1978 AMC 12 Problem 12

Attempt Problem 12 of the 1978 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 1978 AMC 12 solutions, or check the answer key.

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12.

\triangle ADE,

\angle ADE=140^\circ,

points

B

and

C

lie on sides

AD

and

AE,

respectively, and points

A,

B,

C,

D,

E

are distinct. If lengths

AB,

BC,

CD,

and

DE

are all equal, then the measure of

\angle EAD

5^\circ

6^\circ

7.5^\circ

8^\circ

10^\circ

Name

\angle BAC=\angle BCA=x

and use the successive isosceles triangles

Express the other base angles in terms of

x,

then use the angle sum in

\triangle ADE

Let

x=\angle BAC=\angle BCA,

y=\angle CBD=\angle CDB,

and

z=\angle DCE=\angle DEC.

The exterior-angle theorem applied successively gives

y=2x,\qquad z=x+y=3x.

The angles of

\triangle ADE

are

x,

140^\circ,

and

z,

x+140^\circ+3x=180^\circ.

Hence

x=10^\circ.

Therefore, the correct answer is E.

Problem 12 in other years