Which of the following statements is/are correct?
A kite always has two axes of symmetry
A square is both a kite and an isosceles trapezium
Every kite is an orthodiagonal
an isosceles trapezium can have a circumcircle
A kite can always have an incircle
If the sum of two opposite sides is not the same as the sum of the other pair of opposite sides, it can never be a circumscribing quadrilateral.

Options

A. all, except (ii) and (iv)
B. only (ii), (iii) and (v)
C. only (i), (iii), (v)
D. all, except (i)

Answer

To determine which statements about kites, squares, isosceles trapeziums, and quadrilaterals are correct, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement 1: A kite always has two axes of symmetry.**
- A kite has one axis of symmetry that runs along the line connecting the two pairs of equal-length sides. Therefore, this statement is **incorrect**.

2. **Statement 2: A square is both a kite and an isosceles trapezium.**
- A square has all sides equal and opposite sides parallel, fulfilling the properties of both a kite (two pairs of equal-length sides) and an isosceles trapezium (one pair of parallel sides). Thus, this statement is **correct**.

3. **Statement 3: Every kite is an orthodiagonal.**
- In a kite, the diagonals intersect at right angles (90 degrees). Therefore, this statement is **correct**.

4. **Statement 4: An isosceles trapezium can have a circumcircle.**
- An isosceles trapezium can have a circumcircle if the sum of the lengths of the bases is equal to the sum of the lengths of the legs. Thus, this statement is **correct**.

5. **Statement 5: A kite can always have an incircle.**
- A kite can have an incircle if the lengths of the two pairs of opposite sides are equal. Therefore, this statement is **correct**.

6. **Statement 6: If the sum of two opposite sides is not the same as the sum of the other pair of opposite sides, it can never be a circumscribing quadrilateral.**
- For a quadrilateral to be circumscribed (having a circumcircle), the sums of the lengths of opposite sides must be equal. Thus, this statement is **correct**.

### Summary of Statements:
- Statement 1: Incorrect
- Statement 2: Correct
- Statement 3: Correct
- Statement 4: Correct
- Statement 5: Correct
- Statement 6: Correct

### Final Conclusion:
The correct statements are 2, 3, 4, 5, and 6. Therefore, the answer is that all options except the first statement are correct.

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