Which Statement Can You Use to Conclude That Quadrilateral Xyzw Is a Parallelogram

Properties of Parallelogram

Properties of a parallelogram help u.s. to identify a parallelogram from a given fix of figures easily and quickly. Earlier nosotros larn most the properties, let us first know virtually parallelograms. It is a four-sided closed figure with equal and parallel opposite sides and equal opposites angles. Let us learn more about the properties of parallelograms in detail in this commodity.

one.	What are the Properties of Parallelogram?
ii.	Properties of Diagonals of Parallelogram
3.	Theorems on Parallelogram Backdrop
4.	FAQs on Backdrop of Parallelogram

What are the Backdrop of Parallelogram?

A parallelogram is a blazon of quadrilateral in which the opposite sides are parallel and equal. In that location are iv 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 bug relating to the sides and angles of a parallelogram.

The iv important properties of parallelogram angles and sides are as follows:

1. The contrary sides of a parallelogram are equal and parallel to each other.
2. Opposite angles are equal, i.e ∠A = ∠C, and ∠B = ∠D.
3. All the angles of a parallelogram add upward to 360°, i.due east. ∠A + ∠B + ∠C + ∠D = 360°.
4. The consecutive angles of a parallelogram are supplementary, i.due east.,
   ∠A + ∠B = 180°
   ∠B + ∠C = 180°
   ∠C + ∠D = 180°
   ∠D + ∠A = 180°

All the above properties hold true for all types of parallelograms, simply at present let us as well larn about individual properties of some special parallelograms. The three different parallelograms are foursquare, rectangle, and rhombus which are different from each other considering of their backdrop yet they all come nether the category of parallelograms.

Properties of Square:

• All four sides of a square are equal.
• All iv angles are equal and of 90 degrees each.
• Diagonals of a square bifurcate its angles.
• Both the diagonals have the same length.
• Opposite sides are equal and parallel to each other.

Backdrop of Rectangle:

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

Properties of Rhombus:

• All sides are equal in length.
• Diagonals bisect each other at a ninety-degree angle.
• The sum of any 2 adjacent interior angles is 180 degrees.
• Opposite sides are equal and parallel to each other.

Now, permit usa expand our knowledge by learning nigh the properties of diagonals of parallelograms in the section beneath.

Properties of Parallelogram Diagonals

First, we will recall the meaning of a diagonal. Diagonals are line segments that join the opposite vertices. In parallelogram ABCD (refer to the prototype given higher up), Air conditioning and BD are the diagonals. Let us presume that O is the intersecting indicate of the diagonals Air-conditioning and BD. The backdrop of diagonals of a parallelogram are equally follows:

• Diagonals of a parallelogram bisect each other, i.e., OB = OD and OA = OC.
• Each diagonal divides the parallelogram into ii coinciding triangles, i.eastward., 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^two + BC² + CD² + DA² = Air conditioning² + BD².

Theorems on Parallelogram Properties

The theorems on properties of a parallelogram are helpful to define the rules for working beyond the bug on parallelograms. The backdrop relating to the sides and angles of a parallelogram tin can all be easily understood and practical to solve various bug. Farther, these theorems are also supportive of understanding the concepts in other quadrilaterals. 4 important theorems related to the properties of a parallelogram are given below:

• Opposite sides of a parallelogram are equal
• Reverse angles of a parallelogram are equal
• Diagonals of a parallelogram bifurcate each other
• If 1 pair of opposite sides is equal and parallel in a quadrilateral then it is a parallelogram

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

Proof:

Given: ABCD is a parallelogram.

To Prove: The reverse sides are equal, AB = CD and BC = Advertisement.

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

• Ac = CA (mutual side)
• ∠BAC = ∠DCA (alternating 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 take AB = CD, and BC = AD.

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

Proof:

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. At present compare the two triangles ABC, and CDA. Here we take

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

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

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

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

Proof:

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

To Prove: ∠A = ∠C and ∠B = ∠D

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

• AC = CA (common side)
• ∠1 = ∠4 (alternate interior angles)
• ∠two = ∠3 (alternating interior angles)

Thus, by ASA, the 2 triangles are coinciding, which ways that ∠B = ∠D. Similarly, we tin can show that ∠A = ∠C. This proves that contrary angles in whatever parallelogram are equal.

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

Proof:

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

To Prove: ABCD is a parallelogram.

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

∠A + ∠B + ∠C + ∠D = 360º

ii(∠A + ∠B) =360º

∠A + ∠B = 180º.

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

Theorem 3: Diagonals of a Parallelogram Bisect Each Other.

Proof:

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

To Prove: The diagonals PT and RQ bifurcate each other, i.due east. PE = ET and ER = EQ.

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

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

By the ASA criterion, the 2 triangles are congruent, which means by CPCTC, PE = ET, and RE = EQ. Thus, the two diagonals PT and RQ bifurcate 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, and so 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 bifurcate each other. Compare triangle RET, and triangle PEQ once again. We accept:

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

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

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

Proof:

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

To Evidence: ABCD is a parallelogram.

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

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

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

Important Notes:

A quadrilateral is a parallelogram when:

• the opposite sides of a quadrilateral are equal and parallel.
• the reverse angles of a quadrilateral are equal.
• the diagonals bisect each other.
• one pair of contrary sides is equal and parallel.

Do you lot know?

1. Why a kite is non a parallelogram?
2. Is an isosceles trapezoid a parallelogram?

Also Check:

• Parallelogram Formula
• Expanse of Parallelogram
• Properties of Rectangle
• Departure Between Rectangle and Parallelogram

Properties of Parallelogram Examples

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

   Solution:

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

   

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

   As well, in whatsoever parallelogram, the adjacent angles are supplementary. This implies ∠B = 180° - ∠A = 180° - 90° = ninety°.

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

   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 take proved that when 1 angle of a parallelogram is xc°, then it is a rectangle.

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

   Solution:

   Consider the following effigy:

   

   First of all, we note that since the diagonals bisect each other, and so by using the properties of parallelograms, we tin 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:

   • AE = AE
   • Be = ED (diagonals bifurcate each other)
   • ∠AEB = ∠AED = 90° (given)

   Thus, past 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=Advertizing. Clearly, ABCD is a rhombus.

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

3. Example 3: ABCD is a quadrilateral with AB = eight.5 units. The diagonals of ABCD bifurcate each other at right angles. Then find the perimeter of ABCD.

   Solution:

   Past using the properties of parallelograms, if the diagonals of a quadrilateral bifurcate each other at right angles then it is a rhombus.

   So ABCD is a rhombus and hence AB = BC = CD = DA.

   Thus, the perimeter = four(AB) = four(8.5) = 34 units.

   Reply: The perimeter of ABCD is 34 units.

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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 equally follows:

• The reverse sides are equal.
• The opposite angles of a parallelogram are equal.
• The sequent angles are supplementary.
• If i angle of a parallelogram is a right angle, and then all the angles are right angles.
• The diagonals of a parallelogram bisect each other.
• Each diagonal of a parallelogram bisects information technology into two coinciding triangles.
• If one pair of reverse sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

What are the Backdrop of Parallelogram with Respect to Diagonals?

At that place are ii 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 bifurcate each other.

Are the Diagonals of a Parallelogram Equal?

The diagonals of a parallelogram are Non equal. The opposite sides and opposite angles of a parallelogram are equal. Diagonals of square and rectangle are equal which are special types of parallelograms.

What are the Four Important Properties of Parallelogram?

A parallelogram is a quadrilateral with reverse sides equal and parallel. The opposite angles of a parallelogram are also equal. In brusque, a parallelogram tin be considered a twisted rectangle. Information technology is more of a rectangle, but the angles at the vertices need non be correct angles. The four important backdrop of a parallelogram are as follows:

• The opposite sides are equal
• The opposite angles are equal
• The adjacent angles are supplementary
• Diagonals of a parallelogram bisect each other

Tin 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 tin can be called a parallelogram.

What are Unique Properties of Parallelograms?

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

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

What are the Different Backdrop of Each Special Parallelogram?

There are iii special types of parallelograms - square, rectangle, and rhombus. A foursquare 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 mensurate 90 degrees each. A rhombus is a parallelogram with four equal sides, but its angles need non exist right angles.

How Can We Differentiate a Parallelogram and a Quadrilateral by Backdrop of Parallelogram?

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

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Source: https://www.cuemath.com/geometry/properties-of-parallelograms/

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