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
Perpendicular to what? If they are perpendicular, they form 90 degree angles, giving you a rectangle.
To understand whether the diagonals of a parallelogram are perpendicular or not, let's first review the properties of a parallelogram.
A parallelogram is a quadrilateral with two pairs of parallel sides. This means that the opposite sides of a parallelogram are parallel and congruent. Additionally, the opposite angles of a parallelogram are congruent.
Now, consider the diagonals of a parallelogram. Diagonals are line segments that connect two non-adjacent vertices of a polygon. In a parallelogram, there are two diagonals – one connecting the opposite vertices, and another connecting the other two opposite vertices.
To determine whether the diagonals of a parallelogram are perpendicular or not, we need to examine the properties shared by the diagonals.
If the diagonals of a parallelogram are perpendicular, it signifies that they intersect at a 90-degree angle. In other words, the measure of the angle formed at their intersection is 90 degrees.
So, are the diagonals of a parallelogram always perpendicular? The answer is no. The diagonals of a general parallelogram are not always perpendicular.
However, there is a special case where the diagonals of a parallelogram can be perpendicular. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a specific type of parallelogram called a rhombus.
In a rhombus, all four sides are congruent, and the opposite angles are congruent, just like in a parallelogram. But what sets a rhombus apart is that its diagonals are perpendicular to each other.
So, to summarize:
- In a general parallelogram, the diagonals are not always perpendicular.
- If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.
If you want to determine whether a given parallelogram has perpendicular diagonals or not, check if it satisfies the properties of a rhombus.