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# A E D C B Segment AD and AC divide angle EAB into three non

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Intern
Joined: 12 Aug 2010
Posts: 13
A E D C B Segment AD and AC divide angle EAB into three non  [#permalink]

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26 Nov 2010, 09:52
1
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Difficulty:

75% (hard)

Question Stats:

36% (02:50) correct 64% (02:40) wrong based on 32 sessions

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A

E D C B

Segment AD and AC divide angle EAB into three non overlapping angles that are equal in measure. Are AE and AB equal in length?

(2) AC = CB
Math Expert
Joined: 02 Sep 2009
Posts: 49968

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26 Nov 2010, 11:07
chiragatara wrote:
A

E D C B

Segment AD and AC divide angle EAB into three non overlapping angles that are equal in measure. Are AE and AB equal in length?

(2) AC = CB

For such kind of graphic questions you must post the diagram as well. As you OA is given to be A, then I guess points E, D, C and B are collinear as shown below:
Attachment:

untitled1.PNG [ 6.22 KiB | Viewed 1676 times ]

Q: is $$AE=AB$$? Or is triangle AEB is isosceles? Or is $$\angle{AED}=\angle{ABC}$$?

Given: $$\angle{CAB}=\angle{EAD}=\angle{DAC}=x$$.

(1) AD = AC --> triangle ADC is isosceles --> $$\angle{ADC}=\angle{ACD}$$ --> $$\angle{ADE}=\angle{ACB}$$ --> in triangles ADE and ACB two angles are equal so the third angle must be equal too --> $$\angle{AED}=\angle{ABC}$$ --> triangle AEB is isosceles --> $$AE=AB$$. Sufficient.

(2) AC = CB --> so triangle ABC is isosceles and $$\angle{CBA}=\angle{CAB}(=\angle{EAD}=\angle{DAC}=x)$$. Now, if triangle AEB is indeed isosceles then $$\angle{AED}=\angle{ABC}=x$$ and $$x+x+3x=180$$ --> $$x=36$$ degrees but we don't know that, for example if $$\angle{CBA}=\angle{CAB}=\angle{EAD}=\angle{DAC}=30$$ then $$\angle{AED}=180-(30+3*30)=60\neq{\angle{ABC}}$$. Not sufficient.

Note that if we were not told that points E, D, C and B are collinear then the answer would be E.

Hope it's clear.
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Joined: 12 Aug 2010
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27 Nov 2010, 01:40
Thanks Bunuel and apology for not posting the diagram. Actually i tried but somehow couldn't.

Anyway, i have one query on this solution. I dont know how stupid that is but as E, D, C, B are on the same line, and EAD, DAC and DAC are equal angles, are ED = DC = CB?

In a triangle if two lines are equal, angles opposites are equal. Is there any modified equation on this?
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Re: A E D C B Segment AD and AC divide angle EAB into three non  [#permalink]

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30 Sep 2017, 08:27
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Re: A E D C B Segment AD and AC divide angle EAB into three non &nbs [#permalink] 30 Sep 2017, 08:27
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