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Types of Parallelograms: Properties, Definitions, and Proofs

Types of Parallelograms: Properties, Definitions, and Proofs

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<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for

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<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for

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<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for

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Sign up to get unlimited access to thousands of study materials. It's free!

Access to all documents

Join milions of students

Improve your grades

By signing up you accept Terms of Service and Privacy Policy

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for proving that a figure is a parallelogram using different statements and reasons.

There are different types of parallelogram, where their properties can be applied to prove their characteristics. Opposite sides are congruent and opposite angles are congruent. Diagonals bisect each other and adjacent angles are supplementary.

There are different ways to prove that a given figure is a parallelogram:

  1. Show that both pairs of opposite sides are parallel.
  2. Show that both pairs of opposite sides are congruent.
  3. Show that both pairs of opposite angles are congruent.
  4. Show that the diagonals bisect each other.
  5. Show that one angle is supplementary to both adjacent angles.
  6. Show that one pair of opposite sides are congruent and parallel.

Given: ZBAC = ZDCA; ZBCA = ZDAC
Prove: ABCD is a parallelogram
Statements:

  1. <BAC = <DCA; <BCA = <DAC
  2. BA || DC; BC || DA
  3. ABCD is a parallelogram

Given: WXYZ; WZ = YX
Prove: WXYZ is a parallelogram
Statements:

  1. WX ~ UZ; WZ = X
  2. XZZX
  3. AWZX ≈ AYXZ
  4. <WXZ = <YZX; <W ZX =< YXZ
  5. WX || YZ; WZ || YX
  6. WXYZ is a parallelogram

Given: PQRS; PQ || RS
Prove: PQRS is a parallelogram
Statements:

  1. PQ = Rs; P Q || Rs
  2. <Q PRLS RP
  3. PR = RP
  4. AQPR ASRP
  5. LQRPL SPR
  6. QR || SP
  7. PQRS is a parallelogram

Given: JKLM is a parallelogram
Prove: JK = LM and JM = LK
Statements:

  1. JKLM is a parallelogram
  2. JK || LM; JM || LK
  3. AJKL ALMJ
  4. JK = LM and JM = LK

Given: CDEF is a parallelogram
Prove: ZDCF and ZCFE are supplementary
Statements:

  1. CDEF is a parallelogram
  2. CD || EF
  3. < KJL ≈ LM L J; <MJL ≈ <KW
  4. JL = LJ
  5. LDCF and <CFE are supplementary

Given: RSTU is a parallelogram
Prove: ZUZS
Statements:

  1. RSTU is a parallelogram
  2. RS || TU
  3. <SRT = <UTR; <URT=<STR
  4. RT = TR
  5. ARUTATSR
  6. LULS

The definition of a parallelogram is crucial for proving that a given quadrilateral is a parallelogram. Its properties, such as opposite sides and angles being congruent, as well as the diagonals bisecting each other, can be used to demonstrate the characteristics of different types of parallelograms. Through these proofs, it becomes evident that the properties of a parallelogram are sufficient to prove that a quadrilateral is indeed a parallelogram.

Summary - Geometry

  • Definition of a Parallelogram in Math: A parallelogram is a special type of quadrilateral where both pairs of opposite sides are parallel.
  • Special Quadrilaterals and Their Properties: Parallelograms have properties such as congruent opposite sides and angles, as well as bisecting diagonals and supplementary adjacent angles.
  • 6 Ways to Prove a Quadrilateral is a Parallelogram: There are six different ways to prove that a quadrilateral is a parallelogram, including showing that opposite sides and angles are congruent and parallel, and that the diagonals bisect each other.
  • Proving a Quadrilateral is a Parallelogram - Examples: Several examples are provided to demonstrate how to prove that a given quadrilateral is a parallelogram using the properties and definitions of a parallelogram.
  • Conclusion: The properties and definitions of a parallelogram are essential for proving that a given quadrilateral is a parallelogram, and can be utilized to demonstrate the characteristics of different types of parallelograms.
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Uploaded by Bianca Stratford

435 Followers

Frequently asked questions on the topic of Geometry

Q: What is the definition of a parallelogram in math?

A: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Q: What are the 6 ways to prove a quadrilateral is a parallelogram?

A: The 6 ways to prove a quadrilateral is a parallelogram are: show that both pairs of opposite sides are parallel, show that both pairs of opposite sides are congruent, show that both pairs of opposite angles are congruent, show that the diagonals bisect each other, show that one angle is supplementary to both adjacent angles, and show that one pair of opposite sides are congruent and parallel.

Q: How can you prove that a given figure is a parallelogram?

A: You can prove that a figure is a parallelogram by showing that both pairs of opposite sides are parallel, both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, the diagonals bisect each other, one angle is supplementary to both adjacent angles, or one pair of opposite sides are congruent and parallel.

Q: What are some examples of proving a quadrilateral is a parallelogram?

A: Some examples of proving a quadrilateral is a parallelogram include showing that the given figure has congruent opposite sides and angles, and parallel diagonals, or using side-angle-side or angle-side-angle congruence to prove that a figure is a parallelogram.

Q: What are the properties of a parallelogram that can be used to prove its characteristics?

A: The properties of a parallelogram, such as opposite sides and angles being congruent, as well as the diagonals bisecting each other, can be used to prove its characteristics.

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U8L3 Pallelogram Proofs Notes

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Bianca Stratford

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<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for
<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for
<p>A <strong>parallelogram</strong> is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for

Congruence

/

Theorems concerning quadrilateral properties

/

Types of Parallelograms: Properties, Definitions, and Proofs

U8L3 Pallelogram Proofs Notes

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A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. This definition is the basis for proving that a figure is a parallelogram using different statements and reasons.

There are different types of parallelogram, where their properties can be applied to prove their characteristics. Opposite sides are congruent and opposite angles are congruent. Diagonals bisect each other and adjacent angles are supplementary.

There are different ways to prove that a given figure is a parallelogram:

  1. Show that both pairs of opposite sides are parallel.
  2. Show that both pairs of opposite sides are congruent.
  3. Show that both pairs of opposite angles are congruent.
  4. Show that the diagonals bisect each other.
  5. Show that one angle is supplementary to both adjacent angles.
  6. Show that one pair of opposite sides are congruent and parallel.

Given: ZBAC = ZDCA; ZBCA = ZDAC
Prove: ABCD is a parallelogram
Statements:

  1. <BAC = <DCA; <BCA = <DAC
  2. BA || DC; BC || DA
  3. ABCD is a parallelogram

Given: WXYZ; WZ = YX
Prove: WXYZ is a parallelogram
Statements:

  1. WX ~ UZ; WZ = X
  2. XZZX
  3. AWZX ≈ AYXZ
  4. <WXZ = <YZX; <W ZX =< YXZ
  5. WX || YZ; WZ || YX
  6. WXYZ is a parallelogram

Given: PQRS; PQ || RS
Prove: PQRS is a parallelogram
Statements:

  1. PQ = Rs; P Q || Rs
  2. <Q PRLS RP
  3. PR = RP
  4. AQPR ASRP
  5. LQRPL SPR
  6. QR || SP
  7. PQRS is a parallelogram

Given: JKLM is a parallelogram
Prove: JK = LM and JM = LK
Statements:

  1. JKLM is a parallelogram
  2. JK || LM; JM || LK
  3. AJKL ALMJ
  4. JK = LM and JM = LK

Given: CDEF is a parallelogram
Prove: ZDCF and ZCFE are supplementary
Statements:

  1. CDEF is a parallelogram
  2. CD || EF
  3. < KJL ≈ LM L J; <MJL ≈ <KW
  4. JL = LJ
  5. LDCF and <CFE are supplementary

Given: RSTU is a parallelogram
Prove: ZUZS
Statements:

  1. RSTU is a parallelogram
  2. RS || TU
  3. <SRT = <UTR; <URT=<STR
  4. RT = TR
  5. ARUTATSR
  6. LULS

The definition of a parallelogram is crucial for proving that a given quadrilateral is a parallelogram. Its properties, such as opposite sides and angles being congruent, as well as the diagonals bisecting each other, can be used to demonstrate the characteristics of different types of parallelograms. Through these proofs, it becomes evident that the properties of a parallelogram are sufficient to prove that a quadrilateral is indeed a parallelogram.

Summary - Geometry

  • Definition of a Parallelogram in Math: A parallelogram is a special type of quadrilateral where both pairs of opposite sides are parallel.
  • Special Quadrilaterals and Their Properties: Parallelograms have properties such as congruent opposite sides and angles, as well as bisecting diagonals and supplementary adjacent angles.
  • 6 Ways to Prove a Quadrilateral is a Parallelogram: There are six different ways to prove that a quadrilateral is a parallelogram, including showing that opposite sides and angles are congruent and parallel, and that the diagonals bisect each other.
  • Proving a Quadrilateral is a Parallelogram - Examples: Several examples are provided to demonstrate how to prove that a given quadrilateral is a parallelogram using the properties and definitions of a parallelogram.
  • Conclusion: The properties and definitions of a parallelogram are essential for proving that a given quadrilateral is a parallelogram, and can be utilized to demonstrate the characteristics of different types of parallelograms.
user profile picture

Uploaded by Bianca Stratford

435 Followers

Frequently asked questions on the topic of Geometry

Q: What is the definition of a parallelogram in math?

A: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Q: What are the 6 ways to prove a quadrilateral is a parallelogram?

A: The 6 ways to prove a quadrilateral is a parallelogram are: show that both pairs of opposite sides are parallel, show that both pairs of opposite sides are congruent, show that both pairs of opposite angles are congruent, show that the diagonals bisect each other, show that one angle is supplementary to both adjacent angles, and show that one pair of opposite sides are congruent and parallel.

Q: How can you prove that a given figure is a parallelogram?

A: You can prove that a figure is a parallelogram by showing that both pairs of opposite sides are parallel, both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, the diagonals bisect each other, one angle is supplementary to both adjacent angles, or one pair of opposite sides are congruent and parallel.

Q: What are some examples of proving a quadrilateral is a parallelogram?

A: Some examples of proving a quadrilateral is a parallelogram include showing that the given figure has congruent opposite sides and angles, and parallel diagonals, or using side-angle-side or angle-side-angle congruence to prove that a figure is a parallelogram.

Q: What are the properties of a parallelogram that can be used to prove its characteristics?

A: The properties of a parallelogram, such as opposite sides and angles being congruent, as well as the diagonals bisecting each other, can be used to prove its characteristics.

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying