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# M18-15

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 01:03
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Difficulty:

85% (hard)

Question Stats:

54% (01:27) correct 46% (02:04) wrong based on 164 sessions

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In rectangle $$ABCD$$, $$E$$ is the point of intersection of diagonals. If angle $$ABD$$ is twice angle $$EAD$$, what is the value of angle $$CED$$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees
[Reveal] Spoiler: OA

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Math Expert
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16 Sep 2014, 01:03
Expert's post
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Official Solution:

In rectangle $$ABCD$$, $$E$$ is the point of intersection of diagonals. If angle $$ABD$$ is twice angle $$EAD$$, what is the value of angle $$CED$$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Look at the diagram below:

Notice that since $$\angle{BAD}=90$$ degrees then $$\angle{EAB}=90-x$$ degrees.

Next, since diagonals of a rectangle are equal and bisect each other then $$AE=BE$$. So, $$\angle{EAB}=\angle{EBA}$$: $$90-x=2x$$, which gives $$x=30$$. Thus, $$\angle{EAB}=\angle{EBA}=60$$ and $$\angle {AEB}=60$$ degrees.

Finally as $$\angle{AEB}=\angle{CED}$$ then $$\angle{CED}=60$$ degrees.

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03 Mar 2015, 05:44
I think this question is good and helpful.
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Marty Murray
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16 Aug 2016, 05:48
I think this is a high-quality question and I agree with explanation.

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11 Sep 2016, 02:25
Bunuel wrote:
Official Solution:

In rectangle $$ABCD$$, $$E$$ is the point of intersection of diagonals. If angle $$ABD$$ is twice angle $$EAD$$, what is the value of angle $$CED$$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Look at the diagram below:

Notice that since $$\angle{BAD}=90$$ degrees then $$\angle{EAB}=90-x$$ degrees.

Next, since diagonals of a rectangle are equal and bisect each other then $$AE=BE$$. So, $$\angle{EAB}=\angle{EBA}$$: $$90-x=2x$$, which gives $$x=30$$. Thus, $$\angle{EAB}=\angle{EBA}=60$$ and $$\angle {AEB}=60$$ degrees.

Finally as $$\angle{AEB}=\angle{CED}$$ then $$\angle{CED}=60$$ degrees.

Hi,
Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ?
thanks

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Math Expert
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Posts: 41687

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11 Sep 2016, 03:02
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19941010 wrote:
Bunuel wrote:
Official Solution:

In rectangle $$ABCD$$, $$E$$ is the point of intersection of diagonals. If angle $$ABD$$ is twice angle $$EAD$$, what is the value of angle $$CED$$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Look at the diagram below:

Notice that since $$\angle{BAD}=90$$ degrees then $$\angle{EAB}=90-x$$ degrees.

Next, since diagonals of a rectangle are equal and bisect each other then $$AE=BE$$. So, $$\angle{EAB}=\angle{EBA}$$: $$90-x=2x$$, which gives $$x=30$$. Thus, $$\angle{EAB}=\angle{EBA}=60$$ and $$\angle {AEB}=60$$ degrees.

Finally as $$\angle{AEB}=\angle{CED}$$ then $$\angle{CED}=60$$ degrees.

Hi,
Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ?
thanks

The diagonal of a rectangle is a bisector only if the rectangle is a square.
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26 Sep 2016, 07:33
Aren't the diagonals of rectangle perpendicular bisector of each other?

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26 Sep 2016, 07:39
Expert's post
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La1yaMalhotra wrote:
Aren't the diagonals of rectangle perpendicular bisector of each other?

The diagonals of a rectangle always bisect each other but they are perpendicular to each other only if a rectangle is a square.
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12 Dec 2016, 17:04
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. Hi, can you explain more?
Basically, total 3 angles of a triangle is 180. If <AEB = 60, total value of 3 angles in triangle AEB is 150 degree (Because <AEB = 60, <ABE= 60, and <BAE = 30). Moreover, 4 angles AEB + BEC + CED + DEA = 360. As a result, <AEB must be at least 90 degree.
Thank you so much!

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13 Dec 2016, 02:01
hnguyen26 wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. Hi, can you explain more?
Basically, total 3 angles of a triangle is 180. If <AEB = 60, total value of 3 angles in triangle AEB is 150 degree (Because <AEB = 60, <ABE= 60, and <BAE = 30). Moreover, 4 angles AEB + BEC + CED + DEA = 360. As a result, <AEB must be at least 90 degree.
Thank you so much!

The question is fine. You just made mistake in calculations. The measure of angle BAE is 60, not 30. Please re-read the discussion more above carefully this time.
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21 Jun 2017, 07:41
The answer is option C. Well explained above.
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" The few , the fearless "

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22 Aug 2017, 14:46
I think this is a high-quality question and I agree with explanation.

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Re M18-15   [#permalink] 22 Aug 2017, 14:46
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# M18-15

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