$ 4. Simplify to find $x + y$: $$x + y = 180 - 60 = 120$$ 5. Without additional information, we cannot find unique values for $x$ and $y$, but if $x = y$, then: $$2x = 120 \implies x = 60$$ 6. For the second part, solve for $x$ in right triangles with given sides and angles. 7. For question 7, given $x$, $25$, and $45^\circ$, use the sine or cosine rule depending on the triangle. 8. For question 8, given $x$, $20$, and $45^\circ$, similarly use trigonometric ratios. 9. For question 9, given $15$, $22$, and $x$, use the Pythagorean theorem or trigonometric ratios. 10. For question 10, given $15$, $20$, and $x$, use the Pythagorean theorem or trigonometric ratios. Since the user only provided partial data and asked to solve the first problem, we focus on the first problem only. Final answer for the first problem: $$x + y = 120$$ If $x = y$, then: $$x = y = 60$$