### Video Transcript

In this video, weβll learn how to
use the properties of a parallelogram. And weβll also identify the special
cases of parallelograms along with their properties. These special cases are rectangles,
rhombuses, and squares.

But first, letβs begin with
defining what a parallelogram is. A parallelogram is defined as a
quadrilateral, thatβs a four-sided shape, with both pairs of opposite sides
parallel. So, for example, in the diagram of
this parallelogram, we know that π΄π΅ is parallel to π·πΆ and π΄π· is parallel to
π΅πΆ. So lots of the properties that
weβll see in parallelograms come from some of the properties that we see in parallel
lines. When parallel lines are cut by a
transversal, then there are a number of congruent and supplementary angles. Pairs of corresponding angles are
congruent. Pairs of alternate interior angles
are congruent. And consecutive interior angles are
supplementary.

If we then also considered a line
which is parallel to this transversal, then we obtain a similar pattern of congruent
and supplementary angles. This allows us to identify some of
the first properties of a parallelogram. Notice how we have one pair of
opposite angles each measuring π₯ degrees and another pair of opposite angles each
measuring π¦ degrees. The sum of π₯ degrees and π¦
degrees must equal 180 degrees.

We can now note the five important
properties of a parallelogram. Firstly, we have that opposite
sides are parallel. This arises as a result of the
definition. And in this parallelogram π΄π΅πΆπ·,
we could say that π΄π· is parallel to π΅πΆ and π·πΆ is parallel to π΄π΅. The second property tells us that
opposite angles are equal in measure. So here we have that the measure of
angle π΄ is equal to the measure of angle πΆ. And the measure of angle π΅ is
equal to the measure of angle π·.

Property number three tells us that
the sum of any two consecutive interior angles is 180 degrees. For example, the measure of angle
π΄ plus the measure of angle π΅ is equal to 180 degrees. The fourth property tells us that
opposite sides are equal in length. Finally, the fifth property is one
regarding the diagonals. And that is that the diagonals
bisect each other.

We have already seen how we obtain
the properties in one, two, and three. But letβs consider how we know that
properties four and five hold. To do this, letβs take this
slightly different parallelogram π΄π΅πΆπ·. We already know that opposite sides
are parallel, and we can construct the diagonal π·π΅. In order to demonstrate property
four, that opposite sides are equal in length, we can use this diagonal and
demonstrate that the two triangles π΄π΅π· and πΆπ·π΅ are congruent. We can start by recognizing that we
have a pair of congruent angles coming from the parallel lines and the transversals
such that the measure of angle πΆπ·π΅ is equal to the measure of angle π΄π΅π·.

Using the other pair of
transversals then, we identify a second pair of congruent angles. The measure of angle π΄π·π΅ is
equal to the measure of angle πΆπ΅π·. Finally, we can identify that the
diagonal π΅π· is in fact a common or shared side in both of these triangles.

We now have enough information to
fulfill the angle-side-angle congruence criterion, which states that if two angles
and the included side are congruent to two angles and the included side of a second
triangle, then the triangles are congruent. We can therefore prove that
triangle πΆπ·π΅ is congruent to triangle π΄π΅π·. The importance of this congruency
means that we can now say that side π·πΆ and side π΄π΅ are congruent and π΄π· and
πΆπ΅ are congruent. This proves property number
four. And although we wonβt demonstrate
it here, we can use similar properties of congruent triangles to prove property
five.

Weβll now have a look at the first
example. And weβll need to use some of these
properties in order to help us find some unknown lengths in a parallelogram.

Find the lengths of line
segment πΆπ· and line segment π·π΄.

If we look at this
quadrilateral π΄π΅πΆπ·, the first thing we can do is identify that this is a
parallelogram. This is because we can see on
the diagram that we have opposite sides parallel. If we use the properties of a
parallelogram, we can recall that itβs not just simply that opposite sides are
parallel, but opposite sides are equal in length too. The line segment πΆπ· is
opposite to the line segment π΄π΅. So πΆπ· will be 15
centimeters. Next, we need to find the
length of the line segment π·π΄. And it is opposite to the line
segment πΆπ΅. As these two line segments are
the same length, then π·π΄ will also be 13 centimeters.

And so we can give the answer
for the two line segments as πΆπ· is equal to 15 centimeters and π·π΄ is equal
to 13 centimeters.

Weβll now see how we can calculate
an unknown angle in a parallelogram by considering its properties.

π΄π΅πΆπ· is a parallelogram in
which the measure of angle π΅πΈπΆ equals 79 degrees and the measure of angle
πΈπΆπ΅ equals 56 degrees. Determine the measure of angle
πΈπ΄π·.

In this figure, we can see that
we have the two angles π΅πΈπΆ and πΈπΆπ΅ given. And we need to determine the
measure of angle πΈπ΄π·. As we are told that this is a
parallelogram, then that means that we can apply the properties of
parallelograms to help us find this unknown angle. And as we are thinking about an
angle, then the two angle properties of a parallelogram are that opposite angles
are equal in measure and the sum of the measures of two consecutive angles is
180 degrees. As we want to find the measure
of angle πΈπ΄π·, then a consecutive angle which might be useful to know is the
measure of angle πΆπ΅πΈ. If we knew this, then by the
second property here we could calculate the measure of angle πΈπ΄π·.

Letβs consider this triangle
π΅πΈπΆ. As we are given two angles, we
can apply the property that the interior angle measures in a triangle sum to 180
degrees. This means that the measure of
angle πΆπ΅πΈ plus 56 degrees plus 79 degrees is equal to 180 degrees. Rearranging this, we obtain
that the measure of angle πΆπ΅πΈ is equal to 180 degrees subtract 135 degrees,
and that is 45 degrees. And then as previously
mentioned, we can identify this pair of consecutive angles. The measure of angle πΈπ΄π· and
the measure of angle πΆπ΅πΈ must add to be 180 degrees. To find the measure of angle
πΈπ΄π· then, we subtract 45 degrees from 180 degrees, giving us an answer of 135
degrees.

We will now consider a few special
cases of parallelograms. The first of these is a rectangle,
and it is defined as a parallelogram with four congruent angles. Now, of course, since we know that
the sum of the angle measures in any quadrilateral is 360 degrees, then we know that
each of the four angles in a rectangle is 90 degrees. This leads us into the properties
of a rectangle. The important thing to remember is
that since a rectangle is a type of parallelogram, then it will have all the same
properties that parallelograms do. And there are two additional
properties that rectangles have. All the angles are equal in
measure. They are all 90 degrees. And the diagonals are equal in
length.

Notice that in a parallelogram we
only know that the diagonals bisect each other. They will only be equal when the
parallelogram is a rectangle.

Weβll now consider the second
special case of parallelogram, that is, a rhombus. A rhombus is a parallelogram with
four congruent sides. The additional properties of a
rhombus arise from the diagonals. If we draw the diagonals onto the
rhombus, one of the properties we will see is that the diagonals bisect the opposite
angles. We will also have the property that
the diagonals are perpendicular. So letβs make a note of the
properties of a rhombus.

Just as we saw with the rectangle,
because a rhombus is a type of parallelogram, then it inherits all the properties of
a parallelogram. It also has three additional
properties that, firstly, all sides are equal in length. Then, the diagonals bisect opposite
pairs of angles. And thirdly, the diagonals are
perpendicular. Notice with this last property,
because we can say that the diagonals are perpendicular and we know from the
properties of a parallelogram that the diagonals bisect each other, we can say that
the diagonals of a rhombus are perpendicular bisectors.

Weβll now have a look at the last
special case of parallelogram. If a parallelogram has all angles
equal in measure and has all sides congruent, we call it a square. Now we should remember that a
parallelogram that has four congruent angles is called a rectangle and a
parallelogram that has four congruent sides is called a rhombus. We can then say that a square is a
special case of both a rectangle and a rhombus. So it has all the properties we
have seen for a parallelogram, a rectangle, and a rhombus.

We will already of course be
familiar with the easier properties of a square, for example, that there are four
90-degree angles and four congruent sides. It is, however, the diagonals of a
square which are most important to note. The diagonals of a square bisect
each other, they are equal in length, and they are perpendicular. And so now we have seen these three
special cases of parallelograms, we can complete some more examples.

A parallelogram whose what are
equal in length is called a rectangle.

We can recall that a rectangle
is a type of parallelogram with four congruent angles. The angles are all 90
degrees. We can sketch an example of a
rectangle and consider that in fact there is something which we know is equal in
length. The opposite sides in a
rectangle are equal in length. However, if we consider the
statement, we know that all parallelograms have opposite sides which are equal
in length. Therefore, weβre looking for
some additional property of rectangles which distinguish those from just any
parallelogram. That property comes from the
diagonals. The diagonals of a rectangle
are equal in length.

Note that in general the only
diagonal property of a parallelogram is that the diagonals of a parallelogram
bisect each other. Therefore, if we have got a
parallelogram and the diagonals are of equal length, then we know that it must
be a rectangle. We can therefore complete the
statement with the word diagonals. A parallelogram whose diagonals
are equal in length is called a rectangle.

Weβll now see one final
example.

Each of the two diagonals of
the square makes an angle with a measure of what with the adjacent side.

Letβs consider what we know
about a square and its properties. A square is a type of
parallelogram with four congruent sides and four congruent angles. Notice these angles are all 90
degrees. Because squares are a type of
parallelogram, they inherit all the properties of parallelograms plus all the
properties of rectangles and rhombuses.

Now because we are considering
the diagonals of the square, letβs recall the important properties about the
diagonals in a square. Firstly, in any and all
parallelograms, we know that the diagonals bisect each other. Secondly, because a square is a
type of rectangle, we know that the diagonals will be congruent. Thirdly, coming from the
properties of a rhombus, we know that the diagonals bisect opposite pairs of
angles.

In this problem, we need to
consider the measure of the angle that the diagonals make with adjacent
sides. For example, what is the
measure of this angle marked on the diagram? Well, we know that the two
sides of the square make a 90-degree angle. And we also know that the
diagonals bisect opposite pairs of angles. This angle of 90 degrees is
therefore bisected, or divided by two. And we know that that will be
equal to 45 degrees. Because the sides are
congruent, this will be true of every angle created with a diagonal and an
adjacent side. We can therefore give the
complete statement. Each of the two diagonals of
the square makes an angle with a measure of 45 degrees with the adjacent
side.

We will now finish this video by
recapping the key points. We began by defining a
parallelogram, which is a quadrilateral with both pairs of opposite sides
parallel. We saw that there are five
different properties of parallelograms. They are that opposite sides are
parallel. Opposite sides are congruent. The sum of consecutive interior
angles is 180 degrees. Opposite sides are congruent. And the diagonals bisect each
other.

Next, we saw a special case of
parallelogram, a rectangle. A rectangle has all the properties
of a parallelogram plus the angles are congruent and the diagonals are
congruent. The second type of parallelogram we
saw is a rhombus. It again has all the properties of
a parallelogram plus all the sides are congruent. The diagonals bisect opposite pairs
of angles. And the diagonals are
perpendicular. We also saw that a square is a type
of parallelogram with four congruent angles and four congruent sides. A square is a special case of both
a rectangle and a rhombus and inherits the properties of both.