Lesson Explainer: Properties and Special Cases of Parallelograms | Nagwa Lesson Explainer: Properties and Special Cases of Parallelograms | Nagwa

Lesson Explainer: Properties and Special Cases of Parallelograms Mathematics

In this explainer, we will learn how to use the properties of a parallelogram and identify the special cases of parallelograms along with their properties.

Definition: Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Together, we will discover the properties of parallelograms. First, we will recall properties of angles created by intersecting lines, including angles created by parallel lines with a transversal. They can be summarized as follows:

When two lines intersect, then

  • the pairs of opposite vertical angles are congruent,
  • and the linear adjacent pairs of angles are supplementary.

When parallel lines are cut by a transversal, then

  • the pairs of corresponding angles are congruent,
  • the pairs of alternate interior angles are congruent,
  • and the pairs of consecutive interior angles are supplementary.

With these properties in mind, we will construct two pairs of parallel lines intersecting at four points such that 𝑎𝑏 and 𝑑𝑐 as shown below.

These intersecting lines produce 16 angles—eight angles measuring 𝑥 and eight supplementary angles measuring 𝑦. By the definition of supplementary angles, we know that 𝑥+𝑦=180.

Notice how the two pairs of parallel lines form a parallelogram. Within the parallelogram, we have the following:

  • one pair of opposite angles (each measures 𝑥),
  • another pair of opposite angles (each measures 𝑦),
  • and all pairs of consecutive interior angles are supplementary such that 𝑥+𝑦=180.

Properties: Parallelogram

A parallelogram has the following five properties:

  1. Opposite sides are parallel.
  2. Opposite angles are equal in measure.
  3. The sum of any two consecutive interior angles is 180.
  4. Opposite sides are equal in length.
  5. The diagonals bisect each other.

We have already established the reasoning behind properties 2 and 3, but what about properties 4 and 5? Let’s investigate why opposite sides in a parallelogram are not just parallel but also congruent.

Recall the angle-side-angle (ASA) triangle congruence criterion. The ASA triangle congruence criterion says 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 will construct parallelogram 𝐴𝐵𝐶𝐷 with diagonal 𝐷𝐵. The diagonal divides our parallelogram into two triangles, 𝐴𝐵𝐷 and 𝐶𝐷𝐵. We would like to prove that 𝐶𝐷𝐵𝐴𝐵𝐷 and then use that information to show that corresponding sides are congruent.

Because opposite sides in a parallelogram are parallel, we know that 𝐷𝐶𝐴𝐵. We can see that diagonal 𝐷𝐵 is a transversal passing through parallel lines 𝐷𝐶 and 𝐴𝐵. Recall that, under these circumstances, alternate interior angles are congruent. Therefore, 𝐶𝐷𝐵𝐴𝐵𝐷.

We can use the same reasoning but with diagonal 𝐵𝐷 and the other pair of parallel sides to show that 𝐴𝐷𝐵𝐶𝐵𝐷.

𝐷𝐵 is a side shared by our triangles; It is opposite 𝐴 from 𝐴𝐵𝐷 and opposite 𝐶 from 𝐶𝐷𝐵. By the reflexive property of congruence, we can say that 𝐷𝐵𝐷𝐵.

Now that we have demonstrated how 𝐶𝐷𝐵𝐴𝐵𝐷,𝐷𝐵𝐷𝐵,𝐴𝐷𝐵𝐶𝐵𝐷,and we can use the ASA triangle congruence criterion to prove that 𝐶𝐷𝐵𝐴𝐵𝐷.

Then, we will use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, 𝐴𝐷 from 𝐴𝐵𝐷 is congruent to the corresponding side 𝐶𝐵 from 𝐶𝐷𝐵 and 𝐴𝐵 from 𝐴𝐵𝐷 is congruent to the corresponding side 𝐶𝐷 from 𝐶𝐷𝐵. By using properties of congruent triangles, we have shown that opposite sides of a parallelogram are congruent. By the definition of congruence, we can say that opposite sides of a parallelogram are equal in length. This is property 4 of parallelograms.

We can use similar properties of congruent triangles to prove property 5 of parallelograms.

Next, we will use property 4 in the example below.

Example 1: Using the Properties of Parallelograms to Find Missing Lengths

Find the lengths of 𝐶𝐷 and 𝐷𝐴.

Answer

The first thing we notice is that 𝐴𝐵𝐶𝐷 is a parallelogram. That is evident because the opposite sides are marked as parallel. Opposite sides in a parallelogram are not only parallel, but they are also equal in length. We will use this property to write the following equalities: 𝐶𝐷=𝐵𝐴𝐷𝐴=𝐶𝐵.and

The diagram shows 𝐵𝐴=15cm and 𝐶𝐵=13cm. By substitution, 𝐶𝐷=15𝐷𝐴=13.cmandcm

Now, let’s look at how we can calculate an unknown angle measure in a parallelogram.

Example 2: Finding the Measure of an Angle in a Parallelogram given the Other Three Angles’ Measures

𝐴𝐵𝐶𝐷 is a parallelogram in which 𝑚𝐵𝐸𝐶=79 and 𝑚𝐸𝐶𝐵=56. Determine 𝑚𝐸𝐴𝐷.

Answer

We are told 𝐴𝐵𝐶𝐷 is a parallelogram. That allows us to use any of the properties of parallelograms. Since we are considering angle relationships, the relevant properties are the following:

  • Opposite angles are equal in measure.
  • The sum of the measures of two consecutive angles is 180.

We should also be familiar with the fact that the sum of the interior angles in a triangle is always 180. Therefore, in the case of 𝐵𝐸𝐶, 𝑚𝐶𝐵𝐸+𝑚𝐵𝐸𝐶+𝑚𝐸𝐶𝐵=180.

By substitution, we have 𝑚𝐶𝐵𝐸+(79)+(56)=180.

By subtracting 79 and 56 from both sides of the equation, we have 𝑚𝐶𝐵𝐸=180(79)(56).

This leads us to 𝑚𝐶𝐵𝐸=45.

Now, we will take a closer look at parallelogram 𝐴𝐵𝐶𝐷. Any two interior angles that are not opposite are considered consecutive. We can see that 𝐶𝐵𝐸 and 𝐸𝐴𝐷 are consecutive angles. Since the sum of the measures of the two consecutive angles in a parallelogram is 180, we know that 𝑚𝐶𝐵𝐸+𝑚𝐸𝐴𝐷=180.

By substitution, we have (45)+𝑚𝐸𝐴𝐷=180.

This leads us to our final answer, which is 𝑚𝐸𝐴𝐷=135.

There are a couple special cases of parallelograms that we will discuss next—rectangles and rhombuses. These special cases have the same general properties as all parallelograms, but they also have a few unique properties that make them distinct. We will also discover that a square perfectly illustrates all the properties of rectangles and rhombuses combined.

If a parallelogram has all angles equal in measure, we call it a rectangle.

Definition: Rectangle

A rectangle is a parallelogram with four congruent angles.

We should be familiar with the fact that the sum of the interior angles in a quadrilateral is always 360. In a rectangle, all four interior angles are congruent. Therefore, each angle measure can be found by dividing 360 by four. We conclude that each angle measures 90. This means a rectangle has four right angles.

We know that, in any parallelogram, the diagonals bisect each other. Let’s take a closer look at the diagonals in a rectangle named 𝐴𝐵𝐶𝐷. If we construct diagonal 𝐴𝐶, we have created a pair of right triangles. If we construct diagonal 𝐷𝐵, we have created another pair of right triangles. Next, we would like to show that these four triangles are congruent. If the triangles are congruent, then their corresponding hypotenuses must be congruent.

Right triangles 𝐴𝐷𝐶 and 𝐶𝐵𝐴 share hypotenuse 𝐴𝐶. Right triangles 𝐷𝐴𝐵 and 𝐵𝐶𝐷 share hypotenuse 𝐷𝐵. We would like to demonstrate that 𝐴𝐶𝐷𝐵.

First, we will prove 𝐴𝐷𝐶 and 𝐵𝐶𝐷 are congruent by the side-angle-side (SAS) triangle congruence criterion. The SAS triangle congruence criterion says that, if two sides and an included angle in a triangle are congruent to the corresponding sides and angle of a second triangle, then the triangles are congruent.

We already have 𝐴𝐷𝐵𝐶 and 𝐷𝐶𝐶𝐷 because opposite sides of rectangles are congruent. Since the included angles 𝐷 and 𝐶 both equal 90, they are congruent as well. Therefore, by the SAS triangle congruence criterion, 𝐴𝐷𝐶𝐵𝐶𝐷.

We may then use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, the hypotenuse of 𝐴𝐷𝐶 and 𝐶𝐵𝐴 is congruent to the corresponding hypotenuse of 𝐷𝐴𝐵 and 𝐵𝐶𝐷. This demonstrates that, in a rectangle, the diagonals not only bisect each other but are also congruent.

Properties: Rectangle

A rectangle inherits all of the properties of a parallelogram and has the following additional properties:

  1. All angles are equal in measure (each 90):
  2. The diagonals are equal in length:

If a parallelogram is equilateral, we call it a rhombus.

Definition: Rhombus

A rhombus is a parallelogram with four congruent sides.

Let’s take a closer look at the diagonals in a rhombus named 𝐴𝐵𝐶𝐷. We begin by constructing diagonal 𝐴𝐶. We hope to prove that 𝐴𝐵𝐶 and 𝐴𝐷𝐶 are congruent.

All four sides of a rhombus are congruent; therefore, sides 𝐴𝐵 and 𝐵𝐶 from 𝐴𝐵𝐶 are congruent to their corresponding sides 𝐴𝐷 and 𝐷𝐶 from 𝐴𝐷𝐶. So, we concluded that 𝐴𝐵𝐴𝐷 and 𝐵𝐶𝐷𝐶. Because a rhombus inherits all the properties of a parallelogram, opposite angles are congruent. This means corresponding angles 𝐵 and 𝐷 are congruent as well. By the SAS triangle congruence criterion, 𝐴𝐵𝐶𝐴𝐷𝐶.

We may use the fact that these triangles are congruent to prove that corresponding parts are congruent. Therefore, 𝐵𝐴𝐶𝐷𝐴𝐶𝐵𝐶𝐴𝐷𝐶𝐴.

By the definition of an angle bisector, we can say that 𝐵𝐴𝐷 is bisected by diagonal 𝐴𝐶 because 𝐵𝐴𝐶𝐷𝐴𝐶. By the definition of an angle bisector, we can also say that 𝐵𝐶𝐷 is bisected by diagonal 𝐴𝐶 because 𝐵𝐶𝐴𝐷𝐶𝐴.

In our rhombus named 𝐴𝐵𝐶𝐷, opposite angles 𝐵𝐴𝐷 and 𝐵𝐶𝐷 are bisected by diagonal 𝐴𝐶. By the same reasoning, opposite angles 𝐴𝐵𝐶 and 𝐴𝐷𝐶 are bisected by diagonal 𝐵𝐷.

Properties: Rhombus

A rhombus inherits all of the properties of a parallelogram and has the following additional properties:

  1. All sides are equal in length.
  2. The diagonals bisect opposite pairs of angles:
    𝐴𝐶𝐵𝐴𝐷𝐵𝐶𝐷,𝐵𝐷𝐴𝐵𝐶𝐴𝐷𝐶.bisectsandbisectsand
  3. The diagonals are perpendicular.

We have already established the reasoning behind property 2, but what about property 3? Let’s investigate why diagonals in a rhombus are perpendicular.

We will construct a rhombus named 𝐴𝐵𝐶𝐷 and call the point where the diagonals bisect each other 𝐸. We want to find out if 𝑚𝐵𝐸𝐶=90. For now, let’s say 𝑚𝐵𝐸𝐶=𝑧.

We know that, in a rhombus, opposite angles are congruent and bisected by the diagonals. In our rhombus, we have 𝑚𝐴𝐵𝐸=𝑚𝐶𝐵𝐸=𝑚𝐸𝐷𝐴=𝑚𝐸𝐷𝐶=𝑥 and 𝑚𝐵𝐴𝐸=𝑚𝐷𝐴𝐸=𝑚𝐵𝐶𝐸=𝑚𝐷𝐶𝐸=𝑦.

Therefore, the four angles’ measures of our rhombus are 𝑚𝐴=𝑚𝐶=(2𝑦)𝑚𝐵=𝑚𝐷=(2𝑥).and

Since consecutive interior angles are congruent in any parallelogram, 𝑚𝐵+𝑚𝐶=180.

By substitution, (2𝑥)+(2𝑦)=180.

Dividing both sides by 2 gives us 𝑥+𝑦=90.

Now, we will focus our attention on 𝐵𝐸𝐶.

We know that that the sum of the interior angles in a triangle is always 180. Therefore, in the case of 𝐵𝐸𝐶, 𝑥+𝑦+𝑧=180.

By substitution, (90)+𝑧=180.

After subtracting 90 from both sides, we have 𝑧=90.

So, the angle created by the diagonals of a rhombus is a right angle. Therefore, the diagonals of a rhombus are perpendicular. This result confirms property 4 of rhombuses.

If a parallelogram has all angles equal in measure and is equilateral, we call it a square. A square has all the properties of both a rectangle and a rhombus.

Definition: Square

A square is a parallelogram with four congruent angles and four congruent sides.

A square is a special case of both a rectangle and a rhombus. Therefore, it adopts all the properties we have already seen of a parallelogram, a rectangle, and a rhombus.

Example 3: Completing a Sentence About Special Cases of Parallelograms

Fill in the blank: A is a parallelogram with congruent sides and congruent diagonals.

Answer

Every parallelogram has opposite congruent sides. However, all four sides are congruent only in the special case of a rhombus. Every parallelogram has diagonals that bisect each other. However, the diagonals of a parallelogram are congruent only in the special case of a rectangle. In summary, rhombuses have congruent sides, but rectangles generally do not. Rectangles have congruent diagonals, but rhombuses generally do not. Therefore, neither the word rhombus nor rectangle alone is the right answer to fill in the blank.

Recall that squares have all the properties of both a rhombus and a rectangle. Thus, a square fits the description of a parallelogram with congruent sides and congruent diagonals.

A square is a parallelogram with congruent sides and congruent diagonals.

Example 4: Completing a Sentence About Special Cases of Parallelograms

Fill in the blank: A parallelogram whose are equal is called a rectangle.

Answer

Let’s review the properties of a rectangle:

  1. All angles are equal in measure (each 90).
  2. The diagonals are equal in length.

It is worth noting that all parallelograms have opposite sides equal in length, but having congruent diagonals is an additional property unique to rectangles. Therefore, the word diagonals is the best word to complete the sentence. A parallelogram whose diagonals are equal in length is called a rectangle.

Example 5: Completing a Sentence About Special Cases of Parallelograms

Fill in the blank: Each of the two diagonals of the square makes an angle with a measure of with the adjacent side.

Answer

Since squares inherit all the properties of rectangles and rhombuses, there are several things we can conclude about their diagonals:

  • The diagonals bisect each other.
  • The diagonals are congruent.
  • The diagonals bisect opposite angles.

Given that rectangles have four right angles, squares do as well. Because each angle is bisected, we will cut each 90 measure in half. This means that each diagonal makes 45 with the adjacent side.

Each of the two diagonals of the square makes an angle with a measure of 45 with the adjacent side.

Let’s finish by recapping some important points from the explainer.

Key Points

  • A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
  • Parallelograms have the following properties:
    1. Opposite sides are parallel.
    2. Opposite angles are equal in measure.
    3. The sum of any two consecutive interior angles is 180.
    4. Opposite sides are equal in length.
    5. The diagonals bisect each other.
  • A rectangle is a parallelogram with four congruent angles.
  • Rectangles inherit all of the properties of a parallelogram and have the following additional properties:
    1. All angles are equal in measure (each 90).
    2. The diagonals are equal in length.
  • A rhombus is a parallelogram with four congruent sides.
  • Rhombuses inherit all of the properties of a parallelogram and have the following additional properties:
    1. All sides are equal in length.
    2. The diagonals bisect opposite pairs of angles.
    3. The diagonals are perpendicular.
  • A square is a 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.

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