1972 AMC 12 Problem 21

Attempt Problem 21 of the 1972 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1972 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

21.

If the sum of the measures in degrees of angles

A,

B,

C,

D,

E,

and

F

in the figure is

90n,

then

n

is equal to:

2

3

4

5

6

Name the two intersections of

AD

with

BF

and

CE

Combine the angle sums of the central quadrilateral and the two triangles attached to it

Let

P=AD\cap BF

and

Q=AD\cap CE.

In quadrilateral

EFPQ,

E+F+\angle P+\angle Q=360^\circ.

The triangles

BPD

and

AQC

give

B+D=\angle P

and

A+C=\angle Q,

using the supplementary angles at

P

and

Q.

Adding yields

\begin{gathered} A+B+C+D+E+F\\ =360^\circ=90^\circ\cdot4. \end{gathered}

Thus

n=4.

Therefore, the correct answer is C.

Problem 21 in other years