Data Sufficiency

[GMAT math practice question]

What is the measure of ∠BIC in the figure?


1) Point I is the incenter (the point where the three angle bisectors meet) of triangle ABC.
2) ∠BAC = 50.

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Since we have 4 variables (∠BIC, ∠ABC, ∠BCA, and ∠CAB) and 1 equation (∠ABC + ∠BCA + ∠CAB = 180°), C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since point O is the incenter of triangle ABC from condition 1), the segments IA, IB, and IC are bisectors of ∠BAC, ∠ABC and ∠BCA, respectively.

∠BIC = 180° – (∠IBC + ∠ICB), ∠BIC = 180° – (1/2)(180° - ∠BAC), ∠BIC = 180° – (1/2)(180° - 50°), ∠BIC = 180° – 65° = 115°.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

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 in which the answer is A, B, C, or D.