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 [ 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.

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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