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 $ABCD?$

$210$

$220$

$230$

$240$

$250$

Answer: D

Concepts:arithmetic sequence angle chasing casework

Difficulty rating: 1700

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. In increasing order they are either $x,y,180-y,180-x$ or $x,180-y,y,180-x,$ giving $3y=x+180$ or $3y=360-x.$ One of the triangle angles $x,y,180-x-y$ is $60^\circ.$ Substitution leaves the angle sets $(60,80,100,120)$ 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

Answer: D

Difficulty rating: 1700

Solution:

Problem 13 in Other Years