In which of the following quadrilaterals are the diagonals not perpendicular to each other in general?

Introduction / Context:
This question tests your understanding of when diagonals in a quadrilateral are perpendicular. Some special quadrilaterals, like squares and certain kites or rhombuses, have diagonals that intersect at 90°, but others do not. The problem asks which listed quadrilateral does not, in general, have perpendicular diagonals.

Given Data / Assumptions:

• Quadrilaterals considered: parallelogram, kite, rhombus, square.
• “Diagonals are not perpendicular” means that, in the general case for that type, the diagonals do not meet at a right angle.
• We focus on the defining properties of each type, not rare special cases.

Concept / Approach:
We recall diagonal properties:

• Parallelogram: diagonals bisect each other, but they are not generally perpendicular; only in special cases such as a rhombus with particular angles might they be perpendicular.
• Kite: typically, one diagonal is the perpendicular bisector of the other, so diagonals are perpendicular in many kites by definition.
• Rhombus: all sides equal; its diagonals bisect each other and are perpendicular in the standard property list.
• Square: a special rhombus and rectangle; its diagonals are equal and perpendicular.

We are looking for the quadrilateral type where diagonals are not naturally perpendicular for the class as a whole.

Step-by-Step Reasoning:
Step 1: In a square, a well known property is that the diagonals intersect at right angles and are equal in length. So a square definitely has perpendicular diagonals. Step 2: In a rhombus, diagonals bisect each other at right angles, even though they are not equal in length. Thus rhombuses are also known for having perpendicular diagonals. Step 3: Many standard definitions of a kite highlight that one diagonal is the perpendicular bisector of the other, meaning they intersect at 90°. Step 4: In a general parallelogram, however, while the diagonals bisect each other, they are not perpendicular unless the parallelogram has special symmetry (for example, if it is a square or a specific type of rhombus). For typical parallelograms, the diagonals meet at some acute or obtuse angle, not 90°. Step 5: Therefore, the quadrilateral whose diagonals are not perpendicular in the general case is the parallelogram.

Verification / Alternative check:
You can draw a generic slanted parallelogram (not a rectangle or square) and sketch its diagonals. Visually, they intersect at an angle that is clearly not 90°. Measuring with a protractor in a properly scaled drawing will confirm this. Contrast that with a kite or rhombus, where the diagonals visually cross at a right angle in standard textbook diagrams.

Why Other Options Are Wrong:
In a kite, diagonals are typically perpendicular, one being the perpendicular bisector of the other. In a rhombus, diagonals are perpendicular and also bisect the angles. In a square, diagonals are equal and perpendicular, combining properties of a rectangle and a rhombus. Therefore, these three options generally have perpendicular diagonals, not non perpendicular diagonals.

Common Pitfalls:
A common confusion is to assume that because a parallelogram resembles a rhombus or rectangle in some orientation, it must share all their diagonal properties. In reality, only certain special parallelograms (like rectangles and squares) have diagonals with additional properties such as equality or perpendicularity. Understanding that “parallelogram” in a general question means any parallelogram, not necessarily rectangular, helps to avoid this mistake.

Final Answer:
The quadrilateral in which diagonals are not perpendicular in general is the parallelogram.