1. **Stating the problem:** We have a square with two diagonal lines intersecting inside it, creating two triangles. We know two angles: 40° at the top-left corner and 80° near the top-right corner. We need to find the angle $x$ inside the triangle near the bottom side. 2. **Important properties:** A square has four right angles, each 90°. The diagonals of a square intersect at right angles (90°) and bisect each other. 3. **Analyze the left triangle:** It has a 40° angle at the top-left corner and a right angle (90°) at the bottom-left corner. 4. **Calculate the third angle in the left triangle:** Sum of angles in a triangle is 180°. $$x_1 = 180° - 90° - 40° = 50°$$ 5. **Analyze the right triangle:** It has an 80° angle near the top-right corner and a right angle (90°) at the bottom-right corner. 6. **Calculate the third angle in the right triangle:** $$x_2 = 180° - 90° - 80° = 10°$$ 7. **Find angle $x$:** The angle $x$ is adjacent to the 50° and 10° angles formed by the diagonals intersecting inside the square. Since the diagonals intersect at 90°, the sum of these three angles is 90°. $$x + 50° + 10° = 90°$$ 8. **Solve for $x$:** $$x = 90° - 50° - 10° = 30°$$ **Final answer:** $$\boxed{30°}$$