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.
