AMC10 2025 A

AMC10 2025 A · Q6

AMC10 2025 A · Q6. It mainly tests Angle chasing, Triangles (properties).

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?

在一个等边三角形中，每个内角被一对射线三等分。每个顶点处中间20°角内部的交集是一个凸六边形的内部。这个六边形的最小内角的度量是多少度？

(A) 80 80

(B) 90 90

(D) 110 110

(E) 120 120

Answer

Correct choice: (C)

正确答案：（C）

Solution

Assume you have a diagram in front of you. Because each angle of the triangle is trisected, we have 9 $20^\circ$ angles. Using a side of the triangle as a base, we have an isosceles triangle with two $20^\circ$ angles. Using this we can show that the third angle is $140^\circ$. Following that, we use the principle of vertical angles to show that one angle of the hexagon is $140^\circ$. And with rotational symmetry, three. The average of all 6 angles has to be $120^\circ$, so the answer is $\boxed{\textbf{(C) }100}$

假设你面前有示意图。因为每个角被三等分，我们有9个$20^\circ$角。以三角形的一条边为底，我们有一个有两个$20^\circ$角的等腰三角形。利用这个，我们可以证明第三个角是$140^\circ$。然后，我们利用对顶角原理证明六边形的一个角是$140^\circ$。并且由于旋转对称，有三个这样的角。所有6个角的平均值必须是$120^\circ$，因此答案是$\boxed{\textbf{(C) }100}$

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