1999 AMC 12 Problem 7

Below is the professionally curated solution for Problem 7 of the 1999 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.

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Concepts:angle sum bounding to limit cases

Difficulty rating: 1310

What is the largest number of acute angles that a convex hexagon can have?

$2$

$3$

$4$

$5$

$6$

Solution:

Each acute interior angle corresponds to an exterior angle greater than $90^\circ.$ Since the exterior angles of a convex polygon sum to $360^\circ,$ at most three of them can exceed $90^\circ.$ Hence there are at most three acute angles, and a hexagon achieving three acute angles exists.

Thus, the correct answer is B.

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