MathRevolution wrote:
[GMAT math practice question]
(Geometry) The figure shows that \(∠BAD\) is \(30^o\), and \(∠CAE\) is \(40^o.\) What is the measure of \(∠ADE\)?
Attachment:
1.30DS.png
1) Point \(O\) is the circumcenter of triangle \(ABC\).
2) Point \(I\) is the incenter of triangle \(ABC\).
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Since we have \(6\) variables (\(∠ABC, ∠BCA, ∠CAB, ∠ADE, ∠DAE,\) and \(∠DEA\)) and \(2\) equations (\(∠ABC + ∠BCA + ∠CAB = 180°, ∠ADE + ∠DAE + ∠DEA = 180°\)), 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 \(I\) is the incenter of triangle \(ABC\), we have \(∠IAB = ∠IAC = 40°\) and \(∠DAE = ∠IAB - ∠BAD, ∠DAE = 40° - 30°, ∠DAE = 10°.\)
Since point \(\)O is the circumcenter of triangle \(ABC\), we have \(∠BAO = ∠ABO = 30°\) and \(∠OBC = ∠OCB = (\frac{1}{2})(180° – (2*30° + 2*50°)) = 10°.\)
Then \(∠ABC = ∠ABO + ∠OBC = 30° + 10° = 40°.\)
Since \(∠ADE\) is an exterior angle of triangle \(ABD,\) we have \(∠ADE = ∠DAB + ∠ABD = 30° + 40° = 70°.\)
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.
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