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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.
Let’s look at condition 1). It tells us that PA = AB, AC = AQ, and ∠PAC = ∠BAQ.
Since ∠PAC = 60° + ∠A and ∠BAQ = 60° + ∠A, we have ∠PAC = ∠BAQ.
Then triangles △APC and △ABQ are congruent according to the SAS congruency property, so we get yes as an answer.
The answer is unique, yes, so the condition is sufficient according to Common Mistake Type 1, which states that the answers must be in terms of a unique “yes” or “no.”
Let’s look at condition 2). It tells us that it is not sufficient.
If triangles △ABC, △APB, and △ACQ are congruent, then triangles △APC and △ABQ are congruent, so we get yes as an answer.
If triangles △ABC and △APB are congruent with sides 3 and AQ = CQ, then triangles △APC and △ABQ are not congruent, so we get no as an answer.
The answer is not unique, yes and no, so the condition is not sufficient according to Common Mistake Type 1, which states that if we get both yes and no as an answer, it is not sufficient.
Condition 1) ALONE is sufficient.
Therefore, A is the correct answer.
Answer: A
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.
