1990 AMC 12 Problem 3

Attempt Problem 3 of the 1990 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 1990 AMC 12 solutions, or check the answer key.

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

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is

75^\circ,

then the largest angle is

95^\circ

100^\circ

105^\circ

110^\circ

115^\circ

Write the four angles as

75,75+d,75+2d,75+3d

Use the angle sum of a quadrilateral

The four angles sum to

360^\circ,

\begin{aligned} &75+(75+d)\\ &\quad +(75+2d)\\ &\quad +(75+3d)=360. \end{aligned}

Thus

d=10,

and the largest angle is

75+3(10)=105^\circ.

Thus the correct answer is C.

Problem 3 in other years