Question: Find the values of $x$ and $y$. A horizontal line runs through point $F$ from $A$ on the left to $D$ on the right, and a vertical ray goes up from $F$ to $B$; a diagonal line passes through $F$ from $E$ at bottom-left to $C$ at top-right. The angle between the vertical ray $FB$ and the diagonal ray $FC$ is $34^\circ$ in the top-center, and the angle between ray $FC$ and the horizontal ray $FD$ is $(4x - 12)^\circ$ in the center-right. The angle between ray $FA$ and the diagonal ray $FE$ is $(4y)^\circ$ in the bottom-left. A right-angle marker is shown at $F$ between the vertical and horizontal rays.

1. **State the problem:** We need to find the values of $x$ and $y$ given the angles around point $F$ formed by rays $FA$, $FB$, $FC$, $FD$, and $FE$. 2. **Understand the geometry:** At point $F$, the vertical ray $FB$ and horizontal ray $FD$ form a right angle, so: $$\angle BFD = 90^\circ$$ The diagonal ray $FC$ lies between $FB$ and $FD$, creating two angles: - $\angle BFC = 34^\circ$ - $\angle CFD = (4x - 12)^\circ$ Since $FB$ and $FD$ are perpendicular, the sum of these two angles must be $90^\circ$: $$\angle BFC + \angle CFD = 90^\circ$$ 3. **Set up the equation for $x$:** $$34 + (4x - 12) = 90$$ Simplify: $$34 + 4x - 12 = 90$$ $$4x + 22 = 90$$ $$4x = 90 - 22$$ $$4x = 68$$ Divide both sides by 4, showing cancellation: $$\frac{\cancel{4}x}{\cancel{4}} = \frac{68}{4}$$ $$x = 17$$ 4. **Find $y$ using the angles around $F$:** The diagonal ray $FE$ is opposite to $FC$ and $FB$ on the other side of the horizontal ray $FA$. The angles around point $F$ on a straight line sum to $180^\circ$. The angle between $FA$ and $FE$ is $(4y)^\circ$. Since $FA$ and $FD$ lie on the same horizontal line, and $FD$ and $FC$ form angle $(4x - 12)^\circ$, the angle between $FA$ and $FC$ is: $$\angle EFC = \angle EFA + \angle AFC = (4y) + (4x - 12)$$ But $FE$ and $FC$ are opposite rays forming a straight line, so the sum of angles around $F$ on the line $EFC$ is $180^\circ$: $$\angle EFA + \angle AFC = 180^\circ$$ Substitute known values: $$4y + (4x - 12) = 180$$ Substitute $x=17$: $$4y + (4 \times 17 - 12) = 180$$ $$4y + (68 - 12) = 180$$ $$4y + 56 = 180$$ $$4y = 180 - 56$$ $$4y = 124$$ Divide both sides by 4, showing cancellation: $$\frac{\cancel{4}y}{\cancel{4}} = \frac{124}{4}$$ $$y = 31$$ 5. **Final answers:** $$x = 17$$ $$y = 31$$