AMC 10 · 2007 · #7
Grade 8 geometry-2dPick an answer.
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.
Draw it and find a square
The two right angles and equal sides carve out a square.
Two equal sides both standing at right angles on the same base close up into a square.
4.G.A.1Draw A DiagramSpot the equilateral triangle
Its diagonal matches the last two sides, giving an equilateral triangle.
Three sides all the same length can only make an equilateral triangle.
4.G.A.2Draw A DiagramEach triangle angle is 60
Each of its angles is 60.
Equal sides force equal angles, and three equal angles splitting 180° are 60° each.
Equal sides force equal angles, and three equal angles sharing a straight angle are sixty degrees each.
▸ Why?
A triangle with equal sides is its own mirror image, so the angles facing those sides must match.
▸ Why?
The three angles of a triangle always add to a straight angle, so three equal ones are a third of it each.
Add the two pieces at E
Adding the two pieces at that vertex gives 150, choice (E).
Two angles sharing a ray combine by simple addition, just like joining two slices of a pie.
4.MD.C.7Identify SubproblemsTwo 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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