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2007 AMC 10B Problem 15

Problem 15 of 25IntermediateGeometryArithmetic

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=\dfrac{x}{2}, ∠C=x3,\angle C=\dfrac{x}{3}, and ∠D=x4.\angle D=\dfrac{x}{4}. The angles sum to 360∘,360^\circ, so x+x2+x3+x4=25x12=360.x+\dfrac{x}{2}+\dfrac{x}{3}+\dfrac{x}{4}=\dfrac{25x}{12}=360. Thus x=12⋅36025=172.8≈173.x=\dfrac{12\cdot 360}{25}=172.8\approx 173. Thus, the correct answer is D.
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