Problem 7 · AMC 8 Stretch
Stretch
Geometry & Measurement
considering-extreme-casesvisual-representation
Draw a five-pointed star (a pentagram). Add up the five sharp point angles at the tips. What is the total? (It comes out the same for every five-pointed star.)
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Answer: \(180^\circ\)
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Hint 1 of 4
The total is the same for every star, so you may pick a convenient star. But you can also find it with a fact you already know: the angles in a triangle add to \(180^\circ\), and an exterior angle of a triangle equals the sum of the two far interior angles.
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Hint 2 of 4
Each tip of the star is the top angle of a little triangle whose base is one side of the inner pentagon. Build up the five tip angles using triangle facts.
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Hint 3 of 4
Quick way: walk all the way around the star, turning at each of the \(5\) points. Going once around a star, you actually spin around twice, so the turning adds to \(2 \times 360^\circ = 720^\circ\). Each tip angle is \(180^\circ\) minus its turn.
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Approach: Considering a convenient case — total turning around the star
- Since every five-pointed star gives the same total, reason about a nice symmetric one. A clean method uses turning.
- Imagine an ant walking around the outline of the star and back to where it started, facing the same way. At each of the \(5\) sharp tips it makes a turn. For a star (unlike a normal pentagon) the ant spins around TWICE before getting home, so the turns add up to \(2 \times 360^\circ = 720^\circ\).
- At each tip, the turn is \(180^\circ\) minus the tip angle (the sharper the point, the bigger the turn). Adding over all \(5\) tips: \(\sum(180^\circ - \text{tip}) = 720^\circ\).
- That is \(5 \times 180^\circ - (\text{sum of tip angles}) = 720^\circ\), so \(900^\circ - (\text{sum of tip angles}) = 720^\circ\), giving sum of tip angles \(= 180^\circ\) for any five-pointed star.
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