Parallelogram properties

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parallelogram properties

1.>Each diagonal divides it into two congruent triangles.

2.>Diagonals need not bisect angles at the vertices.



I dont understand these two properties.

In property 1:

what does congruent triangles means ? i am not familiar with this term...is this same as "similar triangle" ?

please explain.

In property 2:

it says, diagonal does not bisect vertices. If this is so , that means two halves of the triangle does not have the same area when a diagonal divides the parallelogram......am i correct ?

Or, in other words ,...In mathematical term, if ABCD is the parallelogram and if BD is the diagonal....then Area ABD =! Area BDC ......is this correct ?
 

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  • #2
robphy
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Given a triangle, a congruent triangle is a special type of "similar triangle"... the one which can be superimposed on top of the original triangle (after possible translations, rotations, and reflections). In other words, if you cut it out of the page, you can arrange to lay it on top of the other.

For 2, you probably meant to say "does not bisect angles".
Draw yourself a nice rectangle (a special case of a parallelogram)... preferably not a square.... choose the width to be longer than the height. Now, draw your diagonal. Can you identify which angles are equal in measure?
 
  • #3
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Given a triangle, a congruent triangle is a special type of "similar triangle"... the one which can be superimposed on top of the original triangle (after possible translations, rotations, and reflections). In other words, if you cut it out of the page, you can arrange to lay it on top of the other.
OK....so finally you meant, they are similar and also they must same area ....right ? otherwise they can not sit on top of each other.


For 2, you probably meant to say "does not bisect angles".
Draw yourself a nice rectangle (a special case of a parallelogram)... preferably not a square.... choose the width to be longer than the height. Now, draw your diagonal. Can you identify which angles are equal in measure?
I tried to draw figuare in paper...but its very illusive and not helping.

so, you means diagonals does not bisect the vertex ?



Well, probabily i am confusing you........forget about all these things.

Please answer these questions below. these are the concepts i wanted to know...

Q1 : does diagonal of of a paralleogram bisects vertex ? YES/NO

Q2: if ABCD is the parallelogram and if BD is the diagonal...
a)Area ABD =Area BDC
b)Area ABD !=Area BDC

Q3) congruent triangles are similar triangles having same areas.YES/NO



Please answer these questions . these are the things i want to know . please answer.

Regards
 
  • #4
uart
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momentum said:
Please answer these questions below. these are the concepts i wanted to know...

Q1 : does diagonal of of a paralleogram bisects vertex ? YES/NO
The diagonals of a parallelogram do not generally bisect the angles at the vertex.

Q2: if ABCD is the parallelogram and if BD is the diagonal...
a)Area ABD =Area BDC
b)Area ABD !=Area BDC
a) they are equal.

Q3) congruent triangles are similar triangles having same areas.YES/NO
Yes, if two triangles are both similar and also have the same area then they must be congruent. This is not really the best way to think about the definition of congruent however. Better to think of "congruent" as a special case of "similar" where the scaling ratio is 1:1.
 
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  • #5
arildno
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Remember that "bisect" means to split into two angles EQUALLY BIG.
 

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