Quadrilateral ABCD is a rhombus

[RonPurewal]

The best way to get these sorts of questions answered isn't to post them on a forum.
The best way to get answers to these questions"”and, in general, to just about all math questions"”is to investigate the situation(s) yourself.

For instance, this:

So what I'm getting from this is that if opposite angles of a four sided shape are the same, we can't just assume that opposite sides are the same

If you draw a few rectangles, you'll figure this one out pretty quickly. (Every angle of every rectangle is 90º, so, there you go.)


This:

if opposite sides are the same length (and parallel), we can't assume that opposite angles will be the same

Draw yourself a bunch of parallelograms, and you'll quickly see that this is also true.

[ayush.rastogi.13]

Dear Manhattan Staff,

From statement a, we know that BDC and ABD are equilateral triangle. Hence all the angles of both the triangles are 60 degrees. Now given that C,D & E are on the same like angle ADE = 180 - (angle BDC+angle ADB) i.e. 60 degrees.

As angle BAD and angle ADE are equal lines AE and BD are parallel. Alternate interior angles are equal so the lines are parallel. If this is the case then the second statement becomes redundant and statement a is sufficient.

Could you please clarify my doubt. Thank You!

[RonPurewal]

I'm sorry, but the original picture appears to have disappeared. Could you please post the picture?

(If you don't know how to post an image directly, you can upload it to an image-hosting site and post the link here.)

Thanks.

[ayush.rastogi.13]

http://assets.manhattanprep.com/images/ ... ombus1.gif

[ayush.rastogi.13]

Posted the URL of the image. Hope you can view it.

[RonPurewal]

That image is visible. Thanks.

You're making unjustified assumptions about the placement of point E.
We don't know anything about the location of point E, other than that it's on the same line as C and D. It could be anywhere northeast of D on that line. That's going to give you a whole world of possibilities.

Try this:
"- Just draw ABCD.
"- Now draw the extension of line CD. Draw it way, way, way out in the northeast direction, all the way out to forever-land.
- You can put E anywhere on that extension.
"”"” Try putting it really close to D.
"”"” Try putting it in the location that actually produces another rhombus (the one you originally had in mind).
"”"” Try putting it farther out.

Lots of possibilities.

[ayush.rastogi.13]

Hi,

Thank you for your reply. I think I have not been able to explain my doubt clearly.

The point that I am trying to make is that from statement 1, it is clear that angles BDC, BDA are 60 degrees (equilateral triangles). Even if E is at any place on the north east of D it is said to be on a straight line, hence 180 degrees is the sum of angles BDC, BDA & ADE. Therefore angle ADE is 60 degrees. Also,angle BAD is 60 degrees (equilateral triangle)

Thus, lines BD & AE are parallel ( alternate interior angles being equal). This makes the 2nd point redundant.

Will really appreciate if you can let me know if there is any flaw in my understanding.

Thank You!!

[ayush.rastogi.13]

Hi Manhattan Team,

Will really appreciate if you can clarify my point. Thank You!

[RonPurewal]

Your argument shows that lines BA and DE are parallel. That's already known.
To see why you're showing the parallelism of these lines"”and not the ones you mentioned"”just draw the "Z" shape formed by the two angles you mentioned (BAD and ADE). Note which two parallel lines are in that "Z" shape.

Your argument shows no such thing about lines AE and BD.
To prove those two lines parallel, you'd need information about angle DAE; we have no such information.

[RonPurewal]

More to the point, it seems you haven't made the drawings I mentioned earlier in this thread.

Try making those drawings. If you do so, you'll see very clearly that...
...BA and DE are always parallel;
... angle DAE can have a wide range of possible measures;
... lines AE and BD will be parallel only if angle DAE is exactly 60º, and will be nonparallel in every other possible case.

[ayush.rastogi.13]

Yes. Thank You!

Realized my mistake. Appreciate the help.

[RonPurewal]

Sure.