In the figure, is the line AB perpendicular to the line CD?

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

We have \(x = y\), since \(x°\) and \(y°\) are vertical each other

Since we have 2 variables (\(x\) and \(y\)) and 1 equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since \(x = y\) and \(x + y = 180\), we have \(x = y = 90\).
Thus, \(AB\) and \(CD\) are perpendicular each other and the answer is 'yes'.
Since condition 1) yield a unique answer, it is sufficient.

Condition 2)
If \(AO=OC\), then we have the angle \(<ACO = y/2\) and \(x+y = 180\) or \(x = y = 90\), \(AB\) and \(CD\) are perpendicular each other and the answer is 'yes'.
If \(AO=AC\), then we have the angle \(<ACO = x\) and \(2x+y/2 = 180\) or \(x = y = 72\), \(AB\) and \(CD\) are not perpendicular each other and the answer is 'no'.
Since condition 2) does not yield a unique answer, it is not sufficient.

Therefore, A is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

Asked: In the figure, is the line AB perpendicular to the line CD?
Q. Is x = 90?
x = y

(1) x + y =180
Since x = y ; x = 90
SUFFICIENT

(2) AOC is an isosceles triangle
Case 1:
x + y/2 + y/2 = 180
x +y = 180
2x = 180
x = 90
Case 2:
2x + y/2 = 180
5x/2 = 180
x = 72
line AB may or may not be perpendicular to the line CD
NOT SUFFICIENT

IMO A