# Question Video: Identifying a Special Type of Parallelogram given Its Properties Mathematics

A ＿ is a parallelogram with congruent sides and congruent diagonals.

04:25

### Video Transcript

A what is a parallelogram with congruent sides and congruent diagonals.

Let’s begin this question by recalling that a parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. So, for example, we could draw a parallelogram like this or even like this. They just have to have two pairs of opposite sides parallel. As we are considering the properties of parallelograms that relate to the sides and diagonals, let’s think about the properties relating to these features that we know are true for all parallelograms.

In every parallelogram, we know that opposite sides are congruent. That’s in addition to opposite sides being parallel. And we know that the diagonals bisect each other. However, not every parallelogram has congruent sides or congruent diagonals. We can see, for example, that neither of these properties is true for the parallelogram drawn here. So, the answer for this question will be a different type of quadrilateral that could also be described as a type of parallelogram.

It can be useful to consider parallelograms as part of a Venn diagram covering all types of quadrilaterals. Parallelograms are quite a large classification because quite a few other quadrilaterals share the same properties of having two parallel sides. For example, rectangles also have two pairs of opposite sides parallel so do rhombuses. And the quadrilaterals which have the properties of rectangles and rhombuses and parallelograms are squares. Outside of the parallelogram classification, we have kites and trapezoids because neither of these quadrilateral types have two pairs of sides parallel. But let’s ignore these for the purposes of this question.

We can take a closer look at the three special types of parallelogram: the rectangle, rhombus, and square. Firstly, we can recall that a rectangle is a parallelogram with four congruent angles. In terms of the properties of the diagonals, in a rectangle, the diagonals are congruent, which is one of the properties that our parallelogram type does need to have. However, if we are considering that this unknown type of parallelogram has to have congruent sides, we know that this property would not hold true for all rectangles, since they only have the property that opposite sides are congruent. So the unknown parallelogram type that we need to determine cannot be a rectangle.

Next, let’s recall that a rhombus is a parallelogram with four congruent sides. This is one of the properties that we are looking for. However, we can notice even just by eye that the diagonals in this rhombus on the screen would not be congruent. The only properties for the diagonals of a rhombus that we can say for sure is that they are perpendicular and they bisect opposite pairs of angles. So, we can also rule out the rhombus as the answer for this question, which leaves us with squares, which are defined as parallelograms with four congruent angles and four congruent sides.

Looking at the Venn diagram or the definition of a square, we know that squares are a type of parallelogram. And they are also a type of rectangle and a type of rhombus. Therefore, they inherit all the properties of these quadrilaterals, including the properties that the sides are congruent and that the diagonals are congruent. We can then complete the blank with the word square, since a square is a parallelogram with congruent sides and congruent diagonals.