2025 AMC 10A Problems/Problem 6

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?

$\textbf{(A) } 80 \qquad\textbf{(B) } 90 \qquad\textbf{(C) } 100 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Solution 1

Assume you have a diagram in front of you.

Because each angle of the triangle is trisected, we have 9 20° angles. Using a side of the triagle as a base, we have an isosceles triangle with two 20° angles. Using this we can show that the third angle is 140°.

Following that, we use the vertex angles to show that one angle of the hexagon is 140°. And with rotational symmetry, three. The average of all 6 angles has to be 120°, so the answer is $\boxed{\textbf{(C) }100}$ - SpectralScholar

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2025 AMC 10A Problems/Problem 6

Solution 1