Question

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

$2$

$3$

$4$

$5$

$6$

Answer: B

Concepts:angle sum bounding to limit cases

Difficulty rating: 1310

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 $90^\circ.$ This bound is attainable: take a hexagon with equal side lengths and exterior angles alternating $100^\circ$ and $20^\circ.$ Its edge directions split into two triples $120^\circ$ apart, so it closes, and its interior angles alternate between $80^\circ$ and $160^\circ.$ Hence the largest possible number is three.

Thus, the correct answer is B.

1999 AMC 12 Problem 7

Answer: B

Difficulty rating: 1310

Solution:

Problem 7 in Other Years