1. **State the problem:** We need to find the values of $x$ and $y$ given the angles labeled $(12x)^\circ$, $42^\circ$, and $(6y)^\circ$ around point $K$. 2. **Understand the geometry:** The arrows form angles around point $K$. The right angle between horizontal (D to G) and vertical (F to J) lines is $90^\circ$. 3. **Sum of angles around point $K$:** The total degrees around a point is $360^\circ$. 4. **Identify the angles:** The angles around $K$ are: - $(12x)^\circ$ (angle from $K$ to $E$) - $42^\circ$ (angle between $D$ to $K$ and $K$ to $E$) - $(6y)^\circ$ (angle from $K$ to $H$) - The right angle $90^\circ$ 5. **Write the equation:** $$ (12x) + 42 + (6y) + 90 = 360 $$ 6. **Simplify:** $$ 12x + 6y + 132 = 360 $$ $$ 12x + 6y = 360 - 132 $$ $$ 12x + 6y = 228 $$ 7. **Divide entire equation by 6 to simplify:** $$ \cancel{6} \times 2x + \cancel{6} \times y = \cancel{6} \times 38 $$ $$ 2x + y = 38 $$ 8. **Use the right angle to find relation between $x$ and $y$:** Since the right angle is $90^\circ$, and the angles $(12x)^\circ$ and $(6y)^\circ$ are adjacent to it, we can assume: $$ (12x) + (6y) = 90 $$ 9. **Divide by 6:** $$ 2x + y = 15 $$ 10. **Now we have two equations:** $$ 2x + y = 38 \quad (1) $$ $$ 2x + y = 15 \quad (2) $$ These contradict, so re-examine the problem. 11. **Reconsider the angle sum:** The $42^\circ$ is between $D$ to $K$ and $K$ to $E$, so the angle $(12x)^\circ$ is not added to $42^\circ$ but is the angle from $K$ to $E$ itself. 12. **Assuming the angles around $K$ are:** - $42^\circ$ - $(12x)^\circ$ - $(6y)^\circ$ - $90^\circ$ Sum to $360^\circ$: $$ 42 + 12x + 6y + 90 = 360 $$ $$ 12x + 6y = 360 - 132 $$ $$ 12x + 6y = 228 $$ Divide by 6: $$ 2x + y = 38 $$ 13. **Use the right angle relation:** The right angle is between horizontal and vertical arrows, so the sum of $(12x)^\circ$ and $(6y)^\circ$ must be $90^\circ$: $$ 12x + 6y = 90 $$ Divide by 6: $$ 2x + y = 15 $$ 14. **Solve the system:** $$ \begin{cases} 2x + y = 38 \\ 2x + y = 15 \end{cases} $$ This is impossible unless the problem has a typo or misinterpretation. 15. **Assuming the $42^\circ$ is part of $(12x)^\circ$ angle, then:** $$ 12x = 42 + \text{some other angle} $$ But without more info, we cannot solve uniquely. **Final conclusion:** Given the problem as stated, the only consistent equation is: $$ 2x + y = 38 $$ and $$ 2x + y = 15 $$ which contradicts. **Therefore, assuming the right angle is between $(12x)^\circ$ and $(6y)^\circ$, then:** $$ 12x + 6y = 90 $$ Divide by 6: $$ 2x + y = 15 $$ **From the sum of all angles:** $$ 42 + 90 + 12x + 6y = 360 $$ $$ 132 + 12x + 6y = 360 $$ $$ 12x + 6y = 228 $$ Divide by 6: $$ 2x + y = 38 $$ **Subtract the two equations:** $$ (2x + y) - (2x + y) = 38 - 15 $$ $$ 0 = 23 $$ Contradiction. **Hence, the only way is to solve for $x$ and $y$ from:** $$ 12x + 6y = 90 $$ $$ 12x + 6y = 228 $$ which is impossible unless the problem has missing info. **If we assume the $42^\circ$ is part of $(12x)^\circ$ angle, then:** $$ 12x = 42 $$ $$ x = \frac{42}{12} = 3.5 $$ **Then from right angle:** $$ 12x + 6y = 90 $$ $$ 42 + 6y = 90 $$ $$ 6y = 48 $$ $$ y = 8 $$ **Final answers:** $$ x = 3.5, \quad y = 8 $$