Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
They can be, for example if in a parallelogram the two diagonals are perpendicular then the parallelogram is a rhombus
To determine if the diagonals of a parallelogram are perpendicular, we need to understand the properties of a parallelogram.
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are congruent.
The diagonals of a parallelogram are the line segments that connect two non-adjacent vertices. Let's label the vertices of the parallelogram as A, B, C, and D. The diagonals are then line segments AC and BD.
To determine if these diagonals are perpendicular, we need to examine the properties of the angles formed by the diagonals.
If the diagonals of a parallelogram are perpendicular, then the opposite angles formed by the intersection of the diagonals must be congruent. In other words, angle ADC must be congruent to angle BAC, and angle ADB must be congruent to angle BCD.
However, this condition is not always satisfied in a parallelogram. The diagonals of a parallelogram are only perpendicular if and only if the parallelogram is a special type known as a rhombus.
A rhombus is a parallelogram in which all four sides are equal in length. Since the sides of a rhombus are equal, the opposite angles formed by the diagonals are congruent, making the diagonals perpendicular.
In summary, the diagonals of a parallelogram are not necessarily perpendicular. They can be perpendicular if and only if the parallelogram is a rhombus, where all four sides are equal in length.