Question

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

$80$

$90$

$100$

$110$

$120$

Solution:

Label the equilateral triangle $ABC.$ Each $60^\circ$ angle splits into three $20^\circ$ pieces. Take the outermost trisectors from $A$ and $B$ : they meet at base angles $\tfrac23 \cdot 60^\circ = 40^\circ,$ so the hexagon vertex there has angle $180^\circ - 2\cdot 40^\circ = 100^\circ.$ The innermost trisectors from $A$ and $B$ meet at base angles $20^\circ,$ giving apex $180^\circ - 2\cdot 20^\circ = 140^\circ,$ and by vertical angles that's the opposite hexagon angle. So the six angles alternate $100^\circ$ and $140^\circ.$ The smallest is $100^\circ.$ Therefore, the answer is C.

2025 AMC 10A Problem 6

Solution:

Problem 6 in Other Years