1999 AHSME Problems/Problem 7
Problem
What is the largest number of acute angles that a convex hexagon can have?
Solution
The sum of the interior angles of a hexagon is degrees. In a convex polygon, each angle must be strictly less than degrees.
Six acute angles can only sum to less than degrees, so six acute angles could not form a hexagon.
Five acute angles and one obtuse angle can only sum to less than degrees, so these angles could not form a hexagon.
Four acute angles and two obtuse angles can only sum to less than degrees. This is a strict inequality, so these angles could not form a hexagon. (The limiting figure would be four right angles and two straight angles, which would really be a square with two "extra" points on two sides to form the straight angles.)
Three acute angles and three obtuse angles work. For example, if you pick three acute angles of degrees, the three obtuse angles would be degrees and give a sum of degrees, which is a genuine hexagon. Thus, the answer is
See Also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.