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ABC and CDE are equilateral triangles, as the figure shows.

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ABC and CDE are equilateral triangles, as the figure shows.

by Max@Math Revolution » Mon Jul 27, 2020 12:50 am

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[GMAT math practice question]

\(\triangle\) ABC and \(\triangle\) CDE are equilateral triangles, as the figure shows. What is the measure of the \(\angle\) x?

A. 45°
B.50°
C.55°
D. 60°
E. 65°
7.27PS.png
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Re: ABC and CDE are equilateral triangles, as the figure shows.

by raghav_25 » Mon Jul 27, 2020 2:03 am
Since nothing is given about where E point lies on the line AC, we can assume E to be the mid point of AC to solve the question for a specific case. Why this should work is because no other measurements are given, and this means, the angle x should be same for every position of E on line AC.

So solving for this specific case:
Line BE will bisect the angle B and will be perpendicular to AC.
since we have assumed E to be the mid point of AC, EC=AE. Moreover, since triangle CDE is equilateral, EC = DE.
So triangle ADE becomes isosceles as AE = DE ( = EC)

Since \(\angle AED\) = 120° (as \(\angle AED\) + \(\angle DEC\) = 180°, and \(\angle DEC\) = 60°),
the \(\angle DAE\) and \(\angle EDA\) will both be 30°
Now, as I mentioned earlier, BE is perpendicular to AC, so \(\angle AEF\) = 90°
In triangle AEF, we know two angles, we can find the third.
\(\angle FAE\) + \(\angle AEF\) + \(\angle x\) = 180°
30° + 90° + \(\angle x\) = 180°
\(\angle x\) = 60°
So answer is D
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Re: ABC and CDE are equilateral triangles, as the figure shows.

by Max@Math Revolution » Wed Jul 29, 2020 5:00 am
=>

We have AC = BC, CD = CE and ∠BCE = ∠DCE = 60°.
Triangles ACD and BCE are congruent.
Since ∠EBC = ∠DAC, we have ∠x = ∠ACB = 60°.

Therefore, D is the correct answer.
Answer: D
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