Difference between revisions of "1973 AHSME Problems/Problem 16"
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==See Also== | ==See Also== | ||
| − | {{AHSME | + | {{AHSME 30p box|year=1973|num-b=15|num-a=17}} |
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 13:00, 20 February 2020
Problem
If the sum of all the angles except one of a convex polygon is 
, then the number of sides of the polygon must be
Solution
Let 
be the number of sides in the polygon. The number of interior angles in the polygon is 
. We know that the sum of all but one of them is , so the sum of all the angles is more than that.
![\[180(n-2) > 2190\]](http://latex.artofproblemsolving.com/9/d/9/9d9b325fe94f5d074666d71083d28048ab9c57b1.png)
![\[n-2 > 12 \tfrac{1}{6}\]](http://latex.artofproblemsolving.com/6/9/0/6901979cf1e8765a880ea98b13195f03b41e98ca.png)
![\[n > 14 \tfrac{1}{6}\]](http://latex.artofproblemsolving.com/b/6/4/b64802790b73d74af941cafea28476e7ffca7299.png)
The sum of the angles in a 15-sided polygon is 
, making the remaining angle 
. The angles of a convex polygon are all less than 
, and since adding one more side means adding to the measure of the remaining angle, we can confirm that there are 
sides in the polygon.
See Also
| 1973 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
