The degree measure of angle A is

AMC 8 1999 Problem 21

AMC8 1999 年第 21 题

三角形★★★☆☆

The degree measure of angle

A

角

A

的度数是多少？

正确答案 · Correct Answer

Solution 1

Angle-chasing using the small triangles:

Use the line below and to the left of the

110^\circ

angle to find that the rightmost angle in the small lower-left triangle is

180 - 110 = 70^\circ

Then use the small lower-left triangle to find that the remaining angle in that triangle is

180 - 70 - 40 = 70^\circ

Use congruent vertical angles to find that the lower angle in the smallest triangle containing

A

is also

70^\circ

Next, use line segment

AB

to find that the other angle in the smallest triangle containing

A

180 - 100 = 80^\circ

The small triangle containing

A

has a

70^\circ

angle and an

80^\circ

angle. The remaining angle must be

180 - 70 - 80 = 30^\circ, B

The third angle of the triangle containing the

100^\circ

angle and the

40^\circ

angle is

180^\circ - 100^\circ - 40^\circ = 40^\circ

. It follows that

A

is the third angle of the triangle consisting of the found

40^\circ

angle and the given

110^\circ

angle. Thus,

A

is a

180^\circ - 110^\circ - 40^\circ = 30^\circ

angle, and so the answer is

30^\circ, \textbf{B}

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