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

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

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New post 26 Nov 2010, 09:52
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A
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48% (01:00) correct 52% (01:22) wrong based on 27 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?

(1) AD = AC
(2) AC = CB
[Reveal] Spoiler: OA

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Re: Triangle [#permalink]

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New post 26 Nov 2010, 11:07
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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?

(1) AD = AC
(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
untitled1.PNG [ 6.22 KiB | Viewed 1449 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.

Answer: A.

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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Re: Triangle [#permalink]

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