AMC8 1999

AMC8 1999 · Q21

AMC8 1999 · Q21. It mainly tests Angle chasing, Triangles (properties).

The degree measure of angle A is

角 A 的度量是

(A) 20 20

(B) 30 30

(D) 40 40

(E) 45 45

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Since $\angle 1$ forms a straight line with angle $100^\circ$, $\angle 1 = 80^\circ$. Since $\angle 2$ forms a straight line with angle $110^\circ$, $\angle 2 = 70^\circ$. Angle $3$ is the third angle in a triangle with $\angle E = 40^\circ$ and $\angle 2 = 70^\circ$, so $\angle 3 = 180^\circ - 40^\circ - 70^\circ = 70^\circ$. Angle $4 = 110^\circ$ since it forms a straight angle with $\angle 3$. Then $\angle 5$ forms a straight angle with $\angle 4$, so $\angle 5 = 70^\circ$. (Or $\angle 3 = \angle 5$ because they are vertical angles.) Therefore, $\angle A = 180^\circ - \angle 1 - \angle 5 = 180^\circ - 80^\circ - 70^\circ = 30^\circ$.

答案（B）：由于$\angle 1$与$100^\circ$构成一直线角，所以$\angle 1 = 80^\circ$。由于$\angle 2$与$110^\circ$构成一直线角，所以$\angle 2 = 70^\circ$。角$3$是三角形中的第三个角，其中$\angle E = 40^\circ$且$\angle 2 = 70^\circ$，所以$\angle 3 = 180^\circ - 40^\circ - 70^\circ = 70^\circ$。因为$\angle 4$与$\angle 3$构成一直线角，所以$\angle 4 = 110^\circ$。接着$\angle 5$与$\angle 4$构成一直线角，因此$\angle 5 = 70^\circ$。（或者$\angle 3 = \angle 5$，因为它们是对顶角。）因此，$\angle A = 180^\circ - \angle 1 - \angle 5 = 180^\circ - 80^\circ - 70^\circ = 30^\circ$。

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