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Difficulty: HardPolygon Angles and Properties

An irregular convex hexagon has three interior angles that each measure 110∘110^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the degree measure of the exterior angle corresponding to the smallest interior angle of this hexagon?

  1. A
    17.5∘17.5^\circ
  2. B
    27.5∘27.5^\circ
  3. C
    50.0∘50.0^\circ
  4. 82.5∘82.5^\circAnswer
  5. E
    97.5∘97.5^\circ

Answer

82.5∘82.5^\circ
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the three known angles (330∘330^\circ) leaves 390∘390^\circ for the remaining three angles. Using the ratio 3:4:53:4:5, the smallest of these remaining angles is calculated as 312×390∘=97.5∘\frac{3}{12} \times 390^\circ = 97.5^\circ. Since 97.5∘97.5^\circ is smaller than the other angles (110∘110^\circ, 130∘130^\circ, 162.5∘162.5^\circ), it is the smallest interior angle of the hexagon. The exterior angle is supplementary to the interior angle, so it measures 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.

Step-by-Step Solution

1
Calculate the sum of all interior angles of the hexagon.
The sum is (6−2)×180∘=4×180∘=720∘(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
This establishes the total measure of the interior angles of a hexagon.
2
Subtract the sum of the three known interior angles from the total sum.
The remaining sum is 720∘−(3×110∘)=720∘−330∘=390∘720^\circ - (3 \times 110^\circ) = 720^\circ - 330^\circ = 390^\circ.
This determines the combined measure of the three remaining angles.
3
Determine the measures of the remaining three angles using the ratio.
Setting up the equation 3y+4y+5y=390∘3y + 4y + 5y = 390^\circ gives 12y=390∘12y = 390^\circ, or y=32.5∘y = 32.5^\circ. The individual angles are 3×32.5∘=97.5∘3 \times 32.5^\circ = 97.5^\circ, 4×32.5∘=130∘4 \times 32.5^\circ = 130^\circ, and 5×32.5∘=162.5∘5 \times 32.5^\circ = 162.5^\circ.
This finds each of the remaining interior angle measures.
4
Find the smallest interior angle and compute its supplementary exterior angle.
The smallest interior angle of the hexagon is 97.5∘97.5^\circ. The corresponding exterior angle is 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.
An interior angle and its adjacent exterior angle sum to 180∘180^\circ.

Key Concept

Polygon Angles and Properties
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Topics

Polygon Angles and Properties

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