Lesson Video: Properties of Parallelograms Mathematics

Video Transcript

In this video, we’ll learn what it means for a shape to be a parallelogram. We’ll then discover the important properties that a parallelogram has. We can use these properties to help us find unknown angles or lengths.

Let’s begin with a mathematical definition of a parallelogram. This will be a quadrilateral or four-sided shape with both pairs of opposite sides parallel. So we could draw a parallelogram like this or one like this or even like this. All they have to have is both pairs of opposite sides parallel. You might wonder why we’ve drawn a rectangle here, but remember that a rectangle is just a special type of parallelogram. It will still have both pairs of opposite sides parallel. Squares and rhombuses would also be two other special types of parallelogram.

We’ll now take a closer look at some of the important properties. When we’re investigating the properties of a parallelogram, we can just use a general parallelogram rather than a special one, like a square or a rectangle. So here we have parallelogram 𝐴𝐵𝐶𝐷. The first property of a parallelogram is that opposite sides are parallel. And we know that from the very definition of a parallelogram. The second property of parallelograms are that opposite sides are equal or congruent. Let’s look at this parallelogram and see if we can understand why this would be the case. We can split our parallelogram along the diagonal 𝐴𝐶. We can then think about the angles that we would create.

Looking at angle 𝐵𝐴𝐶 because we have two parallel lines and a transversal, then we also have an alternate angle here at 𝐴𝐶𝐷. So these two angles would be equal. For the same reasons, we can say that angle 𝐷𝐴𝐶 will be equal in size to angle 𝐵𝐶𝐴, once again alternate angles. Looking at the lengths, the line 𝐴𝐶 is common to both of these triangles. Don’t worry if you haven’t done too much on the congruency of triangles. But we’ve actually got enough information here to prove that the two triangles 𝐷𝐴𝐶 and 𝐵𝐶𝐴 are congruent using the angle-side-angle rules. We have two pairs of corresponding angles congruent and a corresponding side congruent. When two triangles are congruent, that means they’re exactly the same shape and size.

In terms of helping us to prove the property that opposite sides are equal, as we have two congruent sides, we could say that this length 𝐵𝐴 in triangle 𝐵𝐶𝐴 is congruent or equal to this length of 𝐷𝐶 in triangle 𝐷𝐴𝐶. So these two sides of the parallelogram are equal in length. In triangle 𝐵𝐶𝐴, the length 𝐵𝐶 will be congruent with the length 𝐷𝐴. And so we’ve shown that the other two sides are also equal in length, and so proving the second property.

The third property of parallelograms is that opposite angles are equal. We can use the congruency of the two triangles within it to demonstrate this. We could see how the sum of the angles here at angle 𝐴 would be equal to the sum of the angles here at angle 𝐶. We could write that angle 𝐷𝐴𝐵 is equal to angle 𝐵𝐶𝐷. In our congruent triangles, this angle at 𝐴𝐷𝐶 will correspond with this angle at 𝐶𝐵𝐴. So these two angles will also be equal, and so proving the third property.

The next property of parallelograms is that the sum of two adjacent angles is 180 degrees. We could alternatively write this as adjacent angles are supplementary. What we mean by this is if, for example, we took the angle 𝐴 and 𝐵 and added those together, they would add up to 180 degrees. Angle 𝐵 and 𝐶 would also add to 180 degrees, and so would angle 𝐶 and 𝐷. We can prove this by remembering that both pairs of opposite sides are parallel. Let’s look at angles 𝐵 and 𝐶.

We can recall that 𝐴𝐵 and 𝐷𝐶 are parallel and 𝐵𝐶 would be a transversal of these. We can recall that the sum of the interior angles on the same side of a transversal is 180 degrees. And therefore, the angles 𝐵 and 𝐶 would add up to 180 degrees. If we look at angles 𝐶 and 𝐷 instead, then our two parallel lines here would be 𝐵𝐶 and 𝐴𝐷. The transversal would be the line 𝐶𝐷. And so we can see again how the sum of these two angles would add up to 180 degrees.

The final property we’re going to look at in this video is that the diagonals of a parallelogram are bisectors. If they’re bisectors, that means they will cut each other exactly in half. So let’s have a look at why this would happen. We can label this point where they cross with the letter 𝐸. Let’s have a look at this angle 𝐷𝐴𝐸. We should be comfortable now with recognizing that this angle 𝐴𝐶𝐵 will be congruent to it as we have our parallel lines and a transversal. In the same way, this angle 𝐴𝐷𝐸 would be equal to the angle 𝐸𝐵𝐶.

You might realize we’re working towards another pair of congruent triangles. But we’d need to show that there’s a corresponding side congruent. And we’ve already shown that the property of parallelograms is that opposite sides are equal in length. So the length 𝐴𝐷 will be equal to the length 𝐵𝐶. And so we can say that triangle 𝐴𝐷𝐸 is congruent to triangle 𝐶𝐵𝐸 using the angle-side-angle rule. Therefore, the length 𝐴𝐸 in triangle 𝐴𝐷𝐸 corresponds with the length 𝐶𝐸 in triangle 𝐶𝐵𝐸. And so we can say that these two lengths are equal, and we’ve also shown that this diagonal 𝐴𝐶 has been bisected.

If we look at this length 𝐷𝐸, we know that it corresponds to 𝐵𝐸 in triangle 𝐶𝐵𝐸. And therefore, these two lengths are equal and show that the second diagonal has also been bisected. It’s worthwhile making a note of these properties of a parallelogram. They’re useful for exams and we’ll need them as we go through the following questions. Let’s have a look at our first question.

Which of the following statements must be true about a parallelogram? Option (A) it has four sides of equal length. Option (B) it has four right angles. Option (C) it has four congruent sides. Option (D) it has exactly one pair of parallel sides. Or option (E) it has exactly two pairs of parallel sides.

In order to answer this question, we’ll need to remember what exactly a parallelogram is. It’s defined as a quadrilateral or four-sided shape, with both pairs of opposite sides parallel. We could therefore draw a range of different parallelograms. The important thing is that, in each one, it will have both pairs of opposite sides parallel. So let’s have a look at the different statements we’re given.

Looking at option (A), which says it has four sides of equal length, we can see that the parallelograms we’ve drawn definitely don’t have four sides of equal length. We could’ve drawn a square or even a rhombus, and that would have four sides of equal length. But we can’t say that about every single parallelogram. And so option (A) is incorrect.

Option (C) is phrased differently. It has the word congruent here, referring to the sides, which means that it’d say it would have four equal sides. We can see that this would not be the case. Option (B) says it has four right angles. Well, we know that a square or a rectangle does have four right angles, but it doesn’t apply to every single parallelogram. Therefore, option (B) isn’t correct. Option (D) says it has exactly one pair of parallel sides. Well, we know from the definition that both pairs of opposite sides have to be parallel, so option (D) is incorrect, which leaves us with option (E). It has exactly two pairs of parallel sides.

The definition of a parallelogram tells us that both pairs or two pairs of the opposite sides will be parallel. So option (E) is our correct answer. It has exactly two pairs of parallel sides.

In our next question, we’ll find some unknown lengths.

Find the lengths of line segment 𝐶𝐷 and line segment 𝐷𝐴.

If we look at this shape 𝐴𝐵𝐶𝐷, we can see that it’s a parallelogram, but how do we know that for sure? Well, a parallelogram is a quadrilateral with both pairs of opposite sides parallel. We can see from the line markings that 𝐶𝐵 and 𝐴𝐷 are parallel, and so are 𝐷𝐶 and 𝐴𝐵. The question asks us to find the length of this line segment 𝐶𝐷 and the line segment of 𝐷𝐴. In order to do this, we’ll need to remember a key property of parallelograms, that is, that opposite sides are equal. To find the length of 𝐶𝐷 then, it’s going to be the same length as 𝐴𝐵, which is the opposite side. So that’s 15 centimeters.

To find our next length of 𝐷𝐴, we can look at the opposite side, which is 𝐵𝐶, and that’s 13 centimeters. We can answer the question then with our two lengths, 𝐶𝐷 equals 15 centimeters and 𝐷𝐴 equals 13 centimeters.

In the next question, we’ll find an unknown angle in a parallelogram.

Given that 𝐴𝐵𝐶𝐷 is a parallelogram and the measure of angle 𝐶 equals 68 degrees, find the measure of angle 𝐴.

In this question, we’re told that we have a parallelogram. And we can in fact see that we do have both pairs of opposite sides parallel. We’re given this angle 𝐶 is 68 degrees. But in order to find the angle 𝐴, we’ll need to recall an important property of parallelograms. In a parallelogram, opposite angles are equal. Once we know this, it’s very simple to see that the angle 𝐴 and angle 𝐶 are opposite angles, meaning that they’re both 68 degrees. So we can give our answer that angle 𝐴 is 68 degrees.

Let’s have a look at another question.

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.

Because we’re told that 𝐴𝐵𝐶𝐷 is a parallelogram, this means we can say that the line 𝐷𝐶 is parallel to the line 𝐴𝐵 and the line 𝐴𝐷 is parallel to the line 𝐵𝐶. We’re given the two angle measurements of 𝐵𝐸𝐶 and 𝐸𝐶𝐵, and we’re asked to find this angle 𝐸𝐴𝐷. In order to work out this angle measurement, we’ll need to remember some of the properties of the angles in parallelograms.

Firstly, we can remember that opposite angles are equal, and we could also remember that the sum of two adjacent angles is 180 degrees. In this parallelogram, the angle that’s opposite to 𝐴 will be the angle 𝐶. However, we don’t know this total angle 𝐷𝐶𝐵. We only know that 𝐸𝐶𝐵 is 56 degrees. If we look at our second property, if we found the angle measurement of 𝐵, then that would help us to work out 𝐴. So how can we find the measurement of angle 𝐵?

As well as being part of a parallelogram, this angle at 𝐵 is also part of a triangle. We know that the angles in a triangle add up to 180 degrees. So this angle of 𝐶𝐵𝐸 is equal to 180 degrees subtract 79 degrees subtract 56 degrees. 180 subtract 79 gives us 101 degrees, and subtracting 56 from that gives us 45 degrees. Remember that this is not our answer as we still need to work out the angle 𝐸𝐴𝐷. Using the property that the sum of two adjacent angles is 180 degrees, then we calculate 180 subtract 45 degrees, giving us our answer of 135 degrees.

Let’s look at one final question.

In the figure, 𝐴𝐵𝐶𝐷 and 𝐶𝐵𝐻𝑂 are parallelograms. Find the measure of obtuse angle 𝐴𝐵𝐻.

Because we’re told that 𝐴𝐵𝐶𝐷 and 𝐶𝐵𝐻𝑂 are parallelograms, that means in 𝐴𝐵𝐶𝐷, the line 𝐴𝐷 and the line 𝐶𝐵 are parallel and 𝐶𝐵 and 𝑂𝐻 will also be parallel. 𝐶𝐷 and 𝐴𝐵 are parallel. And in the lower parallelogram, we know that 𝐶𝑂 and 𝐵𝐻 are parallel. Knowing the properties of a parallelogram will help us to find our unknown angle. We’re asked to find 𝐴𝐵𝐻, the obtuse angle, which means that it’s this angle marked in orange.

In order to find this obtuse angle, we’ll begin by seeing if we can find this reflex angle 𝐴𝐵𝐻. Let’s consider the parts of this angle in each parallelogram, and we’ll begin with trying to find this angle at 𝐴𝐵𝐶. We should remember that in a parallelogram, adjacent angles are supplementary. We have got two adjacent angles to this angle at 𝐵. We’ve got this angle at 𝐴 and the angle at 𝐶. Either of these would be supplementary. However, as we’re actually given the measure of this angle 𝐷𝐴𝐵, let’s use this angle to help us find our unknown. The angle 𝐴𝐵𝐶 can be found by calculating 180 degrees subtract 72 degrees, which gives us 108 degrees.

Now let’s see if we can find this angle 𝐶𝐵𝐻, and once again we’ll use the fact that adjacent angles add up to 180 degrees. So we subtract 51 degrees from 180 degrees. And we can remember that 180 subtract 50 gives us 130, and subtracting another one would give us 129 degrees. Now, when we look at the diagram, at point 𝐵, we’ve got an angle of 108 degrees, an angle of 129 degrees, and we want to find the remaining portion of this angle. We’ll need to remember that the angles about a point add up to 360 degrees. We’ll need to calculate 360 degrees subtract 129 degrees subtract 108 degrees. We can therefore give our answer that the obtuse angle 𝐴𝐵𝐻 is 123 degrees.

We’ll now summarize what we’ve learned in this video. We began with the definition of a parallelogram that it’s a quadrilateral with both pairs of opposite sides parallel. We can draw many different types of parallelogram, and they even include squares, rectangles, and rhombuses. We saw some important properties of parallelograms. By definition, opposite sides are parallel, but we also saw how opposite sides are equal in length.

We saw two angle properties of parallelograms. Firstly, opposite angles are equal, and secondly the sum of any two adjacent angles is 180 degrees. Finally, we saw that the diagonals of a parallelogram are bisectors. As we see in many geometry problems, we also need to recall key facts, for example, the angles in a triangle add up to 180 degrees or the angles about a point sum to 360 degrees. Learning the properties of parallelograms will help us with these specific types of problems, but also with many more.