1. **State the problem:** We have a polygon with edges and angles, and three pairs of congruent edges: - $15x + 17 = 3z - 5$ - $4x - 2 = 8y + 13$ - $8x - 29 = 10z - 3$ We need to solve for variables $x$, $y$, and $z$. 2. **Write the equations from congruent edges:** $$\begin{cases} 15x + 17 = 3z - 5 \\ 4x - 2 = 8y + 13 \\ 8x - 29 = 10z - 3 \end{cases}$$ 3. **Simplify each equation:** - Equation 1: $$15x + 17 = 3z - 5 \implies 15x - 3z = -5 - 17 = -22$$ - Equation 2: $$4x - 2 = 8y + 13 \implies 4x - 8y = 13 + 2 = 15$$ - Equation 3: $$8x - 29 = 10z - 3 \implies 8x - 10z = -3 + 29 = 26$$ 4. **Rewrite the system:** $$\begin{cases} 15x - 3z = -22 \\ 4x - 8y = 15 \\ 8x - 10z = 26 \end{cases}$$ 5. **Solve for $z$ from the first equation:** $$15x - 3z = -22 \implies -3z = -22 - 15x \implies z = \frac{22 + 15x}{3}$$ 6. **Substitute $z$ into the third equation:** $$8x - 10z = 26 \implies 8x - 10 \times \frac{22 + 15x}{3} = 26$$ Multiply both sides by 3 to clear denominator: $$3 \times 8x - 10(22 + 15x) = 3 \times 26$$ $$24x - 220 - 150x = 78$$ Simplify: $$24x - 150x = 78 + 220$$ $$-126x = 298$$ Divide both sides by -126: $$x = \frac{298}{-126} = -\frac{149}{63}$$ 7. **Calculate $z$ using $x$:** $$z = \frac{22 + 15x}{3} = \frac{22 + 15 \times \left(-\frac{149}{63}\right)}{3} = \frac{22 - \frac{2235}{63}}{3} = \frac{\frac{22 \times 63}{63} - \frac{2235}{63}}{3} = \frac{\frac{1386 - 2235}{63}}{3} = \frac{-849/63}{3} = -\frac{849}{189} = -\frac{283}{63}$$ 8. **Solve for $y$ using the second equation:** $$4x - 8y = 15 \implies -8y = 15 - 4x \implies y = \frac{4x - 15}{8}$$ Substitute $x = -\frac{149}{63}$: $$y = \frac{4 \times \left(-\frac{149}{63}\right) - 15}{8} = \frac{-\frac{596}{63} - 15}{8} = \frac{-\frac{596}{63} - \frac{945}{63}}{8} = \frac{-\frac{1541}{63}}{8} = -\frac{1541}{504}$$ **Final answers:** $$x = -\frac{149}{63}, \quad y = -\frac{1541}{504}, \quad z = -\frac{283}{63}$$