1. **State the problem:** We have two intersecting lines forming an X at point P with angles labeled as follows: angle APK = $(8x-7)^\circ$, the angle on the right is $y^\circ$, and the angle below is $(6x+3)^\circ$. We need to find the value of $x$, the measure of angle APK, and the value of $y$. 2. **Use the property of vertical angles:** Vertical angles formed by two intersecting lines are equal. 3. **Set up equations:** Since $(8x-7)^\circ$ and $(6x+3)^\circ$ are vertical angles, they are equal: $$8x - 7 = 6x + 3$$ 4. **Solve for $x$:** $$8x - 7 = 6x + 3$$ $$8x - 6x = 3 + 7$$ $$2x = 10$$ $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{10}{2} = 5$$ 5. **Find angle APK:** Substitute $x=5$ into $(8x-7)^\circ$: $$8(5) - 7 = 40 - 7 = 33^\circ$$ 6. **Find $y$:** Since $y^\circ$ and $(6x+3)^\circ$ are adjacent angles on a straight line, they are supplementary: $$y + (6x + 3) = 180$$ Substitute $x=5$: $$y + (6(5) + 3) = 180$$ $$y + 30 + 3 = 180$$ $$y + 33 = 180$$ $$y = 180 - 33 = 147^\circ$$ **Final answers:** - $x = 5$ - Angle APK = $33^\circ$ - $y = 147^\circ$