1. **State the problem:** We have two intersecting lines forming an X shape with angles labeled as follows: the top angle is $(8x-7)^\circ$, the bottom angle is $(6x+3)^\circ$, and the right angle is $y^\circ$. 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. Therefore, the top angle and the bottom angle are equal. 3. **Set up the equation:** $$ 8x - 7 = 6x + 3 $$ 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 the measure of angle $APK$:** Angle $APK$ is the top angle, so substitute $x=5$: $$ 8(5) - 7 = 40 - 7 = 33^\circ $$ 6. **Find the value of $y$:** Since the right angle $y^\circ$ is adjacent to the top angle and they form a straight line, their sum is $180^\circ$: $$ y + 33 = 180 \\ y = 180 - 33 = 147^\circ $$ **Final answers:** - $x = 5$ - Angle $APK = 33^\circ$ - $y = 147^\circ$