AMC 10 · 2009 · #24

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
angle-sum-triangleisosceles-trianglesupplementary-angles physical-representation ↑ Prerequisites: angle-sum-triangle
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A keystone arch is built from 9 congruent isosceles trapezoids joined leg-to-leg. Together they curve from the ground on one side, over the top, back down to the ground on the other side. The bottom sides of the two end trapezoids rest flat on the ground. Each trapezoid has two equal (larger) interior angles and two equal (smaller) interior angles. Find the measure x, in degrees, of the larger interior angle.

Pick an answer.

(A)
100
(B)
102
(C)
104
(D)
106
(E)
108

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The picture hides a clean triangle. If you extend the two legs of every trapezoid (Tool #1, an auxiliary construction), symmetry forces all of them to meet at one shared point O in the middle of the base line, and each trapezoid becomes a thin triangle with its tip cut off at O. Tool #17 (Visualize Spatial Relationships) sees that the 9 tip-triangles fan out to fill the half-turn above the base, so each tip angle is a fair share of 180°. Tool #7 (Identify Subproblems) then chains three easy facts: tip angle → base angles of the isosceles tip-triangle → the trapezoid's larger angle by the same-side-interior rule.

1STEP 1

Extend the legs to a shared point

Extend every trapezoid's slanted legs outward; by mirror symmetry all the extensions meet at one point O in the middle of the base.

extend all legs → one shared tip O on the base line
2STEP 2

Share the half-turn among 9 tips

The arch turns from flat ground to flat ground — a half-turn of 180° — and the 9 equal tip angles at O share it evenly, so each is 20°.

9 × (tip angle) = 180° → tip angle = 180°/9 = 20°
3STEP 3

Find the trapezoid's smaller angle

The tip triangle at O is isosceles, so its two base angles split the leftover 160°: the trapezoid's smaller angle is 80°.

(180° - 20°)/2 = 160°/2 = 80°
4STEP 4

Turn the smaller angle into the larger angle

Along one leg the smaller and larger angles are same-side interior angles, so x = 180° - 80° = 100°, choice (A).

x = 180° - 80° = 100° → (A)
Answer
100
A quick sanity check: the smaller and larger angles of an isosceles trapezoid must add to 180° along each leg, and 80° + 100° = 180° works. The larger angle should be a bit more than a right angle because the pieces lean only slightly, and 100° is just past 90°, which matches a gently curving arch. If each larger angle were as big as 108° the smaller angle would be 72°, forcing a tip angle of 180° - 2(72°) = 36° and only 180° / 36° = 5 pieces per half circle, not 9. Choice (A) is the only one that fits 9 trapezoids.
💡Key takeaway

Slide the slanted sides until they meet at one center point, share the half-turn of 180° among the nine pieces, and the rest is just triangle and straight-line angles.

  • Extend the legs to a shared point
  • Share the half-turn among 9 tips
  • Find the trapezoid's smaller angle
  • Turn the smaller angle into the larger angle

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