Solution:
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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Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of \(\angle x\ +\angle y\).
Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.
Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.
If \(\angle ABD\ \ =\ \angle DBE\ =\ \angle EBC\ =\ 20^o\ ,\ and\ \angle y\ =20^o,\ we\ have\ \angle x\ +\ \angle y\ =\ 60^o\)
If \(\angle ABD\ =\angle DBE\ =\ \angle EBC\ =15^o,\ \angle x\ =30^o,\ \angle y\ =15^o,\ we\ have\ \angle x\ +\ \angle y\ =45^o\)
The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) & 2) together are not sufficient.
Therefore, E is the correct answer.
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
