# 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 and as shown below.

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

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 ),
• another pair of opposite angles (each measures ),
• and all pairs of consecutive interior angles are supplementary such that .

### 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 .
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 with diagonal . The diagonal divides our parallelogram into two triangles, and . We would like to prove that and then use that information to show that corresponding sides are congruent.

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

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

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

Now that we have demonstrated how we can use the ASA triangle congruence criterion to prove that .

Then, we will use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, from is congruent to the corresponding side from and from is congruent to the corresponding side from . 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 and .

The first thing we notice is that 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:

The diagram shows and . By substitution,

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

is a parallelogram in which and . Determine .

We are told 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 .

We should also be familiar with the fact that the sum of the interior angles in a triangle is always . Therefore, in the case of ,

By substitution, we have

By subtracting and from both sides of the equation, we have

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

By substitution, we have

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 . In a rectangle, all four interior angles are congruent. Therefore, each angle measure can be found by dividing by four. We conclude that each angle measures . 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 . If we construct diagonal , we have created a pair of right triangles. If we construct diagonal , 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 and share hypotenuse . Right triangles and share hypotenuse . We would like to demonstrate that .

First, we will prove and 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 and because opposite sides of rectangles are congruent. Since the included angles and both equal , they are congruent as well. Therefore, by the SAS triangle congruence criterion, .

We may then use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, the hypotenuse of and is congruent to the corresponding hypotenuse of and . 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 ):
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 . We begin by constructing diagonal . We hope to prove that and are congruent.

All four sides of a rhombus are congruent; therefore, sides and from are congruent to their corresponding sides and from . So, we concluded that and . Because a rhombus inherits all the properties of a parallelogram, opposite angles are congruent. This means corresponding angles and are congruent as well. By the SAS triangle congruence criterion, .

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

By the definition of an angle bisector, we can say that is bisected by diagonal because . By the definition of an angle bisector, we can also say that is bisected by diagonal because .

In our rhombus named , opposite angles and are bisected by diagonal . By the same reasoning, opposite angles and are bisected by diagonal .

### 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:
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 and call the point where the diagonals bisect each other . We want to find out if . For now, let’s say .

We know that, in a rhombus, opposite angles are congruent and bisected by the diagonals. In our rhombus, we have and

Therefore, the four angles’ measures of our rhombus are

Since consecutive interior angles are congruent in any parallelogram,

By substitution,

Dividing both sides by 2 gives us

Now, we will focus our attention on .

We know that that the sum of the interior angles in a triangle is always . Therefore, in the case of ,

By substitution,

After subtracting from both sides, we have

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.

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.

Let’s review the properties of a rectangle:

1. All angles are equal in measure (each ).
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.

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 measure in half. This means that each diagonal makes with the adjacent side.

Each of the two diagonals of the square makes an angle with a measure of 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 .
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 ).
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.