AMC 10 · 2002 · #5

Grade 8 algebrageometry-2d

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

angle-sum-polygonsequences-arithmeticlinear-equations-one-var convert-to-algebrasymmetry-argument ↑ Prerequisites: angle-sum-polygonsequences-arithmetic

📏 Medium solution 💡 2 insights

Problem

A pentagon has five interior angles, measured in degrees. All five are different, and written from smallest to largest they are v ＜ w ＜ x ＜ y ＜ z. In that order they form an arithmetic sequence, meaning each angle is a fixed amount larger than the one before it. Find the middle angle x.

Pick an answer.

(A)

(B)

(C)

(D)

108

(E)

120

How to solve

Strategy Introduce a Variable

Two separate facts have to meet: the five angles add to a fixed total, and they are evenly spaced. Tool #1 (Draw a Diagram) supplies the total — cutting the pentagon into triangles from one vertex converts a pentagon question into a triangle question already known. Tool #4 (Introduce a Variable) supplies the spacing, and here the choice of variable is the whole trick: naming the terms around the middle one, as x-2d through x+2d, rather than around the first one. Tool #15 (Organize Information in More Ways) then adds the five terms by pairing the outer ones inward instead of left to right, which makes every d cancel and leaves an equation in x alone. The last thing to check is one the problem invites you to skip: that some legal d actually exists, so the pentagon being described is a real pentagon and not an empty description.

1STEP 1

Cut the pentagon into three triangles

Cutting the pentagon into three triangles from one vertex gives a total of 540°.

v+w+x+y+z = (5-2) · 180° = 3 · 180° = 540°

2STEP 2

Name the middle angle, not the first

Centre the sequence on the asked angle: x-2d, x-d, x, x+d, x+2d.

v = x-2d, w = x-d, y = x+d, z = x+2d, d ＞ 0

3STEP 3

Pair the outer terms inward

Pairing the ends inward cancels every d, so the five terms total 5x.

(x-2d)+(x+2d) = 2x, (x-d)+(x+d) = 2x, v+w+x+y+z = 2x + 2x + x = 5x

4STEP 4

Solve, then check a pentagon exists

So 5x = 540 gives x = 108, and d = 10 shows such a pentagon really exists, choice (D).

5x = 540 → x = 108. d=10: 88+98+108+118+128 = 540 ✓ → (D)

Answer

108

108° is exactly the interior angle of a regular pentagon, and that is the right sanity check: x is the average of the five angles, and averaging does not care how spread out the angles are, so the spread-out pentagon must have the same middle angle as the perfectly even one. The wrong choices are all recognisable strays — 72 is the exterior angle of a regular pentagon, 90 is the angle of a square, 120 is the interior angle of a regular hexagon, and 84 is what you get from mistakenly using 5 · 84 = 420. A second check on the size: x must sit strictly between v and z, and v = 108-2d stays positive while z = 108+2d stays below 180 for every d in the working range, so nothing about the answer is on the edge of being impossible.

💡Key takeaway

When an odd number of values are evenly spaced, the middle one is the average — so the total alone tells you the middle, no matter how far apart the values are spread.

Cut the pentagon into three triangles
Name the middle angle, not the first
Pair the outer terms inward
Solve, then check a pentagon exists

Cut the pentagon into three triangles

Name the middle angle, not the first

Pair the outer terms inward

Solve, then check a pentagon exists

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