Problem 10 · 2015 Math Kangaroo
Easy
Geometry & Measurement
sum-constraintcasework
A pentagon is called convex if all its internal angles are less than 180°. The number of right angles in a convex pentagon is n. Which of the following lists is a complete listing of all possible values of n?
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Answer: C — 0, 1, 2, 3
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Hint 1 of 2
The five interior angles of a pentagon add to 540°, and each must stay below 180°.
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Hint 2 of 2
How many 90° angles can you use while the remaining angles each stay under 180°?
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Approach: use the angle-sum bound
- A convex pentagon's angles sum to 540°, each < 180°.
- 0, 1, 2, or 3 right angles all leave the rest achievable (e.g. 3 right angles leave 270° over two angles, each < 180°).
- 4 right angles would force the 5th to be 540 − 360 = 180°, not allowed.
- So the complete list is 0, 1, 2, 3.
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