1990 AMC 12 第 3 题

Question

一个梯形的四个连续内角构成等差数列。若最小角为

75^\circ

，则最大角为

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

将四个角写成

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

Write the four angles as

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

利用四边形的内角和

Use the angle sum of a quadrilateral

四个角的和为

360^\circ

，因此

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

所以

d=10

，最大角为

75+3(10)=105^\circ

。

所以正确答案是 C。

The four angles sum to

360^\circ,

so

\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.

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