Problem 27 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
sum-constraintcasework
In the quadrilateral PQRS, PQ = 2006, QR = 2008, RS = 2007 and SP = 2009. At which corners must the interior angle definitely be smaller than 180°?
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Answer: D — P, R, S
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Hint 1 of 2
A simple quadrilateral has at most one reflex (>180) angle - find which vertex could be it.
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Hint 2 of 2
A vertex folds inward only if its two sides together are shorter than the other two together.
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Approach: find the only vertex that could be reflex
- A vertex can be reflex only if the two sides meeting there sum to less than the other two sides.
- At Q: 2006+2008 = 4014 < 2009+2007 = 4016, so only Q might be reflex; at P, R, S the two sides are not shorter than the rest, so those angles are under 180.
- The corners definitely below 180 degrees are P, R and S.
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