Question

The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD.$ 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 $ABCD?$

$210$

$220$

$230$

$240$

$250$

Solution:

The angles of a triangle form an arithmetic progression exactly when the middle one is $60^\circ.$ With $\angle DBA = x$ and $\angle ADB = y,$ the four angles of $ABCD$ are $x, y, 180 - y, 180 - x,$ which must itself be an arithmetic progression. Combined with a $60^\circ$ angle in the triangles, this forces either $x = 60$ or $x + y = 120.$ Working through the cases, the possible angle sets are $(60, 80, 100, 120)$ and $(45, 75, 105, 135).$ The two largest angles sum to at most $105 + 135 = 240.$ Thus, the correct answer is D.

2013 AMC 12B Problem 13

Solution:

Problem 13 in Other Years