1. **Stating the problem:** We have two pairs of parallel lines crossed by transversals, creating pairs of corresponding or alternate interior angles that are equal. 2. **First pair of angles:** Given angles are $80^\circ$ and $(x+25)^\circ$. Since these angles are corresponding or alternate interior angles, they are equal: $$80 = x + 25$$ 3. **Solve for $x$:** $$x = 80 - 25$$ $$x = 55$$ 4. **Second pair of angles:** Given angles are $4(x+1)^\circ$ and $(7x + 3)^\circ$. These angles are also equal: $$4(x+1) = 7x + 3$$ 5. **Expand and simplify:** $$4x + 4 = 7x + 3$$ 6. **Bring all terms to one side:** $$4x + 4 - 7x - 3 = 0$$ $$-3x + 1 = 0$$ 7. **Solve for $x$:** $$-3x = -1$$ $$x = \frac{\cancel{-1}}{\cancel{-3}} = \frac{1}{3}$$ 8. **Summary:** - From the first pair, $x = 55$ - From the second pair, $x = \frac{1}{3}$ Since the problem states $x=112^\circ$ initially, but solving the pairs gives different values, the consistent solution for each pair is as above. **Final answers:** - For the first pair: $x = 55$ - For the second pair: $x = \frac{1}{3}$