AMC 10 · 2017 · #11
Grade 8 geometry-2dPick an answer.
Two things are unknown — the missing angle and the number of sides — and the problem hands us only one number, 2017. Tool #4 (Introduce a Variable) names both: let x be the forgotten angle and n the number of sides, giving 2017 + x = (n-2) · 180. One equation with two unknowns normally has many solutions, so the equation alone is not enough. The rescue is the word convex: it forces 0 ＜ x ＜ 180, which is exactly Tool #14 (Extreme Principle) — pushing x to its two boundaries traps the true angle sum inside a window only 180 wide. Consecutive multiples of 180 are 180 apart, so such a window holds at most one of them. That converts the whole problem into Tool #3 (Eliminate Possibilities): list the nearby multiples of 180, throw out the ones outside the window, and one survivor is left.
Name the forgotten angle
The true total is the reported one plus the missing angle.
Putting the skipped angle back has to land exactly on a legal polygon total, and every legal total is a multiple of 180.
8.G.A.5Introduce A VariableLet convexity squeeze the total
Convexity keeps that angle below 180.
The multiples of 180 are rungs on a ladder spaced 180 apart, so an opening only 180 wide cannot show two rungs at once.
The legal totals are rungs on a ladder spaced a straight angle apart, so a window that narrow shows only one.
▸ Why?
A polygon splits into triangles, each contributing a straight angle, so the totals climb by that fixed step.
▸ Why?
An evenly spaced ladder cannot show two rungs inside a gap narrower than its own step.
Find the one multiple of 180 inside
Only one valid total fits that window.
Dividing 2017 by 180 tells you which rung of the ladder you are standing under, and only the next rung up fits in the window.
6.NS.B.2Eliminate PossibilitiesSubtract to recover the angle
Subtracting gives 143, choice (D).
What she counted plus what she skipped equals the true total, so the skipped angle is just the difference.
8.EE.C.7Introduce A VariableA polygon's angles always add to a multiple of 180, and every angle of a convex polygon is under 180 — so the forgotten angle is exactly the distance from 2017 up to the next multiple of 180.
- Name the forgotten angle
- Let convexity squeeze the total
- Find the one multiple of 180 inside
- Subtract to recover the angle