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2007 AMC 12B Problem 11

Problem 11 of 25IntermediateAlgebraGeometry

The angles of quadrilateral ABCDABCD satisfy ∠A=2∠B=3∠C=4∠D.\angle A=2\angle B=3\angle C=4\angle D. What is the degree measure of ∠A,\angle A, rounded to the nearest whole number?

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Solution

Let x=∠A.x=\angle A. Then ∠B=x2,\angle B=\tfrac x2, ∠C=x3,\angle C=\tfrac x3, and ∠D=x4.\angle D=\tfrac x4. The angle sum gives x+x2+x3+x4=25x12=360, x+\dfrac x2+\dfrac x3+\dfrac x4=\dfrac{25x}{12}=360, so x=12⋅36025=172.8≈173.x=\dfrac{12\cdot360}{25}=172.8\approx173. Thus, the correct answer is D.

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Concepts: angle sum · ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.