1. **Problem Statement:** Given an isosceles trapezoid front face of a sign with sides WX = $2x - 2$, YZ = $2x + 6$, WZ = $4x + 5$, and XY = $5x - 3$, prove $x=8$, find $m\angle Z$ if $m\angle W=106^\circ$, and find the perimeter. 2. **Step 1: Prove $x=8$** Since the trapezoid is isosceles, the legs are equal: $WZ = XY$. Write the equation: $$4x + 5 = 5x - 3$$ Subtract $4x$ from both sides: $$\cancel{4x} + 5 = 5x - \cancel{4x} - 3 \implies 5 = x - 3$$ Add 3 to both sides: $$5 + 3 = x - 3 + 3 \implies 8 = x$$ So, $x=8$ is proven. 3. **Step 2: Find $m\angle Z$ given $m\angle W=106^\circ$** In an isosceles trapezoid, the base angles are equal in pairs. Since $m\angle W=106^\circ$, the angle adjacent to it on the same base, $m\angle X$, is also $106^\circ$. The consecutive angles between the bases are supplementary, so: $$m\angle W + m\angle Z = 180^\circ$$ Substitute $m\angle W=106^\circ$: $$106^\circ + m\angle Z = 180^\circ$$ Solve for $m\angle Z$: $$m\angle Z = 180^\circ - 106^\circ = 74^\circ$$ 4. **Step 3: Find the perimeter of the trapezoid** Substitute $x=8$ into each side length: $$WX = 2(8) - 2 = 16 - 2 = 14$$ $$YZ = 2(8) + 6 = 16 + 6 = 22$$ $$WZ = 4(8) + 5 = 32 + 5 = 37$$ $$XY = 5(8) - 3 = 40 - 3 = 37$$ Calculate perimeter $P$: $$P = WX + YZ + WZ + XY = 14 + 22 + 37 + 37 = 110$$ **Final answers:** - $x=8$ - $m\angle Z = 74^\circ$ - Perimeter $= 110$ inches