1. **State the problem:** We need to find values of $w$ and $x$ such that triangles $\triangle HIJ$ and $\triangle FEG$ are congruent. Given the sides: - $HI = x - 2w + 49$ - $HJ = 9x + w - 12$ - $FE = 19w - 8x - 8$ - $FG = 6w + 6x - 23$ Since both triangles have a right angle, congruence can be established by the Hypotenuse-Leg (HL) theorem or by matching corresponding sides. 2. **Set corresponding sides equal:** Assuming $HI$ corresponds to $FE$ and $HJ$ corresponds to $FG$, we write the system: $$ \begin{cases} x - 2w + 49 = 19w - 8x - 8 \\ 9x + w - 12 = 6w + 6x - 23 \end{cases} $$ 3. **Simplify the first equation:** $$ x - 2w + 49 = 19w - 8x - 8$$ Bring all terms to one side: $$ x - 2w + 49 - 19w + 8x + 8 = 0$$ Combine like terms: $$ ( x + 8x ) + ( -2w - 19w ) + (49 + 8) = 0 \\ 9x - 21w + 57 = 0 $$ Rewrite: $$ 9x - 21w = -57 $$ 4. **Simplify the second equation:** $$ 9x + w - 12 = 6w + 6x - 23 $$ Bring all terms to one side: $$ 9x + w - 12 - 6w - 6x + 23 = 0 $$ Combine like terms: $$ (9x - 6x) + (w - 6w) + (-12 + 23) = 0 \\ 3x - 5w + 11 = 0 $$ Rewrite: $$ 3x - 5w = -11 $$ 5. **Solve the system:** $$ \begin{cases} 9x - 21w = -57 \\ 3x - 5w = -11 \end{cases} $$ Multiply the second equation by 3 to align $x$ coefficients: $$ 9x - 15w = -33 $$ Subtract this from the first equation: $$ (9x - 21w) - (9x - 15w) = -57 - (-33) \\ 9x - 21w - 9x + 15w = -57 + 33 \\ -6w = -24 $$ Divide both sides by $-6$: $$ \cancel{-6}w = \cancel{-6}4 \\ w = 4 $$ 6. **Find $x$ using $w=4$ in $3x - 5w = -11$:** $$ 3x - 5(4) = -11 \\ 3x - 20 = -11 \\ 3x = -11 + 20 \\ 3x = 9 \\ \cancel{3}x = \cancel{3}3 \\ x = 3 $$ **Final answer:** $$w = 4, \quad x = 3$$