Lesson Video: Properties and Special Cases of Parallelograms Mathematics

Video Transcript

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

But first, let’s begin with defining what a parallelogram is. A parallelogram is defined as a quadrilateral, that’s a four-sided shape, with both pairs of opposite sides parallel. So, for example, in the diagram of this parallelogram, we know that 𝐴𝐵 is parallel to 𝐷𝐶 and 𝐴𝐷 is parallel to 𝐵𝐶. So lots of the properties that we’ll 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 𝑥 degrees and another pair of opposite angles each measuring 𝑦 degrees. The sum of 𝑥 degrees and 𝑦 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 𝐴𝐵𝐶𝐷, we could say that 𝐴𝐷 is parallel to 𝐵𝐶 and 𝐷𝐶 is parallel to 𝐴𝐵. The second property tells us that opposite angles are equal in measure. So here we have that the measure of angle 𝐴 is equal to the measure of angle 𝐶. And the measure of angle 𝐵 is equal to the measure of angle 𝐷.

Property number three tells us that the sum of any two consecutive interior angles is 180 degrees. For example, the measure of angle 𝐴 plus the measure of angle 𝐵 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 let’s consider how we know that properties four and five hold. To do this, let’s take this slightly different parallelogram 𝐴𝐵𝐶𝐷. We already know that opposite sides are parallel, and we can construct the diagonal 𝐷𝐵. In order to demonstrate property four, that opposite sides are equal in length, we can use this diagonal and demonstrate that the two triangles 𝐴𝐵𝐷 and 𝐶𝐷𝐵 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 𝐶𝐷𝐵 is equal to the measure of angle 𝐴𝐵𝐷.

Using the other pair of transversals then, we identify a second pair of congruent angles. The measure of angle 𝐴𝐷𝐵 is equal to the measure of angle 𝐶𝐵𝐷. Finally, we can identify that the diagonal 𝐵𝐷 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 𝐶𝐷𝐵 is congruent to triangle 𝐴𝐵𝐷. The importance of this congruency means that we can now say that side 𝐷𝐶 and side 𝐴𝐵 are congruent and 𝐴𝐷 and 𝐶𝐵 are congruent. This proves property number four. And although we won’t demonstrate it here, we can use similar properties of congruent triangles to prove property five.

We’ll now have a look at the first example. And we’ll 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 𝐶𝐷 and line segment 𝐷𝐴.

If we look at this quadrilateral 𝐴𝐵𝐶𝐷, 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 it’s not just simply that opposite sides are parallel, but opposite sides are equal in length too. The line segment 𝐶𝐷 is opposite to the line segment 𝐴𝐵. So 𝐶𝐷 will be 15 centimeters. Next, we need to find the length of the line segment 𝐷𝐴. And it is opposite to the line segment 𝐶𝐵. As these two line segments are the same length, then 𝐷𝐴 will also be 13 centimeters.

And so we can give the answer for the two line segments as 𝐶𝐷 is equal to 15 centimeters and 𝐷𝐴 is equal to 13 centimeters.

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

𝐴𝐵𝐶𝐷 is a parallelogram in which the measure of angle 𝐵𝐸𝐶 equals 79 degrees and the measure of angle 𝐸𝐶𝐵 equals 56 degrees. Determine the measure of angle 𝐸𝐴𝐷.

In this figure, we can see that we have the two angles 𝐵𝐸𝐶 and 𝐸𝐶𝐵 given. And we need to determine the measure of angle 𝐸𝐴𝐷. 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 𝐸𝐴𝐷, then a consecutive angle which might be useful to know is the measure of angle 𝐶𝐵𝐸. If we knew this, then by the second property here we could calculate the measure of angle 𝐸𝐴𝐷.

Let’s consider this triangle 𝐵𝐸𝐶. 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 𝐶𝐵𝐸 plus 56 degrees plus 79 degrees is equal to 180 degrees. Rearranging this, we obtain that the measure of angle 𝐶𝐵𝐸 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 𝐸𝐴𝐷 and the measure of angle 𝐶𝐵𝐸 must add to be 180 degrees. To find the measure of angle 𝐸𝐴𝐷 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.

We’ll 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 let’s 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.

We’ll 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, we’re 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.

We’ll 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.

Let’s 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, let’s 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.