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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 π‘Žβ«½π‘ and 𝑑⫽𝑐 as shown below.

These intersecting lines produce 16 anglesβ€”eight angles measuring π‘₯∘ and eight supplementary angles measuring π‘¦βˆ˜. By the definition of supplementary angles, we know that π‘₯+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 π‘₯∘),
  • another pair of opposite angles (each measures π‘¦βˆ˜),
  • and all pairs of consecutive interior angles are supplementary such that π‘₯+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 𝐴𝐡𝐢𝐷 with diagonal 𝐷𝐡. The diagonal divides our parallelogram into two triangles, △𝐴𝐡𝐷 and △𝐢𝐷𝐡. We would like to prove that △𝐢𝐷𝐡≅△𝐴𝐡𝐷 and then use that information to show that corresponding sides are congruent.

Because opposite sides in a parallelogram are parallel, we know that 𝐷𝐢⫽𝐴𝐡. We can see that diagonal 𝐷𝐡 is a transversal passing through parallel lines ⃖⃗𝐷𝐢 and ⃖⃗𝐴𝐡. Recall that, under these circumstances, alternate interior angles are congruent. Therefore, βˆ πΆπ·π΅β‰…βˆ π΄π΅π·.

We can use the same reasoning but with diagonal 𝐡𝐷 and the other pair of parallel sides to show that βˆ π΄π·π΅β‰…βˆ πΆπ΅π·.

𝐷𝐡 is a side shared by our triangles; It is opposite ∠𝐴 from △𝐴𝐡𝐷 and opposite ∠𝐢 from △𝐢𝐷𝐡. By the reflexive property of congruence, we can say that 𝐷𝐡≅𝐷𝐡.

Now that we have demonstrated how △𝐢𝐷𝐡≅△𝐴𝐡𝐷,𝐷𝐡≅𝐷𝐡,△𝐴𝐷𝐡≅△𝐢𝐡𝐷,and we can use the ASA triangle congruence criterion to prove that △𝐢𝐷𝐡≅△𝐴𝐡𝐷.

Then, we will use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, 𝐴𝐷 from △𝐴𝐡𝐷 is congruent to the corresponding side 𝐢𝐡 from △𝐢𝐷𝐡 and 𝐴𝐡 from △𝐴𝐡𝐷 is congruent to the corresponding side 𝐢𝐷 from △𝐢𝐷𝐡. 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 𝐢𝐷 and 𝐷𝐴.

Answer

The first thing we notice is that 𝐴𝐡𝐢𝐷 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: 𝐢𝐷=𝐡𝐴𝐷𝐴=𝐢𝐡.and

The diagram shows 𝐡𝐴=15cm and 𝐢𝐡=13cm. By substitution, 𝐢𝐷=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

𝐴𝐡𝐢𝐷 is a parallelogram in which π‘šβˆ π΅πΈπΆ=79∘ and π‘šβˆ πΈπΆπ΅=56∘. Determine π‘šβˆ πΈπ΄π·.

Answer

We are told 𝐴𝐡𝐢𝐷 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 △𝐡𝐸𝐢, π‘šβˆ πΆπ΅πΈ+π‘šβˆ π΅πΈπΆ+π‘šβˆ πΈπΆπ΅=180.∘

By substitution, we have π‘šβˆ πΆπ΅πΈ+(79)+(56)=180.∘∘∘

By subtracting 79∘ and 56∘ from both sides of the equation, we have π‘šβˆ πΆπ΅πΈ=180βˆ’(79)βˆ’(56).∘∘∘

This leads us to π‘šβˆ πΆπ΅πΈ=45.∘

Now, we will take a closer look at parallelogram 𝐴𝐡𝐢𝐷. Any two interior angles that are not opposite are considered consecutive. We can see that ∠𝐢𝐡𝐸 and ∠𝐸𝐴𝐷 are consecutive angles. Since the sum of the measures of the two consecutive angles in a parallelogram is 180∘, we know that π‘šβˆ πΆπ΅πΈ+π‘šβˆ πΈπ΄π·=180.∘

By substitution, we have (45)+π‘šβˆ πΈπ΄π·=180.∘∘

This leads us to our final answer, which is π‘šβˆ πΈπ΄π·=135.∘

There are a couple special cases of parallelograms that we will discuss nextβ€”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 𝐴𝐡𝐢𝐷. If we construct diagonal 𝐴𝐢, we have created a pair of right triangles. If we construct diagonal 𝐷𝐡, 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 △𝐴𝐷𝐢 and △𝐢𝐡𝐴 share hypotenuse 𝐴𝐢. Right triangles △𝐷𝐴𝐡 and △𝐡𝐢𝐷 share hypotenuse 𝐷𝐡. We would like to demonstrate that 𝐴𝐢≅𝐷𝐡.

First, we will prove △𝐴𝐷𝐢 and △𝐡𝐢𝐷 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 𝐴𝐷≅𝐡𝐢 and 𝐷𝐢≅𝐢𝐷 because opposite sides of rectangles are congruent. Since the included angles ∠𝐷 and ∠𝐢 both equal 90∘, they are congruent as well. Therefore, by the SAS triangle congruence criterion, △𝐴𝐷𝐢≅△𝐡𝐢𝐷.

We may then use the fact that these triangles are congruent to prove that their corresponding parts are congruent. Therefore, the hypotenuse of △𝐴𝐷𝐢 and △𝐢𝐡𝐴 is congruent to the corresponding hypotenuse of △𝐷𝐴𝐡 and △𝐡𝐢𝐷. 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 𝐴𝐡𝐢𝐷. We begin by constructing diagonal 𝐴𝐢. We hope to prove that △𝐴𝐡𝐢 and △𝐴𝐷𝐢 are congruent.

All four sides of a rhombus are congruent; therefore, sides 𝐴𝐡 and 𝐡𝐢 from △𝐴𝐡𝐢 are congruent to their corresponding sides 𝐴𝐷 and 𝐷𝐢 from △𝐴𝐷𝐢. So, we concluded that 𝐴𝐡≅𝐴𝐷 and 𝐡𝐢≅𝐷𝐢. Because a rhombus inherits all the properties of a parallelogram, opposite angles are congruent. This means corresponding angles ∠𝐡 and ∠𝐷 are congruent as well. By the SAS triangle congruence criterion, △𝐴𝐡𝐢≅△𝐴𝐷𝐢.

We may use the fact that these triangles are congruent to prove that corresponding parts are congruent. Therefore, βˆ π΅π΄πΆβ‰…βˆ π·π΄πΆβ‰…βˆ π΅πΆπ΄β‰…βˆ π·πΆπ΄.

By the definition of an angle bisector, we can say that ∠𝐡𝐴𝐷 is bisected by diagonal 𝐴𝐢 because βˆ π΅π΄πΆβ‰…βˆ π·π΄πΆ. By the definition of an angle bisector, we can also say that ∠𝐡𝐢𝐷 is bisected by diagonal 𝐴𝐢 because βˆ π΅πΆπ΄β‰…βˆ π·πΆπ΄.

In our rhombus named 𝐴𝐡𝐢𝐷, opposite angles ∠𝐡𝐴𝐷 and ∠𝐡𝐢𝐷 are bisected by diagonal 𝐴𝐢. By the same reasoning, opposite angles ∠𝐴𝐡𝐢 and ∠𝐴𝐷𝐢 are bisected by diagonal 𝐡𝐷.

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:
    𝐴𝐢∠𝐡𝐴𝐷∠𝐡𝐢𝐷,𝐡𝐷∠𝐴𝐡𝐢∠𝐴𝐷𝐢.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 𝐴𝐡𝐢𝐷 and call the point where the diagonals bisect each other 𝐸. We want to find out if π‘šβˆ π΅πΈπΆ=90∘. For now, let’s say π‘šβˆ π΅πΈπΆ=π‘§βˆ˜.

We know that, in a rhombus, opposite angles are congruent and bisected by the diagonals. In our rhombus, we have π‘šβˆ π΄π΅πΈ=π‘šβˆ πΆπ΅πΈ=π‘šβˆ πΈπ·π΄=π‘šβˆ πΈπ·πΆ=π‘₯∘ and π‘šβˆ π΅π΄πΈ=π‘šβˆ π·π΄πΈ=π‘šβˆ π΅πΆπΈ=π‘šβˆ π·πΆπΈ=𝑦.∘

Therefore, the four angles’ measures of our rhombus are π‘šβˆ π΄=π‘šβˆ πΆ=(2𝑦)π‘šβˆ π΅=π‘šβˆ π·=(2π‘₯).∘∘and

Since consecutive interior angles are congruent in any parallelogram, π‘šβˆ π΅+π‘šβˆ πΆ=180.∘

By substitution, (2π‘₯)+(2𝑦)=180.∘∘∘

Dividing both sides by 2 gives us π‘₯+𝑦=90.∘∘∘

Now, we will focus our attention on △𝐡𝐸𝐢.

We know that that the sum of the interior angles in a triangle is always 180∘. Therefore, in the case of △𝐡𝐸𝐢, π‘₯+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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