AMC 10 · 2025 · #6

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
equilateral-triangleisosceles-triangleangle-sum-trianglepolygon-angle-sum symmetry-argument ↑ Prerequisites: equilateral-triangleangle-sum-triangle
📏 Medium solution 💡 3 insights
Problem
An equilateral triangle has each of its three corners split into three equal slices by two rays. At every corner keep only the middle slice, a 20-degree wedge. The three middle wedges overlap in a convex hexagon. Find the smallest interior angle of that hexagon.

Pick an answer.

(A)
80
(B)
90
(C)
100
(D)
110
(E)
120

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The words hide a picture, so the first job is to draw the trisecting rays and see where the three middle wedges overlap. Once the hexagon is drawn, the triangle's 120-degree rotational symmetry shows its angles must repeat in a pattern, which cuts six unknown angles down to two. One angle comes from a tiny triangle pinched between the triangle's side and two rays (a clean subproblem), and the last angle drops out of the hexagon's total-angle equation once we name the two repeating values.

1STEP 1

Draw the trisected corners

Draw the triangle, cut each 60-degree corner with two rays into 20-degree slices, and shade the middle wedges — they overlap in the hexagon.

60° = 20° + 20° + 20°
2STEP 2

Use the 120-degree symmetry

A 120-degree turn about the center maps the hexagon onto itself, so its angles alternate: three large b and three small s.

angles = b, s, b, s, b, s (three b, three s)
3STEP 3

Pinch a small triangle at a side

On side AB the inner rays lean in 20 degrees from each end and meet at P, so angle APB is 180 - 20 - 20 = 140 degrees.

∠ APB = 180° - 20° - 20° = 140°
4STEP 4

Read off the larger hexagon angle

The hexagon lies past P, so its angle there is the vertical angle of APB: b = 140 degrees, and symmetry gives three such angles.

b = 140° (vertical angle of ∠ APB)
5STEP 5

Solve for the smaller angle

A hexagon's angles total 720 degrees, so 3(140) + 3s = 720 gives s = 100 degrees, the smallest angle — choice (C).

3(140°) + 3s = 720° → 3s = 300° → s = 100°
Answer
100
The six angles read 140, 100, 140, 100, 140, 100, which sum to 3(140) + 3(100) = 720 degrees, exactly the required total for a hexagon, so the count is consistent. Their average is 120 degrees, matching a regular hexagon; the trisection just stretches that regular shape into alternating 140 and 100 corners. The tempting answer 120 is the average, not the smallest, and 100 is safely below 140, so (C) is the genuine minimum.
💡Key takeaway

Draw the rays to reveal the hexagon, use the triangle's one-third-turn symmetry to shrink six angles to two, and let the 720-degree total finish the job.

  • Draw the trisected corners
  • Use the 120-degree symmetry
  • Pinch a small triangle at a side
  • Read off the larger hexagon angle
  • Solve for the smaller angle

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