Question

A convex, 11-sided polygon can have at most how many acute interior angles?

Note: Convex means that each interior angle measure is less than 180 degrees

Answer 1

Answer: At most 18 acute angles

Explanation:

The sum of the interior angles of an n-sided polygon is (n-2) × 180 degrees. In a convex polygon, each interior angle is less than 180 degrees.

Let a₁, a₂, ..., a₁₁ be the interior angles of the 11-sided polygon. Then the sum of the interior angles is:

a₁ + a₂ + ... + a₁₁ = (11-2) × 180 = 1620 degrees

Since each angle is acute, we know that each angle is less than 90 degrees. Let A be the number of acute angles. Then the sum of the acute angles is at most:

A × 90

So we have:

a₁ + a₂ + ... + a₁₁ ≤ A × 90

Substituting the sum of the interior angles, we get:

1620 ≤ A × 90

Solving for A, we get:

A ≤ 18

Therefore, the polygon can have at most 18 acute angles.