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Difficulty: HardPolygon Angles and Properties

For a certain convex polygon with nn sides, the sum of the measures of its interior angles is exactly 2424 times the measure of one exterior angle of a regular polygon with nn sides. If this polygon is regular, what is the measure, in degrees, of each of its interior angles?

  1. A
    45
  2. B
    120
  3. 135Answer
  4. D
    165
  5. E
    1080

Answer

135
By translating the problem description into an algebraic relationship, we establish that the sum of the interior angles, (n−2)×180∘(n-2) \times 180^\circ, equals 2424 times the measure of a single exterior angle, 360∘n\frac{360^\circ}{n}. Simplifying this equation by dividing both sides by 180180 yields n−2=48nn - 2 = \frac{48}{n}. Multiplying by nn creates the quadratic equation n2−2n−48=0n^2 - 2n - 48 = 0, which factors as (n−8)(n+6)=0(n-8)(n+6) = 0. Since the number of sides of a polygon must be positive, n=8n = 8. For a regular octagon (n=8n=8), the measure of each interior angle is (8−2)×180∘8=135∘\frac{(8-2) \times 180^\circ}{8} = 135^\circ.

Step-by-Step Solution

1
Set up the equation based on the geometric properties of polygons.
(n−2)×180=24×360n(n-2) \times 180 = 24 \times \frac{360}{n}
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. The measure of one exterior angle of a regular polygon with nn sides is 360∘n\frac{360^\circ}{n}.
2
Simplify the equation and solve for the number of sides nn.
n2−2n−48=0n^2 - 2n - 48 = 0, which gives n=8n = 8.
Dividing both sides by 180180 simplifies the equation to n−2=48nn - 2 = \frac{48}{n}. Multiplying by nn yields the quadratic equation n(n−2)=48n(n - 2) = 48, or n2−2n−48=0n^2 - 2n - 48 = 0. Factoring gives (n−8)(n+6)=0(n - 8)(n + 6) = 0. Since the number of sides must be a positive integer, n=8n = 8.
3
Calculate the measure of each interior angle of a regular octagon (n=8n = 8).
135∘135^\circ
The measure of each interior angle of a regular polygon is given by (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}. Substituting n=8n = 8 yields (8−2)×180∘8=6×180∘8=135∘\frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ.

Key Concept

The relationship between the sum of interior angles, the number of sides, and the exterior angles of regular polygons.
Estimated Time:2m 30s
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Topics

Polygon Angles and Properties

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