properties of parallelograms assignment

Free Printable properties of parallelograms Worksheets for 9th Grade

Math properties of parallelograms come to life with these free printable worksheets for Grade 9 students. Discover the world of geometry and enhance your students' understanding of parallelogram properties through our comprehensive collection provided by Quizizz.

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Explore printable properties of parallelograms worksheets for 9th Grade

Properties of parallelograms worksheets for Grade 9 are essential resources for teachers looking to enhance their students' understanding of geometry concepts in Math. These worksheets provide a variety of exercises and problems that challenge students to apply their knowledge of parallelograms, including their angles, sides, and diagonals. By incorporating these worksheets into their lesson plans, teachers can ensure that their Grade 9 students develop a strong foundation in geometry, which will be crucial for their success in higher-level Math courses. Additionally, these properties of parallelograms worksheets for Grade 9 can be easily adapted to suit different learning styles and classroom environments, making them a versatile and valuable teaching tool.

Quizizz is an excellent platform for teachers to access a wide range of educational resources, including properties of parallelograms worksheets for Grade 9, Math, and geometry lessons. With Quizizz, teachers can create interactive quizzes and games that engage students and reinforce their understanding of key concepts. This platform also offers a variety of other resources, such as flashcards, presentations, and lesson plans, which can be easily integrated into the classroom to support learning and make teaching more effective. Furthermore, Quizizz provides teachers with valuable insights into their students' progress, allowing them to identify areas where additional support may be needed. By utilizing Quizizz in conjunction with properties of parallelograms worksheets for Grade 9, teachers can create a comprehensive and engaging learning experience for their students.

Properties of Parallelogram

The properties of a parallelogram help us to identify a parallelogram from a given set of figures easily and quickly. Before we learn about the properties, let us first know about parallelograms. It is a four-sided closed figure with equal and parallel opposite sides and equal opposite angles. Let us learn more about the properties of parallelograms in detail in this article.

What are the Properties of Parallelogram?

A parallelogram is a type of quadrilateral in which the opposite sides are parallel and equal. There are four angles in a parallelogram at the vertices. Understanding the properties of parallelograms helps to easily relate its angles and sides. Also, the properties are helpful for calculations in problems relating to the sides and angles of a parallelogram.

Parallelogram Angle Properties

The important properties of parallelograms related to angles are as follows:

• The opposite angles of a parallelogram are equal, i.e., ∠A = ∠C, and ∠B = ∠D.
• All the angles of a parallelogram add up to 360°, i.e., ∠A + ∠B + ∠C + ∠D = 360°.
• The consecutive angles of a parallelogram are supplementary , i.e., ∠A + ∠B = 180° ∠B + ∠C = 180° ∠C + ∠D = 180° ∠D + ∠A = 180°

Parallelogram Side Properties

The opposite sides of a parallelogram are equal and parallel to each other.

Observe the following figure to understand the properties of a parallelogram.

All the above properties hold true for all types of parallelograms, but now let us also learn about the individual properties of some special parallelograms . The three different parallelograms are square, rectangle, and rhombus which are different from each other because of their properties yet they all come under the category of parallelograms.

Properties of a Square

• All four sides of a square are equal.
• All four angles are equal and of 90 degrees each.
• The diagonals of a square bisect its angles.
• Both the diagonals of a square have the same length.
• The opposite sides of a square are equal and parallel to each other.

Properties of a Rectangle

• The opposite sides of a rectangle are equal and parallel.
• All four angles of a rectangle are equal and measure 90° each.
• Both the diagonals of a rectangle are of the same length.

Properties of Rhombus

• All sides of a rhombus are equal in length.
• The diagonals of a rhombus bisect each other at 90°.
• The sum of any two adjacent interior angles is 180°.
• The opposite sides of a rhombus are equal and parallel to each other.

Now, let us expand our knowledge by learning about the properties of diagonals of parallelograms in the following section..

Properties of Parallelogram Diagonals

First, we will recall the meaning of a diagonal. Diagonals are line segments that join the non-adjacent vertices of any polygon. In parallelogram ABCD (refer to the figure given above), AC and BD are the diagonals. Let us assume that O is the intersecting point of the diagonals AC and BD. The properties of diagonals of a parallelogram are as follows:

• The diagonals of a parallelogram bisect each other, i.e., OB = OD and OA = OC.
• Each diagonal divides the parallelogram into two congruent triangles, i.e., ΔCDA ≅ ΔABC and ΔBAD ≅ ΔDCB.
• Parallelogram Law: The sum of the squares of the sides is equal to the sum of the squares of the diagonals, i.e., AB 2 + BC 2 + CD 2 + DA 2 = AC 2 + BD 2

Theorems on Parallelogram Properties

The theorems on properties of a parallelogram are helpful to define the rules for working across the problems on parallelograms. The properties relating to the sides and angles of a parallelogram can all be easily understood and applied to solve various problems. Further, these theorems are also supportive of understanding the concepts in other quadrilaterals. Four important theorems related to the properties of a parallelogram are given below:

• The opposite sides of a parallelogram are equal.
• The opposite angles of a parallelogram are equal.
• The diagonals of a parallelogram bisect each other.
• If one pair of opposite sides is equal and parallel in a quadrilateral then it is a parallelogram.

Theorem 1: In a Parallelogram the Opposite Sides are Equal.

Given: ABCD is a parallelogram.

To Prove: The opposite sides are equal, AB = CD and BC = AD.

In parallelogram ABCD, compare triangles ABC and CDA. In these triangles:

• AC = CA (common side)
• ∠BAC = ∠DCA ( alternate interior angles )
• ∠BCA = ∠DAC (alternate interior angles)

Hence by the ASA criterion, both the triangles are congruent and the corresponding sides are equal. Therefore we have AB = CD, and BC = AD.

Converse of Theorem 1: If the opposite sides in a quadrilateral are equal, then it is a parallelogram.

Given: The opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD.

To Prove: ABCD is a parallelogram.

n the quadrilateral ABCD we are given that AB = CD and AD = BC. Now compare the two triangles ABC, and CDA. Here we have

• AC = AC (Common sides)
• AB = CD (since alternate interior angles are equal)
• AD = BC (given).

Thus by the SSS criterion , both the triangles are congruent, and the corresponding angles are equal. Hence we can conclude that ∠BAC = ∠DCA, and ∠BCA = ∠DAC.

Therefore AB || CD, BC || AD, and ABCD is a parallelogram.

Theorem 2: In a Parallelogram, the Opposite Angles are Equal.

Given: ABCD is a parallelogram, and ∠A, ∠B, ∠C, ∠D are the four angles.

To Prove: ∠A = ∠C and ∠B = ∠D

Let us assume that ABCD is a parallelogram. Now compare triangles ABC and CDA. Here we have,

• ∠1 = ∠4 (alternate interior angles)
• ∠2 = ∠3 (alternate interior angles)

Thus, by ASA, the two triangles are congruent, which means that ∠B = ∠D. Similarly, we can show that ∠A = ∠C. This proves that opposite angles in any parallelogram are equal.

Converse of Theorem 2: If the opposite angles in a quadrilateral are equal, then it is a parallelogram.

Given: ∠A = ∠C and ∠B = ∠D in the quadrilateral ABCD.

Assume that ∠A = ∠C and ∠B = ∠D in the parallelogram ABCD given above. We have to prove that ABCD is a parallelogram. We have:

∠A + ∠B + ∠C + ∠D = 360º

2(∠A + ∠B) =360º

∠A + ∠B = 180º.

This must mean that AD || BC. Similarly, we can show that AB || CD. Hence, AD || BC, and AB || CD. Therefore ABCD is a parallelogram.

Theorem 3: Diagonals of a Parallelogram Bisect Each Other.

Given: PQTR is a parallelogram. PT and QR are the diagonals of the parallelogram.

To Prove: The diagonals PT and RQ bisect each other, i.e., PE = ET and ER = EQ.

First, let us assume that PQTR is a parallelogram. Compare triangles TER and triangle PEQ. We have,

• PQ = RT (opposite sides of the parallelogram PQTR)
• ∠QRT = ∠PQR (alternate interior angles)
• ∠PTR = ∠QPT (alternate interior angles).

By the ASA criterion, the two triangles are congruent, which means by CPCTC, PE = ET, and RE = EQ. Thus, the two diagonals PT and RQ bisect each other, and PE = ET and ER = EQ.

Converse of Theorem 3: If the diagonals in a quadrilateral bisect each other, then it is a parallelogram. In the quadrilateral PQTR, if PE=ET and ER=EQ, then it is a parallelogram.

Given: The diagonals PT and QR bisect each other.

To Prove: PQRT is a parallelogram.

Proof: Suppose that the diagonals PT and QR bisect each other. Compare triangle RET, and triangle PEQ once again. We have:

• ET = PE (Diagonals bisect each other)
• ∠RET =∠PEQ (vertically opposite angles).

Hence by the SAS criterion , the two triangles are congruent. This means that ∠QRT = ∠PQR, and ∠PRT = ∠QPT. Hence, PQ || RT, and RT || QT. Thus PQRT is a parallelogram.

Theorem 4: If One Pair of Opposite Sides is Equal and Parallel in a Quadrilateral then it is a Parallelogram.

Given: It is given that AB = CD and AB || CD.

Let us compare the triangle AEB and triangle CED. We have,

• AB = CD (given)
• ∠1 = ∠3 (alternate interior angles)
• ∠2 = ∠4 (alternate interior angles)

Thus, by ASA criterion , the two triangles are congruent. Hence we can conclude that by CPCTC , AE = EC, and BE = ED. Therefore, the diagonals AC and BD bisect each other, and this further means that ABCD is a parallelogram.

Important Notes:

A quadrilateral is a parallelogram when:

• The opposite sides of a quadrilateral are equal and parallel.
• The opposite angles of a quadrilateral are equal.
• The diagonals bisect each other.
• One pair of opposite sides is equal and parallel.
• Parallelogram Formula
• Area of Parallelogram
• Properties of Rectangle
• Difference Between Rectangle and Parallelogram

Properties of Parallelogram Examples

Example 1: If one angle of a parallelogram is 90°, show that all its angles will be equal to 90°.

Consider the parallelogram ABCD in the following figure, in which ∠A is a right angle.

We know that in any parallelogram, the opposite angles are equal. This implies ∠C = 90°.

Also, in any parallelogram, the adjacent angles are supplementary. This implies ∠B = 180° - ∠A = 180° - 90° = 90°.

Similarly, ∠D = 180° - ∠C = 180° - 90° = 90°.

Hence, ∠A =∠B =∠C =∠D = 90°.

Clearly, all the angles in this parallelogram are equal to 90°.

Answer: By using the properties of parallelograms, we have proved that when one angle of a parallelogram is 90°, then it is a rectangle.

Example 2: In a quadrilateral ABCD, the diagonals AC and BD bisect each other at right angles. Show that the quadrilateral is a rhombus.

Consider the following figure:

First of all, we note that since the diagonals bisect each other, then by using the properties of parallelograms, we can conclude that ABCD is a parallelogram. So the opposite sides are equal.

• AB = CD ... (1)
• AD = BC ... (2)

Now, let us compare ΔAEB and ΔAED:

• BE = ED (diagonals bisect each other)
• ∠AEB = ∠AED = 90° (given)

Thus, by the SAS criterion, the two triangles are congruent, which means that AB = AD.

Also, by (1) and (2), we can conclude that AB=BC=CD=AD. Clearly, ABCD is a rhombus.

Answer: We have proved that the quadrilateral in which the diagonals bisect each other at right angles is a rhombus.

Example 3: State true or false:

a.) As per the parallelogram rules, all the angles of a parallelogram add up to 180°,

b.) In a parallelogram opposite sides are equal.

c.) The opposite angles in a parallelogram are equal.

Solution: Observing all the properties of parallelograms, we can write 'true' and 'false' for the following statements.

a.) False, as per the parallelogram rules, all the angles of a parallelogram add up to 360°,

b.) True, in a parallelogram opposite sides are equal.

c.) True, the opposite angles in a parallelogram are equal.

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Practice Questions on Properties of Parallelogram

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FAQs on Properties of Parallelogram

What are the 7 properties of parallelogram.

The seven properties of a parallelogram are as follows:

• The consecutive angles of a parallelogram are supplementary.
• If one angle of a parallelogram is a right angle , then all the angles are right angles.
• Each diagonal of a parallelogram bisects it into two congruent triangles .
• If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

What are the Properties of Diagonals of a Parallelogram?

There are two important properties of the diagonals of a parallelogram. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. And the diagonals of a parallelogram bisect each other.

Are the Diagonals of a Parallelogram Equal?

The diagonals of a parallelogram are NOT equal. The opposite sides and opposite angles of a parallelogram are equal. The diagonals of a square and a rectangle are equal which are special types of parallelograms.

Write the Properties of a Parallelogram.

A parallelogram is a quadrilateral with opposite sides equal and parallel. The opposite angles of a parallelogram are also equal. In short, a parallelogram can be considered a twisted rectangle. It is more of a rectangle, but the angles at the vertices need not be right angles. The four important properties of a parallelogram are as follows:

• The adjacent angles of a parallelogram are supplementary.

Can a Rectangle be called a Parallelogram?

The opposite sides of a rectangle are equal and parallel. So a rectangle satisfies all the properties of a parallelogram and hence a rectangle can be called a parallelogram.

What are the Unique Properties of Parallelograms?

The unique characteristics of parallelograms that make it different from other quadrilaterals are given below:

• The opposite sides of every parallelogram are equal and parallel.
• The opposite angles are always equal.
• The sum of consecutive interior angles is always equal to 180°.

What are the Different Properties of Each Special Parallelogram?

There are three special types of parallelograms - square, rectangle, and rhombus . A square is a four-sided polygon in which all sides and angles are equal. A rectangle has opposite sides equal and parallel. All the angles of squares and rectangles are equal and measure 90° each. A rhombus is a parallelogram with four equal sides, but its angles need not be right angles.

How to Differentiate a Parallelogram and a Quadrilateral by Properties of Parallelogram?

Every parallelogram can be called a quadrilateral, but every quadrilateral cannot be called a parallelogram. A trapezium and a kite can be called quadrilaterals, but they do not fully satisfy the properties of a parallelogram and hence cannot be called a parallelogram.

What do the Opposite Angles in a Parallelogram add up to?

The opposite angles in a parallelogram are always equal. However, it is to be noted that the consecutive interior angles of a parallelogram always add up to 180°.

What are the Rules of a Parallelogram?

The rules of a parallelogram are the characteristics of a parallelogram that make it distinct from the other polygons . In other words, they are all the properties of a parallelogram that help us identify it. The basic characteristics of a parallelogram are given below:

• Each diagonal of a parallelogram bisects it into two congruent triangles.

Properties of Parallelograms

Related Topics: Lesson Plans and Worksheets for Geometry Lesson Plans and Worksheets for all Grades More Lessons for Geometry Common Core For Geometry

New York State Common Core Math Geometry, Module 1, Lesson 28

Worksheets for Geometry

Student Outcomes

• Students complete proofs that incorporate properties of parallelograms.

Opening Exercise

a. If the triangles are congruent, state the congruence. b. Which triangle congruence criterion guarantees part 1? c. 𝑇𝐺 corresponds with

How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms?

To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we have assumed other details about parallelograms to be true, too. We assume that:

• Opposite sides are congruent.
• Opposite angles are congruent.
• Diagonals bisect each other.

Let us examine why each of these properties is true.

If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why opposite sides and angles of a parallelogram are congruent.

Given: _____ Prove: _____

Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as parallel. Draw diagonal 𝐵𝐷.

If a quadrilateral is a parallelogram, then the diagonals bisect each other. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a parallelogram bisect each other. Remember, now that we have proved opposite sides and angles of a parallelogram to be congruent, we are free to use these facts as needed (i.e., 𝐴𝐷 = 𝐶𝐵, 𝐴𝐵 = 𝐶𝐷, ∠𝐴 ≅ ∠𝐶, ∠𝐵 ≅ ∠𝐷).

Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷. Mark opposite sides as parallel. Draw diagonals 𝐴𝐶 and 𝐵𝐷.

Now we have established why the properties of parallelograms that we have assumed to be true are in fact true. By extension, these facts hold for any type of parallelogram, including rectangles, squares, and rhombuses. Let us look at one last fact concerning rectangles. We established that the diagonals of general parallelograms bisect each other. Let us now demonstrate that a rectangle has congruent diagonals.

If the parallelogram is a rectangle, then the diagonals are equal in length. Complete the diagram, and develop an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a rectangle are congruent. As in the last proof, remember to use any already proven facts as needed.

Construction: Label the rectangle 𝐺𝐻𝐼𝐽. Mark opposite sides as parallel, and add small squares at the vertices to indicate 90° angles. Draw diagonals 𝐺𝐼 and 𝐻𝐽

Converse Properties: Now we examine the converse of each of the properties we proved. Begin with the property, and prove that the quadrilateral is in fact a parallelogram.

If both pairs of opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case.

Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷. Mark opposite angles as congruent. Draw diagonal 𝐵𝐷. Label the measures of ∠𝐴 and ∠𝐶 as 𝑥°. Label the measures of the four angles created by 𝐵𝐷 as 𝑟°, 𝑠°, 𝑡°, and 𝑢°

If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case.

Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as equal. Draw diagonal 𝐵𝐷.

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Draw an appropriate diagram, and provide the relevant Given and Prove for this case. Use triangle congruence criteria to demonstrate why the quadrilateral is a parallelogram.

Construction: Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as equal. Draw diagonals 𝐴𝐶 and 𝐵𝐷.

If the diagonals of a parallelogram are equal in length, then the parallelogram is a rectangle. Complete the diagram, and develop an appropriate Given and Prove for this case.

Construction: Label the quadrilateral 𝐺𝐻𝐼𝐽. Draw diagonals 𝐺𝐼 and 𝐻𝐽.

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Parallelograms

Properties, shapes, and diagonals.

A parallelogram is a quadrilateral made from two pairs of intersecting parallel lines . There are several rules involving:

• the angles of a parallelogram
• the sides of a parallelogram
• the diagonals of a parallelogram

• Interactive Parallelogram

Two Pairs of Parallel Lines

To create a parallelogram just think of 2 different pairs of parallel lines intersecting. ABCD is a parallelogram.

Click on the button below to turn the pure parallel lines into a parallelogram.

Angles of Parallelogram

Opposite angles are congruent.

$$ \angle D \cong \angle B \\ \angle A \cong \angle C $$

Triangles can be used to prove this rule about the opposite angle.

Consecutive angles are supplementary

The following pairs of angles are supplementary

$$ \angle C $$ and $$ \angle D $$ $$ \angle C $$ and $$ \angle B $$ $$ \angle A $$ and $$ \angle B $$ $$ \angle A $$ and $$ \angle D $$

To explore these rules governing the angles of a parallelogram use Math Warehouse's interactive parallelogram.

$$\angle Y = 40 ^{\circ}$$. What is the measure of angles X,W, and Z in parallelogram WXYZ?

There are many different ways to solve this question. You know that the opposite angles are congruent and the adjacent angles are supplementary.

$$ \angle \red W = 40^{\circ} $$ since it is opposite $$ \angle Y $$ and opposite angles are congruent.

Since consecutive angles are supplementary $$ m \angle Y + m \angle Z = 180 ^{\circ} \\ 40^{\circ} + m \angle Z = 180 ^{\circ} \\ m \angle Z = 180 ^{\circ} - 40^{\circ} \\ m \angle \red Z = 140 ^{\circ} $$

What is the measure of x, y, z in parallelogram below?

Sides of A Parallelogram

The opposite sides of a parallelogram are congruent.

Triangles can be used to prove this rule about the opposite sides.

To explore these rules governing the sides of a parallelogram use Math Warehouse's interactive parallelogram .

What is the length of side BD and side CD in parallelogram ABCD?

What is x in the parallelogram on the left?

What is the value of x and y in the parallelogram below?

Since opposite sides are congruent you can set up the following equations and solve for $$x $$: $ \text{ Equation 1} \\ 2x − 10 = x + 80 \\ x - 10 = 80 \\ x = 90 $

Since opposite sides are congruent you can set up the following equations and solve for $$y $$: $ \text{ Equation 2} \\ 3y − 4 = y + 20 \\ 2y − 4 = 24 \\ 2y = 24 \\ y = 12 $

The diagonals of a parallelogram bisect each other.

AO = OD CO = OB

To explore these rules governing the diagonals of a parallelogram use Math Warehouse's interactive parallelogram .

What is x and Y?

Since the diagonals bisect each other, y = 16 and x = 22

$$ x + 40 = 2x + 18 \\ 40 = x +18 \\ 40 = x + 18 \\ 22 = x $$

A classic activity: have the students construct a quadrilateral and its midpoints, then create an inscribed quadrilateral . Then ask the students to measure the angles , sides etc.. of inscribed shape and use the measurements to classify the shape (parallelogram). Designed with Geometer's Sketchpad in mind .

• Rectangle vs. Square
• Comparison of properties of each type of parallelogram
• Criteria prove quadrilateral is a parallelogram

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Properties of parallelograms

• Quadrilateral I
• Quadrilateral II
• Quadrilateral III

One special kind of polygons is called a parallelogram. It is a quadrilateral where both pairs of opposite sides are parallel.

There are six important properties of parallelograms to know:

• Opposite sides are congruent (AB = DC).
• Opposite angels are congruent (D = B).
• Consecutive angles are supplementary (A + D = 180°).
• If one angle is right, then all angles are right.
• The diagonals of a parallelogram bisect each other.

$$\triangle ACD\cong \triangle ABC$$

If we have a parallelogram where all sides are congruent then we have what is called a rhombus. The properties of parallelograms can be applied on rhombi.

If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid. The parallel sides are called bases while the nonparallel sides are called legs. If the legs are congruent we have what is called an isosceles trapezoid.

In an isosceles trapezoid the diagonals are always congruent. The median of a trapezoid is parallel to the bases and is one-half of the sum of measures of the bases.

$$EF=\frac{1}{2}(AD+BC)$$

Video lesson

Find the length of EF in the parallelogram

• Common types of transformation
• Transformation using matrices
• Basic information about circles
• Inscribed angles and polygons
• Advanced information about circles
• Parallelogram, triangles etc
• The surface area and the volume of pyramids, prisms, cylinders and cones
• SAT Overview
• ACT Overview

Calcworkshop

Properties of Parallelograms Detailed w/ 21 Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

Did you know that there are 6 distinct properties of parallelograms?

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

It’s true!

In today’s geometry lesson we’re going to learn to use those properties to uncover missing sides and angles from known parallelograms.

Then we’re going to dive into the associated two-column proofs!

Let’s get started!

A parallelogram is a special type of quadrilateral.

And just as its name suggests, a parallelogram is a figure with two pairs of opposite sides that are parallel.

But there are even more attributes of parallelograms that enable us to determine angle and side relationships.

Properties of Parallelograms Explained

6 Properties of Parallelograms Defined

1. Opposite sides are parallel

Segment AB is parallel to segment DC, and segment AD is parallel to segment BC.

2. Opposite sides are congruent

Segment AB is congruent to segment DC, and segment AD is congruent to segment BC.

3. Opposite angles are congruent

Angle A is congruent to angle C, and angle D is congruent to angle B.

4. Same-Side interior angles (consecutive angles) are supplementary

Angles A and D are supplementary, angles B and C are supplementary, angles A and B are supplementary, and angles D and C are supplementary.

5. Each diagonal of a parallelogram separates it into two congruent triangles

Triangle DAB is congruent to triangle DCB.

6. The diagonals of a parallelogram bisect each other

Segment AE is congruent to segment CE, and segment DE is congruent to segment BE.

And as Math Planet accurately points out, if one angle in a parallelogram is a right angle, then all angles are right angles.

This means that if we know the properties of parallelograms we can identify missing angles and sides.

We will be reminded of our angle pair relationships from our previous study of parallel lines cut by a transversal to aid us on our quest. Remember, all those rules for alternate interior angles, corresponding angles, and even vertical angles? They are going to come in handy!

And what we will discover is that if we have congruent polygons, then corresponding angles and sides are also congruent. This fact enables us to prove two parallelograms are congruent, all while using our properties.

In the video below:

• We will use our new properties of parallelograms to find unknown measures.
• Prove corresponding parts of congruent parallelograms are congruent.

Parallelogram Properties – Lesson & Examples (Video)

• Introduction to Parallelograms
• 00:00:23 – Complete the missing statements for the given parallelogram (Examples #1-8)
• Exclusive Content for Member’s Only
• 00:08:56 – Use properties of parallelograms to find the missing side and angle measurements (Examples #9-18)
• 00:17:34 – Use the parallelogram properties to find the indicated measures (Example #19)
• 00:20:47 – Complete the two-column proof for the given parallelograms (Examples #20-21)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Parallelogram Properties: A Comprehensive Guide for Geometry Enthusiasts

Parallelogram properties refer to the characteristics and attributes of this specific type of quadrilateral. A parallelogram is a four-sided polygon with opposite sides that are parallel and equal in length. It also has opposite angles that are congruent. Understanding the properties of parallelograms is essential in geometry and can help solve various mathematical problems . Here are some key takeaways regarding parallelogram properties:

These properties provide a foundation for understanding and working with parallelograms in geometry.

Understanding Parallelogram Properties

Definition of a parallelogram.

A parallelogram is a type of quadrilateral that has several unique characteristics . One of the defining properties of a parallelogram is that opposite sides are equal in length. This means that if we measure the length of one side of a parallelogram, we can automatically determine the length of the opposite side. Additionally, the opposite angles of a parallelogram are also equal, which means that if we measure one angle , we can find the measure of the angle directly across from it.

Basic Properties of a Parallelogram

In addition to the definition of a parallelogram, there are several other important properties that are worth exploring. Let’s take a closer look at some of these properties:

Consecutive Angles : The consecutive angles of a parallelogram are supplementary, which means that the sum of any two consecutive angles is equal to 180 degrees . This property can be useful when solving geometry problems involving parallelograms.

Diagonals Bisect : The diagonals of a parallelogram bisect each other. This means that the point where the diagonals intersect divides each diagonal into two equal segments . This property can be proven using congruent triangles.

Rectangle Properties : A rectangle is a special type of parallelogram that has additional properties. In a rectangle, all angles are right angles, and the diagonals are equal in length. These properties make rectangles useful in various applications , such as constructing buildings or designing furniture.

Rhombus Properties : A rhombus is another special type of parallelogram. In a rhombus, all sides are equal in length, and opposite angles are equal. The diagonals of a rhombus are perpendicular bisectors of each other. These properties make rhombuses useful in fields such as jewelry design or tiling patterns.

Base and Height : The base and height of a parallelogram are important measurements when calculating its area . The base is any of the parallel sides , and the height is the perpendicular distance between the base and the opposite side. The formula for finding the area of a parallelogram is base multiplied by height.

Trapezoid Properties : A trapezoid is a quadrilateral with one pair of parallel sides. Although a trapezoid is not a parallelogram, it shares some properties with parallelograms. For example, the opposite angles of a trapezoid are equal, and the diagonals bisect each other.

These are just a few of the basic properties of parallelograms. Understanding these properties can help us solve geometry problems, complete missing information , and even improve our GMAT scores . Whether we are preparing for a math test or simply curious about the properties of quadrilaterals, knowing the characteristics of a parallelogram can be beneficial.

Parallelograms can also be found in real-life situations. For example, when measuring a wall to hang a picture , we can use the properties of a parallelogram to determine if the wall is straight. Similarly, when planning the layout of a garden , we can use the properties of a parallelogram to ensure that the sides are parallel and the angle s are equal.

In physics, parallelograms are used to represent vector addition. By drawing vectors as sides of a parallelogram, we can determine the resultant vector by connecting the opposite corners of the parallelogram. This application of parallelograms is particularly useful in analyzing forces and motion.

Detailed Analysis of Parallelogram Properties

Parallelogram Properties in Geometry

In geometry, a parallelogram is a four-sided polygon with opposite sides that are parallel. It has several unique properties that distinguish it from other quadrilaterals. One of the key characteristics of a parallelogram is that its opposite sides are equal in length. This means that if we measure the length of one side of a parallelogram, we can automatically determine the length of the opposite side.

Another important property of a parallelogram is that its consecutive angles are supplementary, meaning they add up to 180 degrees . This property allows us to find the measure of an angle in a parallelogram if we know the measure of one of its angles .

Parallelograms also have interesting relationships between their diagonals . The diagonals of a parallelogram bisect each other, meaning they divide each other into two equal parts. This property can be useful when solving problems involving the diagonals of a parallelogram.

Parallelogram Properties and Formulas

In addition to the properties mentioned above, parallelograms have specific formulas for calculating their area and perimeter. The area of a parallelogram can be found by multiplying the length of its base by its height . The base of a parallelogram is any one of its sides , and the height is the perpendicular distance between the base and the opposite side.

The perimeter of a parallelogram is simply the sum of all its sides . Since opposite sides of a parallelogram are equal in length, we can find the perimeter by multiplying the length of one side by 4.

Parallelograms also have connections to other types of quadrilaterals. For example, a rectangle is a special type of parallelogram where all angles are right angles. A rhombus is another special type of parallelogram where all sides are equal in length.

In geometry, parallelogram properties are often used in proofs and problem-solving. By understanding the properties and formulas associated with parallelograms, we can confidently solve various geometry problems and provide complete solutions .

Parallelogram in Real Life and Physics

Parallelograms can be found in various real-life scenarios . For instance, consider a rectangular room where the opposite walls are parallel. The shape of the room can be represented as a parallelogram. Similarly, when a force is applied to an object in a specific direction , the resulting motion can be represented by a parallelogram. This concept is used in physics to analyze vector addition.

Proving Parallelogram Properties

Steps to Prove Parallelogram Properties

To prove the properties of a parallelogram, we need to follow a series of steps. These steps involve analyzing the characteristics of a quadrilateral and using various geometric concepts . Let’s take a look at the steps involved in proving parallelogram properties:

Step 1: Identify the given information: Begin by identifying the given information about the quadrilateral. This may include the lengths of the sides, the measures of the angle s, or any other relevant details .

Step 2: Determine the properties to be proven: Based on the given information, determine which properties of a parallelogram need to be proven. These properties may include the equality of opposite sides, the congruence of opposite angles, or the diagonals bisecting each other.

Step 3: Use the properties of a parallelogram: Utilize the known properties of a parallelogram to establish relationships between the angle s and sides of the quadrilateral. These properties include the fact that opposite sides are equal, consecutive angles are supplementary, and the diagonals bisect each other.

Step 4: Apply relevant theorems and concepts: Apply relevant theorems and concepts from geometry to support the proof. These may include theorems related to congruent triangles, the properties of rectangles and rhombuses, or the parallelogram law of vector addition.

Step 5: Provide a complete solution: Present a complete solution that includes all the necessary steps and justifications. Clearly explain each step of the proof, making sure to use proper mathematical notation and terminology.

Proof of Parallelogram Properties

Now, let’s delve into the proof of some specific parallelogram properties . We will explore examples that demonstrate the equality of angles and sides in a parallelogram.

Example 1: Proving Opposite Angles are Equal

In this example , we will prove that the opposite angles of a parallelogram are equal. Let’s consider a parallelogram ABCD . To prove this property , we can use the fact that opposite sides of a parallelogram are equal and the concept of congruent triangles.

First, we can show that triangle ABC is congruent to triangle CDA by using the side-angle-side (SAS) congruence criterion . This implies that angle ABC is congruent to angle CDA.

Next, we can prove that triangle BCD is congruent to triangle DAB using the same congruence criterion . This implies that angle BCD is congruent to angle DAB.

Since angle ABC is congruent to angle CDA and angle BCD is congruent to angle DAB, we can conclude that the opposite angles of parallelogram ABCD are equal.

Example 2: Proving Opposite Sides are Equal

In this example , we will prove that the opposite sides of a parallelogram are equal. Consider a parallelogram PQRS . To prove this property , we can use the concept of congruent triangles and the fact that opposite angles of a parallelogram are equal.

First, we can show that triangle PQS is congruent to triangle SRP by using the side-angle-side (SAS) congruence criterion . This implies that side PQ is congruent to side SR.

Next, we can prove that triangle PSR is congruent to triangle QRP using the same congruence criterion . This implies that side PS is congruent to side QR.

Since side PQ is congruent to side SR and side PS is congruent to side QR, we can conclude that the opposite sides of parallelogram PQRS are equal.

By following these steps and utilizing the properties of parallelograms, we can prove various properties of these four-sided polygons . These proofs not only help us understand the geometry of parallelograms but also have practical applications in fields such as physics and engineering.

Parallelogram as a Special Quadrilateral

When can a parallelogram also be called a rectangle.

A parallelogram is a special type of quadrilateral that has some unique properties . One interesting case is when a parallelogram can also be called a rectangle. In order for a parallelogram to be classified as a rectangle, it must meet certain conditions .

Firstly, a rectangle is a parallelogram with four right angles. This means that all four angles of the parallelogram must be equal to 90 degrees . Additionally, the opposite sides of the parallelogram must be equal in length. These conditions ensure that the parallelogram has all the properties of a rectangle, making it possible to classify it as such.

Why is a Parallelogram a Parallelogram

A parallelogram is a four-sided polygon with opposite sides that are parallel to each other. This unique property sets it apart from other quadrilaterals. The opposite sides of a parallelogram are also equal in length, which means that the lengths of the adjacent sides are also equal.

Another important characteristic of a parallelogram is that the opposite angles are equal. This means that if we measure the angle s formed by the intersection of the sides, we will find that they are congruent. Additionally, the consecutive angles of a parallelogram are supplementary, meaning that they add up to 180 degrees .

The diagonals of a parallelogram also have interesting properties . They bisect each other, meaning that they divide each other into two equal parts. This property holds true for all parallelograms , regardless of their size or shape.

In terms of area and perimeter, the formula s for a parallelogram are different from those of other quadrilaterals. The area of a parallelogram can be calculated by multiplying the base by the height, while the perimeter is the sum of all four sides .

Parallelograms also have connections to other special quadrilaterals . For example, a rectangle is a type of parallelogram with four right angles. A rhombus is a parallelogram with all sides of equal length . These relationships between different types of quadrilaterals help us understand their properties and how they are related to each other.

In real life, parallelograms can be found in various objects and structures. For example, the shape of a door or a window can be a parallelogram. The design of a soccer field or a basketball court also often incorporates parallelograms. Understanding the properties of parallelograms can help us analyze and solve problems related to these real-life examples .

In geometry, there are several theorems and proofs related to parallelograms. These theorems help us understand and prove various properties of parallelograms. They involve concepts such as congruent triangles, vector addition, and the properties of parallel lines .

Parallelogram in Real Life

Parallelograms are not just abstract shapes we learn about in geometry class . They actually have real-life applications and can be found in various objects and situations around us. Let’s explore some examples of parallelograms in real life and understand the importance of their properties.

Real Life Examples of Parallelograms

Walls and Floors : Many buildings have walls and floors that are shaped like parallelograms. The sides of these structures are parallel to each other, and the opposite sides are equal in length. This property allows for stability and structural integrity .

Road Signs : Have you ever noticed that some road signs , such as yield signs or stop signs, are shaped like parallelograms? The sides of these signs are parallel, and the opposite angles are equal. This shape makes them easily recognizable and distinguishable from other signs .

Tabletops : The tabletops of many desks , dining tables , and countertops are often parallelograms. This shape provides ample space for placing objects and allows for comfortable seating arrangements . The opposite sides of the tabletop are parallel, ensuring stability.

Bookshelves : Bookshelves are often designed with parallelogram-shaped compartments . This arrangement allows for efficient use of space and provides stability to hold books and other items . The sides of the compartments are parallel, and the opposite sides are equal in length.

Windows : Some windows , especially those with slanted tops , have a parallelogram shape . This design not only adds aesthetic appeal but also allows for maximum natural light to enter the room. The sides of the window frame are parallel, and the opposite angles are equal.

Importance of Parallelogram Properties in Real Life

Understanding the properties of parallelograms is essential in various real-life scenarios . Here are a few reasons why these properties are important:

Construction and Architecture : Architects and engineers rely on the properties of parallelograms when designing structures. The knowledge of parallel sides, equal opposite angles , and equal opposite sides helps in creating stable and balanced buildings .

Interior Design : Interior designers often consider the shape and properties of parallelograms when planning furniture layouts and room designs . By utilizing parallelogram-shaped objects , they can optimize space and create visually appealing arrangements .

Mathematical Problem Solving : Parallelograms are frequently used in mathematical problem-solving , especially in geometry. Understanding their properties and theorems helps in solving equations, finding missing angles or sides, and proving geometric relationships .

Physics and Engineering : Parallelograms are also relevant in the field of physics, particularly when studying vector addition. The parallelogram law states that if two vectors are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the resultant vector.

By recognizing and applying the properties of parallelograms in real-life situations, we can better appreciate their significance and understand their role in various fields . Whether it’s in construction, design, mathematics, or physics, parallelograms play a crucial role in our everyday lives .

Learning Activities for Parallelogram Properties

In this section , we will explore various learning activities that will help you understand the properties of parallelograms. By engaging in these activities , you will gain a deeper understanding of quadrilateral characteristics , parallelogram angles , opposite sides being equal, consecutive angles, and much more. Let’s dive in!

Parallelogram Properties Worksheet

To reinforce your understanding of parallelogram properties, we have prepared a worksheet for you. This worksheet will provide you with a series of questions and problems related to parallelograms. By solving these exercises , you will be able to practice identifying and applying different properties of parallelograms. Here are some key topics covered in the worksheet :

• Identifying the properties of a parallelogram
• Finding the measures of angles in a parallelogram
• Determining if a given quadrilateral is a parallelogram
• Calculating the area and perimeter of a parallelogram
• Exploring the properties of diagonals in a parallelogram

Feel free to use any relevant formulas or theorems to solve the problems in the worksheet . Remember to show your work and provide complete solutions . This will help you reinforce your understanding of parallelogram properties and improve your problem-solving skills .

Parallelogram Properties Quiz

Now, let’s put your knowledge of parallelogram properties to the test with a quiz ! This quiz will assess your understanding of the various properties and characteristics of parallelograms. It will consist of multiple-choice and short-answer questions that cover a wide range of topics related to parallelograms. Here’s what you can expect from the quiz :

• Identifying the properties of different types of parallelograms (rectangle, rhombus, square)
• Determining if a given quadrilateral is a parallelogram based on its properties
• Solving problems involving the base and height of a parallelogram
• Applying theorems and properties to find missing angles and sides in parallelograms
• Analyzing real-life examples where parallelogram properties are applicable

This quiz will not only test your knowledge but also help you identify any areas where you may need further practice . Make sure to read each question carefully and double-check your answers before submitting. Remember, practice makes perfect, so don’t hesitate to retake the quiz to improve your score .

By actively participating in the Parallelogram Properties Worksheet and taking the Parallelogram Properties Quiz , you will enhance your understanding of parallelogram properties and develop your problem-solving skills . These activities will provide you with a solid foundation in geometry and prepare you for more advanced concepts in the future . So, let’s get started and have fun exploring the fascinating world of parallelograms!

What are some unique properties of parallelograms and where can I learn more about them?

A parallelogram is a four-sided polygon with opposite sides that are parallel and equal in length. It has several interesting properties, such as opposite angles being congruent and diagonals bisecting each other. To learn more about parallelograms and their properties, you can visit the Learn more about parallelograms and their properties. page on LambdaGeeks.com. Explore this page to gain a comprehensive understanding of the concept and delve deeper into the fascinating characteristics of parallelograms.

Frequently Asked Questions

Q1: what are the key characteristics of a quadrilateral.

A1: A quadrilateral is a four-sided polygon. Its key characteristics include having four vertices , four angles , and four sides . The sum of its internal angles is always 360 degrees . Examples of quadrilaterals include squares, rectangles, trapezoids, and parallelograms.

Q2: Can you explain parallelogram properties?

A2: Parallelogram properties are unique characteristics that define a parallelogram. These include opposite sides being equal, opposite angles being equal, consecutive angles being supplementary, and diagonals bisecting each other.

Q3: How does the property of diagonals bisect apply to a parallelogram?

A3: In a parallelogram, the diagonals bisect each other, meaning they cut each other into two equal parts. This is a unique property of parallelograms, and it is used in many geometry problems .

Q4: What are some examples of properties of a parallelogram?

A4: Some examples of properties of a parallelogram include: 1. Opposite sides are equal in length. 2. Opposite angles are equal in measure. 3. Consecutive angles are supplementary. 4. The diagonals bisect each other.

Q5: Is there a formula to calculate the area of a parallelogram?

A5: Yes, the formula to calculate the area of a parallelogram is base times height (A = bh). This formula is important in geometry and is often practiced for improvement.

Q6: When can a parallelogram be called a rectangle?

A6: A parallelogram can be called a rectangle when all of its angles are right angles ( 90 degrees ). This is a specific property of rectangles, which are a type of quadrilateral and a special case of a parallelogram.

Q7: How many properties does a parallelogram have?

A7: A parallelogram has several unique properties , including: 1. Opposite sides are equal. 2. Opposite angles are equal. 3. Consecutive angles are supplementary. 4. Diagonals bisect each other. So, there are at least four main properties of a parallelogram.

Q8: How do the properties of a rhombus differ from a parallelogram?

A8: A rhombus is a specific type of parallelogram where all four sides are of equal length . In addition to the properties of a parallelogram, a rhombus also has diagonals that are perpendicular bisectors of each other.

Q9: What are the properties of quadrilaterals?

A9: Quadrilaterals have several properties , including having four sides , four angles , and the sum of the internal angles being 360 degrees . Depending on the type of quadrilateral (rectangle, square, parallelogram, trapezoid, etc.), there may be additional properties.

Q10: Where can you find a parallelogram in real life?

A10: Parallelograms can be found in various aspects of real life. Examples include the shape of a book , a door , a desk , and even some types of tiles. They are also used in physics, especially when dealing with vector addition and the parallelogram law.

• Parallelogram diagonals
• Is a rhombus a parallelogram
• Parallelogram
• Parallelogram shape
• Is a trapezoid a parallelogram
• Is a square a parallelogram

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Parallelogram

A parallelogram is a two-dimensional geometrical shape whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length. The  Sum of adjacent angles of a parallelogram is equal to 180 degrees. In geometry, you must have learned about many 2D shapes and sizes such as circles, squares, rectangles, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary from each other and are used to solve many problems. Let us learn here the definition, formulas and properties of a parallelogram.

Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.  Also, the interior angles on the same side of the transversal are supplementary. The Sum of all the interior angles equals 360 degrees.

A three-dimensional shape that has its faces in a parallelogram shape is called a parallelepiped. The area of the parallelogram depends on the base (one of its parallel sides) and its height (altitude drawn from top to bottom) of it. The perimeter of a parallelogram depends on the length of its four sides.

A square and a rectangle are two shapes which have similar properties to a parallelogram.

Rhombus: If all the sides of a parallelogram are congruent or equal to each other, then it is a rhombus.

If there is one parallel side and the other two sides are non-parallel, then it is a trapezium.

See the figure below:

In the figure above, you can see, ABCD is a parallelogram, where AB || CD and AD || BC. 

Also, AB = CD and AD = BC

And,  ∠A = ∠C & ∠B = ∠D

Also, ∠A & ∠D are supplementary angles because these interior angles lie on the same side of the transversal. In the same way, ∠B & ∠C are supplementary angles.

∠A + ∠D = 180

∠B + ∠C = 180

Shape of Parellelogram

A parallelogram is a two-dimensional shape. It has four sides, in which two pairs of sides are parallel. Also, the parallel sides are equal in length. If the length of the parallel sides is not equal in measurement, then the shape is not a parallelogram. Similarly, the opposite interior angles of a parallelogram should always be equal. Otherwise, it is not a parallelogram.

Special Parallelograms

Square and Rectangle: A square and a rectangle are two shapes which have similar properties to a parallelogram. Both have their opposite sides equal and parallel to each other. Diagonals of both shapes bisect each other. 

Rhomboid: A special case of a parallelogram that has its opposite sides parallel to each other, but adjacent sides are of unequal lengths. Also, the angles are non-right angles. 

Trapezium: If there is one pair of side parallel and the other two sides are non-parallel, then it is a trapezium. 

Angles of Parallelogram

A parallelogram is a flat 2d shape which has four angles. The opposite interior angles are equal. The angles on the same side of the transversal are supplementary, that means they add up to 180 degrees. Hence, the sum of the interior angles of a parallelogram is 360 degrees.

Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, it’s a special polygon called Parallelogram. The properties of a parallelogram are as follows:

• The opposite sides are parallel and equal
• The opposite angles are equal
• The consecutive or adjacent angles are supplementary
• If any one of the angles is a right angle, then all the other angles will be at right angle
• The two diagonals bisect each other
• Each diagonal bisects the parallelogram into two congruent triangles
• The Sum of the square of all the sides of a parallelogram is equal to the sum of the square of its diagonals. It is also called parallelogram law

Also, read:

Formulas (Area & Perimeter)

The formula for the area and perimeter of a parallelogram is covered here in this section. Students can use these formulas and solve problems based on them.

Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-dimensional plane. Below is the formula to find the parallelogram area:

Area = Base × Height

In the above figure, || gram ABCD,  Area is given by;

where a is the slant length of the side of || gram ABCD and b is the base.

Check here:   Area of a Parallelogram Formula

Perimeter of Parallelogram

The perimeter of any shape is the total distance covered around the shape or the total length of any shape. Similarly, the  perimeter of a parallelogram  is the total distance of the boundaries of the parallelogram. To calculate the perimeter value, we have to know the values of its length and breadth. The parallelogram has its opposite sides equal in length. Therefore, the formula to calculate the perimeter is written as;

Where a and b are the length of the sides of the parallelogram.

Types of Parallelogram

There are mainly four types of parallelograms, depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc.

• If PQ = QR = RS = SP are the equal sides, then it’s a rhombus. All the properties are the same for rhombus as for parallelogram.

Is Square a Parallelogram?

Yes, a square is considered as a parallelogram since the opposite sides are parallel to each other, and the diagonals of the square bisect each other.

Is Rectangle a Parallelogram?

Yes, a rectangle is also a parallelogram, because it satisfies the conditions or meets the properties of parallelogram such as the opposite sides are parallel and diagonals bisect each other.

Parallelogram Theorems

Theorem 1:   Parallelograms on the same base and between the same parallel sides are equal in area.

Proof:  Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Consider the above figure –

In ∆ADF and ∆BCE,

AD = BC (Since, ABCD is a parallelogram ∴ AD=BC)

AF = BE (Since, ABEF is a parallelogram ∴AF=BE)

∠ADF = ∠BCE (Corresponding Angles)

∠AFD = ∠BEC (Corresponding Angles)

∠DAF = ∠CBE (Angle Sum Property)

Therefore, ∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD) = Area(ABED) + Area(BCE)

Area(ABCD) = Area(ABED)+Area(ADF)

Area(ABCD) = Area(ABEF)

Hence, the area of parallelograms on the same base and between the same parallel sides is equal.

A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

Proof:  Since a rectangle is also a parallelogram so, the result is a direct consequence of the above theorem.

Theorem:   The area of a parallelogram is the product of its base and the corresponding altitude.

Given:  In a parallelogram ABCD, AB is the base.

To prove  that Area(|| gm ABCD) = AB×AL

Construction:  Complete the rectangle ALMB by Drawing BM perpendicular to CD.

Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth.

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.

Difference Between Parallelogram and Rhombus

Examples on parallelogram.

Hope this discussion has cleared all your doubts regarding parallelograms and their properties. Register with BYJU’S to learn more about quadrilaterals and other Maths concepts.

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Frequently Asked Questions (FAQs) on Parallelogram

What is a parallelogram, what are the examples of a parallelogram, what are the area and perimeter of a parallelogram, what is the shape of a parallelogram, what are the four important properties of a parallelogram.

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Class viii math, class 8 - parallelogram worksheet 2.

1. Every parallelogram is a rectangle. Mark True / False. a) True                     b) False

2. Every square is a parallelogram. Mark True / False. a) True                     b) False

3. A parallelogram can be constructed uniquely, if both diagonals and the angle between them is given. Mark True / False. a) True                     b) False

4. Parallelograms diagonals bisect each other at a right angle. Mark True / False. a) True                     b) False

5. In a parallelogram ABCD, If ∠A is equal to 60°, then it's adjacent angle would be 110°. Mark True / False. a) True                     b) False

6. If PQRS is a parallelogram, then ∠P − ∠R is equal to 0°. Mark True / False. a) True                     b) False

7. If diagonals of a parallelogram bisect each other, it must be a parallelogram. Mark True / False. a) True                     b) False

8. Two adjacent angles of a parallelogram are in the ratio 1 : 3. What is the measure of the largest angles? a) 45°                     b) 60° c) 90°                     b) 135°

9. The adjacent angles of a parallelogram are (2p − 4)° and (3p − 1)°. find the measures of all angles of the parallelogram. a) 70°, 110°, 70°, 100° b) 80°, 100°, 80°, 100° c) 70°, 110°, 70°, 110° d) None of these

10. In a parallelogram ABCD ∠A = 70°, find ∠B, ∠C and ∠D. a) ∠B = 110°, ∠C = 70° and ∠D = 100° b) ∠B = 70°, ∠C = 100° and ∠D = 100° c) ∠B = 110°, ∠C = 70° and ∠D = 110° d) None of these

11. In a parallelogram ABCD, ∠A is 4 times ∠B. Find the value of ∠A and ∠B. a) ∠A = 136° and ∠B = 45° b) ∠A = 125° and ∠B = 55° c) ∠A = 140° and ∠B = 40° d) ∠A = 144° and ∠A = 36°

14. ABCD is a parallelogram. Point P and Q are taken on the sides AB and AD respectively and PRQA parallelogram is formed. If ∠C = 60°, then find ∠R. a) 72°                     b) 70° c) 80°                     d) 60°

17. The shorter side of a parallelogram is 5.2 cm and the longer side is twice the length of shorter side. Find the perimeter of the parallelogram. a) 31 cm                     b) 32 cm c) 35 cm                     d) 31.2 cm

If you want to download the above worksheet, please click below link.

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Chapter 1: Number System

• Number System in Maths
• Natural Numbers | Definition, Examples, Properties
• Whole Numbers - Definition, Properties and Examples
• Rational Number: Definition, Examples, Worksheet
• Irrational Numbers: Definition, Examples, Symbol, Properties
• Real Numbers
• Decimal Expansion of Real Numbers
• Decimal Expansions of Rational Numbers
• Representation of Rational Numbers on the Number Line | Class 8 Maths
• Represent √3 on the number line
• Operations on Real Numbers
• Rationalization of Denominators
• Laws of Exponents for Real Numbers

Chapter 2: Polynomials

• Polynomials in One Variable - Polynomials | Class 9 Maths
• Polynomial Formula
• Types of Polynomials
• Zeros of Polynomial
• Factorization of Polynomial
• Remainder Theorem
• Factor Theorem
• Algebraic Identities

Chapter 3: Coordinate Geometry

• Coordinate Geometry
• Cartesian Coordinate System
• Cartesian Plane

Chapter 4: Linear equations in two variables

• Linear Equations in One Variable
• Linear Equation in Two Variables
• Graph of Linear Equations in Two Variables
• Graphical Methods of Solving Pair of Linear Equations in Two Variables
• Equations of Lines Parallel to the x-axis and y-axis

Chapter 5: Introduction to Euclid's Geometry

• Euclidean Geometry
• Equivalent Version of Euclid’s Fifth Postulate

Chapter 6: Lines and Angles

• Lines and Angles
• Types of Angles
• Pairs of Angles - Lines & Angles
• Transversal Lines
• Angle Sum Property of a Triangle

Chapter 7: Triangles

• Triangles in Geometry
• Congruence of Triangles |SSS, SAS, ASA, and RHS Rules
• Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 Maths
• Triangle Inequality Theorem, Proof & Applications

Chapter 8: Quadrilateral

• Angle Sum Property of a Quadrilateral
• Quadrilateral - Definition, Properties, Types, Formulas, Examples
• Introduction to Parallelogram: Properties, Types, and Theorem
• Rhombus: Definition, Properties, Formula, Examples
• Trapezium in Maths | Formulas, Properties & Examples
• Kite - Quadrilaterals

Properties of Parallelograms

• Mid Point Theorem

Chapter 9: Areas of Parallelograms and Triangles

• Area of Triangle | Formula and Examples
• Area of Parallelogram
• Figures on the Same Base and between the Same Parallels

Chapter 10: Circles

• Circles in Maths
• Radius of Circle
• Tangent to a Circle
• What is the longest chord of a Circle?
• Circumference of Circle - Definition, Perimeter Formula, and Examples
• Angle subtended by an arc at the centre of a circle
• What is Cyclic Quadrilateral
• Theorem - The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths

Chapter 11: Construction

• Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60°
• Construction of Triangles

Chapter 12: Heron's Formula

• Area of Equilateral Triangle
• Area of Isosceles Triangle
• Heron's Formula
• Applications of Heron's Formula
• Area of Quadrilateral
• Area of Polygons

Chapter 13: Surface Areas and Volumes

• Surface Area of Cuboid
• Volume of Cuboid | Formula and Examples
• Surface Area of Cube
• Volume of a Cube
• Surface Area of Cylinder
• Volume of a Cylinder: Formula, Definition and Examples
• Surface Area of Cone
• Volume of Cone: Formula, Derivation and Examples
• Surface Area of Sphere: Formula, Derivation and Solved Examples
• Volume of a Sphere
• Surface Area of a Hemisphere
• Volume of Hemisphere

Chapter 14: Statistics

• Collection and Presentation of Data
• Graphical Representation of Data
• Bar Graphs and Histograms
• Central Tendency
• Mean, Median and Mode

Chapter 15: Probability

• Experimental Probability
• Empirical Probability
• CBSE Class 9 Maths Formulas
• NCERT Solutions for Class 9 Maths: Chapter Wise PDF 2024
• RD Sharma Class 9 Solutions

Parallelogram is a quadrilateral in which opposite sides are parallel and congruent and the opposite angles are equal. A parallelogram is formed by the intersection of two pairs of parallel lines.

Table of Content

Important Properties of a Parallelogram

Theorems on properties of a parallelogram.

A parallelogram is a quadrilateral in which opposite sides are parallel and equal in length. Also, opposite angles are also equal. Few of the important properties of a parallelogram are:

• Opposite sides of a parallelogram are equal in length and are parallel to each other.
• Opposite angles in a parallelogram are equal.
• Sum of all interior angles of a parallelogram is 360°.
• Consecutive angles of a parallelogram are supplementary (180°).

Properties of Parallelogram Diagonals

Various properties of the diagonal of a parallelogram are:

• Both diagonals of a parallelogram bisect each other.
• The parallelogram is bisected into two congruent triangles by each diagonal.
• By parallelogram law, Sum of squares of the diagonal of a parallelogram is equal to the sum of squares of all the sides of a parallelogram.

Few important theorems on properties of a Parallelogram are:

• Opposite sides of a parallelogram are equal.
• If opposite sides are equal in a quadrilateral, then it is a parallelogram
• Opposite angles of a parallelogram are equal.
• If the opposite angles in a quadrilateral are equal, then it is a parallelogram
• Diagonals of a parallelogram bisect each other.
• If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Let’s discuss these theorems in detail with proof.

Theorem 1: Opposite sides of a parallelogram are equal.

Given: ABCD is a parallelogram

To Prove: AB = CD & DA = BC

Given ABCD is a parallelogram. Therefore,  AB || DC  &  AD || BC Now,  AD || BC and AC is intersecting A and C respectively. ∠DAC = ∠BCA…(i)                  [Alternate Interior Angles] Now, AB || DC and AC is intersecting A and C respectively. ∠BAC = ∠D …(ii)                    [Alternate Interior Angles] Now, In ΔADC & ΔCBA ∠DAC = ∠BCA [ From (i) ] AC = AC [ Common Side ] ∠DCA = ∠BAC [ From (ii) ] So, by ASA(Angle-Side-Angle) criterion of congruence ΔADC ≅  ΔCBA AB = CD & DA = BC [ Corresponding part of congruent triangles are equal ] Hence Proved

Converse of Theorem 1: If opposite sides are equal in a quadrilateral, then it is a parallelogram

Given: Opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD.

To Prove: Quadrilateral ABCD is a parallelogram.

In quadrilateral ABCD, AB = CD and AD = BC. In triangles ABC, and CDA we have AC = AC (Common sides) AD = BC (given) AB = CD (since alternate interior angles are equal) So by the SSS congruency criterion, triangles ABC, and CDA are congruent, thus by CPCT corresponding angles of triangles are equal. Thus, ∠BAC = ∠DCA, and ∠BCA = ∠DAC. Now AB || CD, BC || AD and thus ABCD is a parallelogram.

Theorem 2: Opposite angles of a parallelogram are equal.

To Prove:   ∠A = ∠C  and ∠B = ∠D

Given ABCD is a parallelogram. Therefore,  AB || DC  &  AD || BC Now, AB || DC and AD is Intersecting them at A and D respectively. ∠A + ∠D = 180º               …(i)             [ Sum of consecutive interior angles is 180º] Now, AD || BC and DC is Intersecting them at D and C respectively. ∠D + ∠C = 180º              …(ii)            [ Sum of consecutive interior angles is 180º] From (i) and (ii) , we get ∠A + ∠D = ∠D  +  ∠C So,  ∠A = ∠C Similarly, ∠B = ∠D ∠A = ∠C and  ∠B = ∠D Hence Proved

Converse of Theorem 2: If the opposite angles in a quadrilateral are equal, then it is a parallelogram

Given: In the quadrilateral ABCD ∠A = ∠C and ∠B = ∠D 

To Prove: ABCD is a parallelogram.

Given ∠A = ∠C and ∠B = ∠D in quadrilateral ABCD. We have to prove ABCD is a parallelogram ∠A + ∠B + ∠C + ∠D = 360º (given ∠A = ∠C and ∠B = ∠D ) 2(∠A + ∠B) =360º ∠A + ∠B = 180º. Thus AD || BC. Similarly, we can show that AB || CD.  Hence, AD || BC, and AB || CD. Therefore ABCD is a parallelogram.

Theorem 3: Diagonals of a parallelogram bisect each other.

To Prove: OA = OC & OB = OD

AB || DC  &  AD || BC Now, AB || DC and AC is intersecting A and C respectively. ∠BAC = ∠DCA                               [ Alternate Interior Angles are equal ] So, ∠BAO = ∠DCO Now,  AB || DC and BD is intersecting B and D respectively. ∠ABD = ∠CDB                               [ Alternate Interior Angles are equal ] So, ∠ABO = ∠CDO Now, in  ΔAOB &  ΔCOD we have,  ∠BAO = ∠DCO                               [ Opposite sides of a parallelogram are equal ] AB = CD ∠ABO = ∠CDO So, by ASA(Angle-Side-Angle) congruence criterion  ΔAOB ≅  ΔCOD OA = OC and OB = OD Hence Proved

Converse of Theorem 3: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Given: The diagonals AC and BD bisect each other.

If the diagonals AB and CD bisect each other. Then in Δ AOB, and Δ COD AO = CO (Diagonals bisect each other) BO = DO (Diagonals bisect each other) ∠AOB =∠COD (vertically opposite angles) Thus, by SAS congruency criterion, triangles are congruent. So  ∠CAB = ∠DCA, and ∠DBA = ∠CDB. Hence, AB || CD, and BC || AD. Thus ABCD is a parallelogram.

In conclusion, parallelograms are fundamental part of geometry. Their unique properties, such as congruent opposite sides and angles, and diagonals that bisect each other, make them applicable in various fields, from architecture to engineering. Understanding these properties enhances our ability to solve geometric problems and appreciate the mathematical harmony in the world around us.

Related Resources

Area of Parallelogram Congruence of Triangle

Solved Example on Properties of Parallelogram

Example 1: ABCD is a quadrilateral with AB = 10 cm. Diagonals of ABCD bisect each other at right angles. Then find the perimeter of ABCD.

We know that, if diagonals of a quadrilateral bisect each other at right angles then it is a rhombus. Thus, ABCD is a rhombus and  AB = BC = CD = DA. Thus, the perimeter of ABCD = 4(AB) = 4(10) = 40 cm

Example 2: Find area of a parallelogram where the base is 6 cm and the height is 12 cm.

Given, Base = 6 cm and Height = 12 cm. We know, Area = Base x Height Area = 6  × 12 Area = 72 cm 2

FAQs on Properties of a Parallelogram

What is a parallelogram.

A parallelogram is a quadrilateral that has in which opposite sides are parallel and equal. And opposite angles in a parallelogram equal. It is also considered as a cyclic quadrilateral.

Can a Rhombus be called a Parallelogram?

Opposite sides of a rhombus are equal and parallel, and its opposite angles are also equal. So it is considered a parallelogram.

What are the four important properties of a parallelogram?

The four important properties of the parallelogram are: Opposite sides of a parallelogram are parallel and congruent Consecutive angles of a parallelogram add up to 180 degrees Opposite angles of a parallelogram are equal Diagonals of a parallelogram bisect each other.

What is the order of rotational symmetry in a parallelogram?

Order of rotational symmetry is a parallelogram is 2.

Does a parallelogram have reflectional symmetry?

No, reflectional symmetry is not possible in a parallelogram.

Are the Diagonals of a Parallelogram Equal?

Diagonals of a parallelogram are generally NOT equal. In some special cases, parallelograms such as squares and rectangles have equal diagonals.

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