AMC 8 2009 Problem 19

AMC8 2009 年第 19 题

三角形★★★☆☆

Two angles of an isosceles triangle measure

70^\circ

and

x^\circ

. What is the sum of the three possible values of

x

一个等腰三角形其中两个角的度数是 70°和 x°。那么

x

的 3 个可能值之和是多少？

A.95
B.125
C.140
D.165
E.180

正确答案 · Correct Answer

解析 · Solution

Solution 1

There are 3 cases: where

x^\circ

is a base angle with the

70^\circ

as the other angle, where

x^\circ

is a base angle with

70^\circ

as the vertex angle, and where

x^\circ

is the vertex angle with

70^\circ

as a base angle. Case 1:

x^\circ

is a base angle with

70^\circ

as the other angle: Here, x=70, since base angles are congruent. Case 2:

x^\circ

is a base angle with

70^\circ

as the vertex angle: Here, the 2 base angles are both

x^\circ

, so we can use the equation 2x+70=180, which simplifies to x=55. Case 3:

x^\circ

is the vertex angle with

70^\circ

as a base angle: Here, both base angles are

70^\circ

, since base angles are congruent. Thus, we can use the equation x+140=180, which simplifies to x=40. Adding up all the cases, we get 70+55+40=165, so the answer is 165.

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