1. **Stating the problem:** We have a parallelogram ABCD with sides |AB| = 5 cm, |BC| = 5.5 cm, and angle BAC = 45°. 2. **Understanding the figure:** In parallelogram ABCD, diagonal BD connects vertices B and D. We want to find the length |BD| and determine which inequality range it fits. 3. **Using the law of cosines:** Since ABCD is a parallelogram, opposite sides are equal and adjacent sides form the given angle. We consider triangle ABD with sides AB = 5 cm, AD = BC = 5.5 cm, and angle BAD = 45°. 4. **Apply the law of cosines to find |BD|:** $$|BD|^2 = |AB|^2 + |AD|^2 - 2 \times |AB| \times |AD| \times \cos(45^\circ)$$ 5. **Calculate:** $$|BD|^2 = 5^2 + 5.5^2 - 2 \times 5 \times 5.5 \times \cos(45^\circ)$$ $$= 25 + 30.25 - 55 \times \frac{\sqrt{2}}{2}$$ $$= 55.25 - 55 \times 0.7071$$ $$= 55.25 - 38.89 = 16.36$$ 6. **Find |BD|:** $$|BD| = \sqrt{16.36} \approx 4.045 \text{ cm}$$ 7. **Convert to millimeters:** $$4.045 \text{ cm} = 40.45 \text{ mm}$$ 8. **Compare with given options:** - a) 75 ≤ |BD| ≤ 79 mm - b) 53 ≤ |BD| ≤ 57 mm - c) 95 ≤ |BD| ≤ 99 mm - d) 70 ≤ |BD| ≤ 74 mm Our calculated |BD| ≈ 40.45 mm does not fit any of these ranges. **Conclusion:** None of the given options a), b), c), or d) correctly represent the length of diagonal BD based on the given data. **Final answer:** |BD| ≈ 40.45 mm, which is outside all provided ranges.