Competition · AMC preparation · step 4 of 4

AMC 10 · 2007B · #7

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
equilateral-triangleangle-sum-triangleisosceles-triangle identify-subproblems ↑ Prerequisites: angle-sum-triangle
📏 Medium solution 💡 3 insights
Problem
In a convex pentagon ABCDE, all five sides have the same length, and the angles at A and B are both 90°. Find the degree measure of the angle at E.

Pick an answer.

(A)
90
(B)
108
(C)
120
(D)
144
(E)
150

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Angles are hard to see in words but easy to see in a picture, so Tool #1 (Draw a Diagram) comes first: sketch ABCDE and the two right angles at A and B. That drawing exposes two familiar shapes hiding inside — a square and an equilateral triangle. Then Tool #7 (Identify Subproblems) finishes the job: the angle at E is not one angle but two angles stacked together (a corner of the square plus a corner of the triangle), so find each piece separately and add them.

1STEP 1

Draw it and find a square

With AB flat and length 1, the 90° at A and at B send AE and BC straight up, so ABCE is a unit square.

AB = BC = EA = 1, AE ⊥ AB, BC ⊥ AB
2STEP 2

Spot the equilateral triangle

The square makes CE = 1, and the pentagon gives CD = DE = 1, so triangle CDE is equilateral.

EC = 1 = CD = DE → △ CDE equilateral
3STEP 3

Each triangle angle is 60

A triangle's angles sum to 180°, and here all three are equal, so ∠ DEC = 60°.

∠ DEC = 180°/3 = 60°
4STEP 4

Add the two pieces at E

EC splits the angle at E into the square's 90° and the triangle's 60°, so ∠ E = 150°, choice (E).

∠ E = ∠ DEC + ∠ CEA = 60° + 90° = 150° → (E)
Answer
150
Check against the total: the interior angles of any pentagon add to 540°. Here ∠ A = ∠ B = 90° and, by the left-right symmetry of the figure, ∠ C = ∠ E while ∠ D = ∠ CDE = 60° (the triangle's top). So 90 + 90 + 60 + 2∠ E = 540, giving 2∠ E = 300 and ∠ E = 150°, matching the answer. It is also convex (150° ＜ 180°) as required, and larger than 90°, which fits the picture where side ED leans well past straight up.
💡Key takeaway

Two right angles on equal sides build a square, the leftover equal sides build an equilateral triangle, and the corner angle at E is just those two pieces added: 90° + 60° = 150°.

  • Draw it and find a square
  • Spot the equilateral triangle
  • Each triangle angle is 60
  • Add the two pieces at E

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