EXPLANATION:1) Labeling the figure from given information:Let's label the figure from the given information:
1.1) \(∠DAE\) = (180-3x) from lower triangle
1.2) Since E is the intersection point of two lines, \(∠BEC\) = \(∠AED\) = x
1.3) Since BE = BC, \(∠ECB\) = \(∠BCE\) = x
1.4) \(∠EBC\) = 180-2x
1.5) Since \(DE^2 > AD^2 + AE^2\), we know that angle opposite to side DE is > 90 degrees and it is an obtuse triangle. So angle opposite to DE is \(∠DAE\) which from 1.1 is 180-3x.
So 180-3x >90, and x <30
2) Evaluating each statement independently:Now that we have established our pre-thinking, let's evaluate each statement:
I) \(CE^2 > 2BE^2\) which is the same as \(CE^2 > BE^2 + BC^2\) since BE = BC
Now for the above inequality to be true, angle opposing CE should be greater than 90, which means 190-2x should be greater than 90 , we already know that x <30, so 180-2x should be less than 120.
Hence sufficient condition. I is true.
II) AE <AD
we know that angle opposing AD is greater than angle opposing side AE, so the above inequality cannot be true.
III) AE > AD
we know that angles opposing each side is proportional to that side length. So angle opposing AE = 180-3x and angle opposing AD is 180-2x. Under all circumstances. 180-3x < 180-2x,
hence Statement III is not true.
So answer is I only. Hence A is the answer.
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