2013 AMC 12B Problem 13

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

Small hint:

A triangle’s angles form an arithmetic progression exactly when its middle angle is

60^\circ

Big hint:

Let

\angle DBA = x

and

\angle ADB = y;

the four angles of

ABCD

become

x, y, 180-y, 180-x,

which must be an arithmetic progression with one triangle angle equal to

60^\circ

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.