Parallelogram Definition and 126 Threads

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, parallēl-ógrammon, a shape "of parallel lines") reflects the definition.

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  1. M

    Proof of Parallelogram Theorem: 2 Pairs of Opposite Angles Congruent

    Proof of parallelogram theorem "If 2 pairs of opposite angles congruent, then par..." I was just wondering, why exactly does "If 2 pairs of opposite angles congruent" prove that a quadrilateral is a parallelogram? Does it have something to do with the fact that the sum of the interiors equals...
  2. S

    Vectors forming a parallelogram

    The vectors 39.0 cm at 15.0° and 23.0 cm at 68.0° both start from the origin. Both angles are measured counterclockwise from the x axis. The vectors form two sides of a parallelogram. a) Find the area of the parallelogram. 716cm^2 b) Find the length of its longer diagonal. For b) is this...
  3. S

    Help with proving a quadrilateral is a parallelogram.

    [SOLVED] Help with proving a quadrilateral is a parallelogram. Hi this is my first post here and I'm glad to see that this is a well visited board. I'm having trouble with this one proof though that I have to do for geometry due tomorrow. Only a few other people I know have been assigned this...
  4. M

    What is the equation for combining two vectors using the parallelogram rule?

    I tried at "Intro physics" and was warn that my post might be deleted if not stuck to a certain template. So, here I am. I'm an outsider in physics, yet I'm trying to make a 2d game with collisions. I know that I can "split" a vector into two and combine two into one by this rule; I need the...
  5. N

    Vectors that constitute the diagonals of this parallelogram

    this is for my mechanics course, but its just dealing with vectors. sorry if this is in the wrong place. Homework Statement The problem is: You are given two vectors (4,0) and (0,4) that form the sides of a parallelogram. Determine the vectors that constitute the diagonals of this...
  6. J

    Prove Parallelogram ABCD: Triangle APD = ABP + DCP

    Parallelogram proof :( Homework Statement In parallelogram ABCD, P is any point on BC. Prove that triangle APD = triangle ABP + triangle DCP Homework Equations n/a The Attempt at a Solution I really don't know where to start :( any help would be greatly appreciated :)
  7. V

    LINEAR ALGEBRA: Given a linear transformaiton from R^2 to R^2 & parallelogram

    LINEAR ALGEBRA: Given a linear transformation from R^2 to R^2 & parallelogram... Let T be an invertible linear transformation from R^2 to R^2. Let P be a parallelogram in R^2 with one vertex at the origin. Is the image of P a parallelogram as well? Explain. Draw a sketch of the image. My...
  8. mattmns

    Complex Cross Product and Area of a parallelogram

    Hello, just a quick question. I have two complex numbers (say z and z'), and I want to find the area of the parallelogram that is generated by the two complex numbers (written as vectors, ie, if z = x + iy is a complex number, then the vector is (x,y)). Now the area of the parallelogram...
  9. Y

    How Can I Find the 4th Coordinate of a Parallelogram in 3D Space?

    i am given the (x,y,z) coordinates of 3 sides of a parallelogram how do i go about finding the 4 coordinate. do i find the distance between points using the same way to find vectors then try to find direction angles or something like that? because there is no slope or anything to compare too...
  10. R

    Inscribing square into parallelogram

    what is the necessary condition (if there is any) for inscribing a square into a parallelogram. In other words, what should the parallelogram be like - so that we can inscribe a square into it.
  11. M

    What are the properties of parallelograms regarding diagonals?

    parallelogram properties 1.>Each diagonal divides it into two congruent triangles. 2.>Diagonals need not bisect angles at the vertices. I don't understand these two properties. In property 1: what does congruent triangles means ? i am not familiar with this term...is this...
  12. D

    Proving Parallelogram Diagonals Bisect Each Other

    Hey. I've got a vectors test coming up soon, so I've been going through everything. At the moment I am working on vector proofs. here's the question: OABC is a parallelogram where \overrightarrow{OA}=\mathbf{a} and \overrightarrow{OC}=\mathbf{c}. M is the midpoint of the diagonal OB. Prove...
  13. F

    Proving that diagonals of a parallelogram bisect each other

    I am not sure whether this question should be posted here or not. 4. Using vector methods, prove that the diagonals of a parallelogram bisect each other. I have no idea at all how to start. Any clues?
  14. A

    Triangle inequality, parallelogram equality

    what does a triangle have to do with triangle inequality, and what does a paralllelogram have to do with parallelogram equality? @@a
  15. C

    Double integrals: when a parallelogram approximates a curvy area

    Sirs, I request your assistance. I am reading chapter 13.9 of James Stewart Calculus (change of variables in double integrals). Premises.- Let (x,y) = T(u,v). The function T has inverse, both are continuously differentiable, etc. Let’s take a very little rectangle S in the uv plane. The...
  16. N

    How do you calculate the area of a parallelogram using a determinant matrix?

    How co you claculate the are a pallelogram determined by points (-2, -2), (0, 3), (4, -1) and (6, 4)...I've seen an example wher a 2x2 determinant matrix was used, but I don't remember how to do it...
  17. S

    Now equate the coordinates of H to get the fourth point.

    I wasn't sure what category to put this in, but here's my problem. There is a parallelogram with 3 given points for its corners (each with 3 different coordinates). The idea is to find the forth point. I've tried subbing (x,y,z) for the unknown point, and creating 4 vectors. I tried equating...
  18. T

    C/C++ Programming a parallelogram in C++

    I'm having troube getting my program to display a parallelogram. The source code here just displays a triangle with "6" as the speicifed size. * ** *** **** ***** ****** ***** **** *** ** * The output is suppose to be something like this: _____* _____** _____*** _____****...
  19. G

    Parallelogram Equality+Inner Product Spaces

    It is true that if a norm satisfies the parallelogram equality then it must come from an inner product right (i.e. < , > is an inner product).? How in the world could you go about proving/showing this?
  20. B

    Change a triangle to a parallelogram?

    how do i change a triangle to a parallelogram? please help english major assigned a math project for paper
  21. S

    PQRS Parallelogram Help: Find Sin Q with Working

    Hi, New to this forum. Im just doing my school certificate (yr10) and need help with 1 question. PQRS is any parallelogram. If sin P = k, find Sin Q Probably sounds like a stupid question but heck if you can give me the answer with working id be really happy. :biggrin: thanks
  22. T

    Find the area of the original parallelogram

    A Parallelogram has a base length of 12cm. In order to increse teh area of teh paralleogram by 54cm(squared), the length of the base is increased by 2cm and the height is incresed by 3cm . Find the area of the original parallelogram. The solution requires me to use fully defined variables...
  23. T

    How Do You Calculate the Original Area of a Parallelogram?

    Help wanted ! Heres teh question ... A Parallelogram has a base length of 12cm. In order to increse teh area of teh paralleogram by 54cm(squared), the length of the base is increased by 2cm and the height is incresed by 3cm . Find the area of the original parallelogram. The solution...
  24. H

    Combinations for word parallelogram

    How many ways can you select four letters out of the word parallelogram ? The answer on the book is 150. For me I am stuck at using 8C4 and it seems to be the logical way. The reason I chose 8 is because some letters are repeating. So how can they have 150?
  25. R

    Proving Parallelogram PQRS & Quadrilateral ABCD: Help Needed

    I'm having trouble with two parallelogram proofs 1) PQRS is a parallelogram and T is any point inside the parallelogram. Prove that triangle TSR + triangle TQP = 1/2 parallelogram PQRS 2) ABCD is a quadrilateral whose area is bisected by the diagonal AC. Prove that BD is bisected by AC...
  26. F

    How can the parallelogram law be proven?

    Hello everyone! =) This might be a good challenge to everyone here... =) 1.) Let A be an n X n matrix and let x and y be vectors in R^n. Show that Ax .y = x.(A^T)(y), where "." means dot product and T is 'transpose'. 2. show that u.v = (1/4)// u + v //^2 - (1/4)//u - v//^2...
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