We're asked to select all the statements that are true
about the types of quadrilaterals. Remember a quadrilateral
is a four sided polygon.
The first statement is, squares are always rectangles.
Let's begin by reviewing the definition of a square. A square is pictured here,
where a square is a quadrilateral in which all sides have the same length,
and the interior angles are right angles. So again all the sides are the same length,
and the interior angles are right angles, or measure ninety degrees.
So now to see if squares are always rectangles.
Let's sketch a square here and compare it to the definition of a rectangle.
So here's a square,
and let's see if it satisfies the definition of a rectangle.
A rectangle is a quadrilateral in which opposite sides have the same length
and the interior angles are right angles.
Well notice how this square does satisfy the definition of a rectangle,
because this side has the same length as this side,
and this side the same length as the side.
And interior angles are also
ninety degrees, or are right angles. Which means the square is a special rectangle,
but squares are always rectangles is true.
So this first statement is true, so we go ahead and check it.
Next, rectangles are always squares.
So let's go back to the definition of a square,
which is here. Let's sketch a typical rectangle,
and see if it satisfies the definition of a square.
A squares a quadrilateral in which all sides have the same length,
and the interior angles are right angles, or measure ninety degrees.
So looking at the rectangle, while the interior angles
are right angles, all four sides do not have the same length,
which means rectangles are not always squares.
And therefore, we do not check the second statement. We'll move on to the third
statement. The third statement is, rhombuses are always squares.
So let's review the definition of a rhombus, a rhombus is pictured here.
A rhombus is a quadrilateral in which both pair of opposite sides
are parallel and have the same length. So the opposite sides are parallel,
and all sides have the same length. Remember sides are parallel
if they were to be extended and they will never intersect.
So to determine whether rhombuses are always squares
is true. Let's sketch a rhombus
and compare it to the definition a square.
So a typical rhombus might look something like this.
Let's see if this satisfies the definition of a square. The square is a quadrilateral
in which all sides have the same length, and
the interior angles are right angles. So while this rhombus does have four
sides are the same length,
the interior angles are not right angles, which means,
it's not true that rhombuses are always squares.
So we don't check this box, and we go to the next statement.
The next statement is that squares are always rhombuses.
So let's go back to definition of a rhombus.
Sketch a square and see if it satisfies the definition of a rhombus.
So here's our square. And a rhombus
is a quadrilateral in which both pair of opposite sides are parallel,
and all sides have the same length. So notice how for a square,
all sides do have the same length. And opposite
sides are parallel, which means squares are
always rhombuses. So this fourth statement is true,
so we'll go ahead and check this.
And we have two more statements to consider. The next statement, rhombuses are always
parallelograms.
So let's review the definition of a parallelogram. Before we do this though,
let's also sketch a typical rhombus.
So we know for a rhombus, opposite sides are parallel
and all four sides have the same length. So this might be a typical rhombus.
And here we also have a picture of a typical parallelogram.
Where our parallelogram is a quadrilateral in which both pair of opposite sides are
parallel.
Notice how it does not mention anything about the lengths of the sides.
And because a rhombus has four equal sides
and opposite sides that are parallel, notice how all
rhombuses do satisfy the definition of a parallelogram,
and therefore, it is true that rhombuses
are always parallelograms. They're just special, because
not only are opposite sides parallel, all sides also have the same length.
Our last statement is rhombuses are sometimes rectangles.
This statement is a little bit more challenging to see if it's sure not,
because other word - sometimes.
Rhombuses are sometimes rectangles will be true.
If we can sketch a single rhombus that would also be a rectangle.
And notice how in our previous statements we said that squares
are always rhombuses, so a squares is a special rhombus.
And we also said that squares are always rectangles.
So if a square is always a rhombus, and a square is also always a rectangle,
it is going to be true that rhombuses are sometimes rectangles.
So while this will be true, let's go back and review the definition of a rectangle.
If we can sketch a single rhombus, where the opposite sides have the same length,
and the interior angles are right angles, then that rhombus is also going to be a
rectangle.
So again if we were to sketch a quadrilateral where opposite sides are
parallel,
and all four sides are the same length. For example,
a square, which is a special rhombus.
Because this square is rhombus, and squares are also rectangles,
our statement is true that rhombuses
are sometimes rectangles.
I hope you found this helpful.