AMC 10 · 2025 · #6
Grade 8 geometry-2dIn 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?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: The triangle is equilateral, so each interior angle is 60 degrees.; Each 60-degree angle is trisected, making three equal 20-degree slices at every vertex.; At each vertex we take the middle 20-degree wedge (the slice between the two trisecting rays).; The region shared by all three middle wedges is a convex hexagon.
Unknowns: The degree measure of the smallest interior angle of the hexagon.
Understand
Restated: 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.
Givens: The triangle is equilateral, so each interior angle is 60 degrees.; Each 60-degree angle is trisected, making three equal 20-degree slices at every vertex.; At each vertex we take the middle 20-degree wedge (the slice between the two trisecting rays).; The region shared by all three middle wedges is a convex hexagon.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #5 Look for a Pattern, #7 Identify Subproblems, #4 Introduce a Variable
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.
Execute — Answer: C
4.MD.C.7 Step 1 Draw the trisected corners
- Sketch the equilateral triangle and, at each 60-degree corner, draw the two rays that cut it into three equal 20-degree slices.
- Shade the middle slice at each vertex.
- The three shaded wedges cross in the center, and their common overlap is the hexagon we care about.
💡 You cannot see the hexagon until you draw the rays, and each corner just splits into three equal 20-degree pieces.
8.G.A.1 Step 2 Use the 120-degree symmetry
- Turning the whole figure 120 degrees about the triangle's center sends each vertex to the next and maps the hexagon exactly onto itself.
- So the six interior angles cannot all be different: they repeat in a pattern, alternating between a larger value and a smaller value, with three copies of each.
💡 A shape that looks identical after a one-third turn must have its angles come in a repeating threefold pattern.
8.G.A.5 Step 3 Pinch a small triangle at a side
- Look at one side, say AB.
- The ray from A that borders its middle wedge makes 20 degrees with AB, and the matching ray from B makes 20 degrees with BA.
- These two rays meet at a point P, forming a small triangle APB with base angles of 20 degrees each.
- Its top angle is 180 - 20 - 20 = 140 degrees.
💡 Two rays leaning in at 20 degrees each leave 140 degrees for the angle where they meet, since a triangle's angles total 180.
7.G.B.5 Step 4 Read off the larger hexagon angle
- The hexagon lies on the far side of P, and its two edges at P are the same rays continued straight past P.
- The hexagon's angle at P is therefore the vertical angle of angle APB, so it equals 140 degrees.
- By the threefold symmetry, three of the hexagon's angles are 140 degrees.
- This is the larger value b.
💡 Two straight rays crossing at P make equal angles on opposite sides, so the hexagon corner matches the 140-degree triangle corner.
8.G.A.5 Step 5 Solve for the smaller angle
- A convex hexagon's interior angles always sum to (6 - 2) times 180 = 720 degrees.
- With three angles of 140 degrees and three of the smaller value s, we get 3(140) + 3s = 720, so 420 + 3s = 720 and s = 100 degrees.
- Since 100 is less than 140, the smallest angle of the hexagon is 100 degrees, which is choice (C).
💡 Once you know three of the six angles, the fixed total of 720 degrees forces the other three.
4.MD.C.7 Sketch the equilateral triangle and, at each 60-degree corner, draw the two rays 8.G.A.1 Turning the whole figure 120 degrees about the triangle's center sends each vert 8.G.A.5 Look at one side, say AB. The ray from A that borders its middle wedge makes 20 7.G.B.5 The hexagon lies on the far side of P, and its two edges at P are the same rays 8.G.A.5 A convex hexagon's interior angles always sum to (6 - 2) times 180 = 720 degrees Review
Reasonableness: 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.
Alternative: Instead of the angle-sum equation, find the smaller angle directly. At an inner hexagon vertex, two rays from different corners cross; tracing the small triangle they form with a piece of the figure gives base angles that leave a 100-degree opening. Both routes agree that the two angle types are 140 and 100, so the smallest is 100.
CCSS standards used (min grade 8)
4.MD.C.7Recognize angle measure as additive and solve addition and subtraction problems (Splitting each 60-degree corner into three equal 20-degree slices, since 20 + 20 + 20 = 60.)8.G.A.1Verify experimentally the properties of rotations, reflections, and translations (Using the 120-degree rotational symmetry to conclude the hexagon's angles repeat, giving three copies each of two values.)8.G.A.5Use informal arguments to establish facts about angle sum and exterior angles (Getting 140 degrees from the small triangle's angle sum and using the 720-degree hexagon total to solve for the smaller angle.)7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles (Turning the small triangle's 140-degree top angle into the hexagon's angle at that vertex through vertical angles.)
⭐ 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 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.
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