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The question asks if quadrilateral \( ABCD \) can be a parallelogram based on three given conditions:

1. \( \angle D + \angle B = 180^\circ \)
2. \( AB = DC = 8 \, \text{cm}, AD = 4 \, \text{cm}, BC = 4.4 \, \text{cm} \)
3. \( \angle A = 70^\circ \) and \( \angle C = 65^\circ \)

### Let's analyze each part to determine if it satisfies the properties of a parallelogram:

#### (i) \( \angle D + \angle B = 180^\circ \)

- One of the properties of a parallelogram is that opposite angles are supplementary (i.e., their sum is 180°). So, this condition is satisfied and consistent with the property of a parallelogram.

#### (ii) \( AB = DC = 8 \, \text{cm}, AD = 4 \, \text{cm}, BC = 4.4 \, \text{cm} \)

- A parallelogram must have opposite sides that are equal. According to the given values, \( AB = DC \), which satisfies one condition. However, for a parallelogram, it must also hold that \( AD = BC \). In this case, \( AD = 4 \, \text{cm} \) and \( BC = 4.4 \, \text{cm} \), which are not equal. Therefore, this condition does not hold, and the quadrilateral cannot be a parallelogram based on this.

#### (iii) \( \angle A = 70^\circ \) and \( \angle C = 65^\circ \)

- Another property of a parallelogram is that opposite angles are equal. Here, \( \angle A \) and \( \angle C \) are given as 70° and 65°, respectively, which are not equal. This means that this condition does not match the requirements for a parallelogram.

### Conclusion:

- Based on the second and third conditions, the quadrilateral cannot be a parallelogram, because the opposite sides are not equal in length, and the opposite angles are not equal.

Would you like further details or have any questions about this solution?

Here are 5 related questions to explore further:
1. What are the key properties of parallelograms?
2. How can you prove a quadrilateral is a parallelogram using coordinate geometry?
3. What is the difference between a parallelogram and a trapezoid?
4. Can a quadrilateral be a rhombus if its opposite angles are not equal?
5. How would you calculate the perimeter of a parallelogram?

**Tip:** Remember, in any quadrilateral, the sum of the internal angles always equals 360°.

Can a quadrilateral ABCD be a parallelogram if (i) angle D + angle B = 180 degrees? (ii) AB = DC = 8 cm, AD = 4 cm, and BC = 4.4 cm? (iii) angle A = 70 degrees and angle C = 65 degrees?

Explore whether a quadrilateral ABCD can be a parallelogram with given angles and side lengths. This problem addresses the properties of parallelograms, including opposite sides being equal and opposite angles being supplementary.

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