1968 AMC 12 Problem 6

Attempt Problem 6 of the 1968 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 1968 AMC 12 solutions, or check the answer key.

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

Let side

AD

of convex quadrilateral

ABCD

be extended through

D,

and let side

BC

be extended through

C,

to meet in point

E.

Let

S

represent the degree-sum of angles

CDE

and

DCE,

and let

S'

represent the degree-sum of angles

BAD

and

ABC.

r=\frac{S}{S'},

then:

r=1

sometimes,

r\gt1

sometimes

r=1

sometimes,

r\lt1

sometimes

0\lt r\lt1

r\gt1

r=1

Express both sums using angle

E

Apply the triangle angle sum to

ECD

and

EAB

In triangle

ECD,

S=180^\circ-\angle E.

Since

D

lies on

AE

and

C

lies on

BE,

triangle

EAB

gives

S'=180^\circ-\angle E

as well. Thus

S=S'

and

r=1.

Therefore, the correct answer is E.

Problem 6 in other years