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2013 AMC 12B Problem 13

Problem 13 of 25IntermediateAlgebraGeometryProblem-Solving Techniques

The internal angles of quadrilateral ABCDABCD form an arithmetic progression. Triangles ABDABD and DCBDCB are similar with ∠DBA=∠DCB\angle DBA = \angle DCB and ∠ADB=∠CBD.\angle ADB = \angle CBD. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of ABCD?ABCD?

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Solution

The angles of a triangle form an arithmetic progression exactly when the middle one is 60∘.60^\circ. With ∠DBA=x\angle DBA = x and ∠ADB=y,\angle ADB = y, the four angles of ABCDABCD are x,y,180−y,180−x,x, y, 180 - y, 180-x, which must itself be an arithmetic progression. In increasing order they are either x,y,180−y,180−xx,y,180-y,180-x or x,180−y,y,180−x,x,180-y,y,180-x, giving 3y=x+1803y=x+180 or 3y=360−x.3y=360-x. One of the triangle angles x,y,180−x−yx,y,180-x-y is 60∘.60^\circ. Substitution leaves the angle sets (60,80,100,120)(60,80,100,120) and (45,75,105,135).(45,75,105,135). The two largest angles sum to at most 105+135=240.105 + 135 = 240. Thus, the correct answer is D.
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