"If a quadrilateral is a rectangle, then it is a parallelogram."

i) is the converse true?
ii) is the inverse true?
iii) is the contrapositive true?

i) no

ii) no
iii) yes, as always

No

Let's analyze each statement:

i) The converse of the statement "If a quadrilateral is a rectangle, then it is a parallelogram" would be "If a quadrilateral is a parallelogram, then it is a rectangle." This statement is true. In other words, every parallelogram is a rectangle.

ii) The inverse of the original statement would be "If a quadrilateral is not a rectangle, then it is not a parallelogram." This statement is false. There are many quadrilaterals that are parallelograms but not rectangles, such as squares and rhombuses.

iii) The contrapositive of the original statement would be "If a quadrilateral is not a parallelogram, then it is not a rectangle." This statement is also true. In other words, if a quadrilateral is not a parallelogram, then it cannot be a rectangle.

To summarize:

i) The converse is true.
ii) The inverse is false.
iii) The contrapositive is true.

To determine whether the converse, inverse, and contrapositive of a statement are true, we need to understand what each of these terms means.

i) Converse: The converse of a statement is formed by switching the hypothesis and conclusion. In this case, the converse statement would be, "If a quadrilateral is a parallelogram, then it is a rectangle."

To determine if the converse is true, we need to evaluate if all parallelograms are rectangles. This statement is not always true because a parallelogram can have angles other than right angles, while a rectangle has four right angles. Therefore, the converse of the original statement is not true.

ii) Inverse: The inverse of a statement is formed by negating both the hypothesis and conclusion. In this case, the inverse statement would be, "If a quadrilateral is not a rectangle, then it is not a parallelogram."

To determine if the inverse is true, we need to evaluate if all non-rectangular quadrilaterals are also not parallelograms. This statement is also not always true because there are other types of quadrilaterals (such as trapezoids) that are not rectangles but are still parallelograms. Therefore, the inverse of the original statement is not true.

iii) Contrapositive: The contrapositive of a statement is formed by switching and negating both the hypothesis and conclusion. In this case, the contrapositive statement would be, "If a quadrilateral is not a parallelogram, then it is not a rectangle."

To determine if the contrapositive is true, we need to evaluate if all non-parallelogram quadrilaterals are also not rectangles. This statement is always true because if a quadrilateral is not a parallelogram, it cannot have four equal sides and opposite angles, which are characteristics of a rectangle. Therefore, the contrapositive of the original statement is true.

In summary:
i) The converse is not true.
ii) The inverse is not true.
iii) The contrapositive is true.