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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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Since we have a square, we have 1 variable and 0 equations, and D is most likely the answer. So, we should consider each condition on its own first.
Let’s look at condition 1). It tells us that triangles ABP and CBQ are congruent to each other.
Since PD = DQ, we have AP = AD – PD, and CQ = CD – DQ are congruent and ∠A and ∠C are congruent as well. We can prove triangles ABP and CBQ are congruent with the SAS (Side-Angle-Side) triangle congruence property.
The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Let’s look at condition 2). It tells us that triangles ABP and CBQ are congruent.
Since BP and BQ are congruent hypotenuses of two right triangles, and AB and BC are two congruent legs, triangles ABP and CBQ are congruent according to the HL right triangle congruence property.
The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D
Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).
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.
