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# Quadrilateral ABCD is a rhombus and points C, D, and E are

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16 Dec 2013, 00:06
AccipiterQ wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

How do you know that BD=AB from 1?? Is there some hidden calculations done in there? All I see is BCD=60, you don't know the measures of any other angles so it doesn't really tell you anything

Because BC=CD, triangle BCD is isosceles. (1) says that angle BCD is 60 degrees, thus triangle BCD is equilateral. Therefore BD=BC=CD.

Hope it's clear.
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16 Dec 2013, 11:14
Bunuel wrote:
AccipiterQ wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

How do you know that BD=AB from 1?? Is there some hidden calculations done in there? All I see is BCD=60, you don't know the measures of any other angles so it doesn't really tell you anything

Because BC=CD, triangle BCD is isosceles. (1) says that angle BCD is 60 degrees, thus triangle BCD is equilateral. Therefore BD=BC=CD.

Hope it's clear.

Perfectly, I should have read the question more thoroughly. Thanks
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11 May 2014, 12:33
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

Hi Bunuel,

I'm trying to catch my disconnect and tried to explain the reasoning. Appreciate any input.

From the question stem, we know that AB=BC=CD=DA because it's a rhombus. We cannot infer the exact degree's besides stating that the opposite angles will be equal and the adjacent will add up to 180. Is that correct to infer?

Additionally, we are told/asked "Is quadrilateral ABDE a rhombus?" -- It does say that ABDE is a quadrilateral, meaning, that the lines AE and BD should be parallel and AB and DE should be parallel. Is that correct to infer? I'm guessing the answer is NO. Maybe i'm confusing quadrilateral with parallelogram?

Now, according to statement A = If I know that BCD is 60 degrees, I can automatically infer that BAD is 60 degrees since it's a rhombus. This makes ADE 60 degrees but I don't know anything about the other two angles DAE and DEA.

INSUFFICIENT

Additionally, statement two states that (2) AE is parallel to BD. Now I can infer that since AE is parallel to BD and since BA is parallel to DE, it's at least a parallelogram. Correct?

Now, since I know that it's a parallelogram, I can work my way through the angles and realize that it's a parallelogram with all acute angles equal to each other and all obtuse equal to each other. I also know that BC = BD = BA, therefore a Rhombus.
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12 May 2014, 01:11
russ9 wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

Hi Bunuel,

I'm trying to catch my disconnect and tried to explain the reasoning. Appreciate any input.

1. From the question stem, we know that AB=BC=CD=DA because it's a rhombus. We cannot infer the exact degree's besides stating that the opposite angles will be equal and the adjacent will add up to 180. Is that correct to infer?

2, Additionally, we are told/asked "Is quadrilateral ABDE a rhombus?" -- It does say that ABDE is a quadrilateral, meaning, that the lines AE and BD should be parallel and AB and DE should be parallel. Is that correct to infer? I'm guessing the answer is NO. Maybe i'm confusing quadrilateral with parallelogram?

Now, according to statement A = If I know that BCD is 60 degrees, I can automatically infer that BAD is 60 degrees since it's a rhombus. This makes ADE 60 degrees but I don't know anything about the other two angles DAE and DEA.

INSUFFICIENT

3. Additionally, statement two states that (2) AE is parallel to BD. Now I can infer that since AE is parallel to BD and since BA is parallel to DE, it's at least a parallelogram. Correct?

Now, since I know that it's a parallelogram, I can work my way through the angles and realize that it's a parallelogram with all acute angles equal to each other and all obtuse equal to each other. I also know that BC = BD = BA, therefore a Rhombus.

1 is correct.
2 is not correct. Quadrilateral just means "four sides". So, any four-sided shape is a quadrilateral.
3 is correct.
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15 May 2014, 16:20
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Bunuel wrote:
russ9 wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

Hi Bunuel,

I'm trying to catch my disconnect and tried to explain the reasoning. Appreciate any input.

1. From the question stem, we know that AB=BC=CD=DA because it's a rhombus. We cannot infer the exact degree's besides stating that the opposite angles will be equal and the adjacent will add up to 180. Is that correct to infer?

2, Additionally, we are told/asked "Is quadrilateral ABDE a rhombus?" -- It does say that ABDE is a quadrilateral, meaning, that the lines AE and BD should be parallel and AB and DE should be parallel. Is that correct to infer? I'm guessing the answer is NO. Maybe i'm confusing quadrilateral with parallelogram?

Now, according to statement A = If I know that BCD is 60 degrees, I can automatically infer that BAD is 60 degrees since it's a rhombus. This makes ADE 60 degrees but I don't know anything about the other two angles DAE and DEA.

INSUFFICIENT

3. Additionally, statement two states that (2) AE is parallel to BD. Now I can infer that since AE is parallel to BD and since BA is parallel to DE, it's at least a parallelogram. Correct?

Now, since I know that it's a parallelogram, I can work my way through the angles and realize that it's a parallelogram with all acute angles equal to each other and all obtuse equal to each other. I also know that BC = BD = BA, therefore a Rhombus.

1 is correct.
2 is not correct. Quadrilateral just means "four sides". So, any four-sided shape is a quadrilateral.
3 is correct.

Thanks a ton. Very clear now.
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Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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14 Jul 2015, 06:35
AccipiterQ wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

How do you know that BD=AB from 1?? Is there some hidden calculations done in there? All I see is BCD=60, you don't know the measures of any other angles so it doesn't really tell you anything

Important thing to remember here: Rhombus is a symmetrical quadrilateral wrt the diagonals. What this means is that any diagonal will divide the rhombus into 2 triangles of equal areas. Also, because of the symmetry it is easy to prove that $$\angle A = \angle C$$. Additional properties of rhombus are:

•Opposite angles of a rhombus have equal measure.
•The two diagonals of a rhombus are perpendicular
•Its diagonals bisect opposite angles.

Draw BD to complete triangle ABD. Given, $$\angle {BCD}$$ = 60 degrees. ----> $$\angle {BAD}$$ = 60 degrees and $$\angle {ABC}$$ = $$\angle {CDA}$$ = 120 degrees (to complete 360 degree for sum of all internal angles of a quadrilateral).

Now, diagonal BD will bisect $$\angle {CBA}$$ and $$\angle {CDA}$$ such that $$\angle {ABD} = \angle {ADB}$$ = x.

Thus, in triangle ABD, x+x+60 = 180 ----> x = 60 degrees . Thus triangle ABD is equilateral and hence AB = BD.
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Re: Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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05 Aug 2015, 05:49
Bunuel wrote:
russ9 wrote:
Bunuel wrote:

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

Hi Bunuel,

I'm trying to catch my disconnect and tried to explain the reasoning. Appreciate any input.

1. From the question stem, we know that AB=BC=CD=DA because it's a rhombus. We cannot infer the exact degree's besides stating that the opposite angles will be equal and the adjacent will add up to 180. Is that correct to infer?

2, Additionally, we are told/asked "Is quadrilateral ABDE a rhombus?" -- It does say that ABDE is a quadrilateral, meaning, that the lines AE and BD should be parallel and AB and DE should be parallel. Is that correct to infer? I'm guessing the answer is NO. Maybe i'm confusing quadrilateral with parallelogram?

Now, according to statement A = If I know that BCD is 60 degrees, I can automatically infer that BAD is 60 degrees since it's a rhombus. This makes ADE 60 degrees but I don't know anything about the other two angles DAE and DEA.

INSUFFICIENT

3. Additionally, statement two states that (2) AE is parallel to BD. Now I can infer that since AE is parallel to BD and since BA is parallel to DE, it's at least a parallelogram. Correct?

Now, since I know that it's a parallelogram, I can work my way through the angles and realize that it's a parallelogram with all acute angles equal to each other and all obtuse equal to each other. I also know that BC = BD = BA, therefore a Rhombus.

1 is correct.
2 is not correct. Quadrilateral just means "four sides". So, any four-sided shape is a quadrilateral.
3 is correct.

hi bunuel,

could you please clarify ABDE is a parallelogram (as AE||BD and BA||DE). I just want to know on what basis we are saying ABDE is parallelogram. I am not clear about this.

Thanks
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Re: Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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05 Aug 2015, 06:02
PathFinder007 wrote:

hi bunuel,

could you please clarify ABDE is a parallelogram (as AE||BD and BA||DE). I just want to know on what basis we are saying ABDE is parallelogram. I am not clear about this.

Thanks

Any quadrilateral with 2 pairs of parallel lines is a parallelogram. All shapes such as rectangles, rhombuses, squares are parallelograms
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Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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27 Sep 2015, 01:11
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Re: Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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