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Difficulty: MediumPolygon Angles and Properties

An irregular convex heptagon (7-sided polygon) has three interior angles that each measure 140∘140^\circ. The remaining four interior angles have measures in the ratio 2:3:3:42:3:3:4. What is the measure of the largest interior angle of this heptagon?

  1. A
    80∘80^\circ
  2. B
    120∘120^\circ
  3. C
    140∘140^\circ
  4. 160∘160^\circAnswer
  5. E
    300∘300^\circ

Answer

160∘160^\circ
The total sum of the interior angles of a convex heptagon is (7−2)×180∘=900∘(7-2) \times 180^\circ = 900^\circ. Subtracting the three angles that each measure 140∘140^\circ (3×140∘=420∘3 \times 140^\circ = 420^\circ) leaves 480∘480^\circ for the remaining four angles. Using the ratio 2:3:3:42:3:3:4, we represent the angles as 2x2x, 3x3x, 3x3x, and 4x4x, giving the equation 2x+3x+3x+4x=480∘2x + 3x + 3x + 4x = 480^\circ, which simplifies to 12x=480∘12x = 480^\circ and yields x=40∘x = 40^\circ. The largest of these four angles is 4x=4(40∘)=160∘4x = 4(40^\circ) = 160^\circ. Since 160∘160^\circ is larger than 140∘140^\circ, the largest interior angle of the heptagon is 160∘160^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a heptagon.
900∘900^\circ
The sum of the interior angles of a convex polygon with nn sides is given by (n−2)×180∘(n-2) \times 180^\circ. For a heptagon (n=7n=7), the sum is (7−2)×180∘=5×180∘=900∘(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ.
2
Find the sum of the three known interior angles.
420∘420^\circ
Since three interior angles each measure 140∘140^\circ, their sum is 3×140∘=420∘3 \times 140^\circ = 420^\circ.
3
Calculate the sum of the remaining four interior angles.
480∘480^\circ
Subtracting the sum of the three known angles from the total interior sum gives 900∘−420∘=480∘900^\circ - 420^\circ = 480^\circ.
4
Set up and solve an equation for the remaining four angles using the given ratio.
x=40∘x = 40^\circ
Let the measures of the remaining four angles be 2x2x, 3x3x, 3x3x, and 4x4x. Their sum is 2x+3x+3x+4x=12x2x + 3x + 3x + 4x = 12x. Setting this equal to the remaining sum gives 12x=480∘12x = 480^\circ, which simplifies to x=40∘x = 40^\circ.
5
Determine the measures of the remaining angles and identify the largest angle.
160∘160^\circ
The remaining angles measure 2(40∘)=80∘2(40^\circ) = 80^\circ, 3(40∘)=120∘3(40^\circ) = 120^\circ, 3(40∘)=120∘3(40^\circ) = 120^\circ, and 4(40∘)=160∘4(40^\circ) = 160^\circ. The largest of these is 160∘160^\circ. Comparing this with the three 140∘140^\circ angles, the largest interior angle of the entire heptagon is 160∘160^\circ.

Key Concept

Interior angle sum of polygons and ratio division
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Topics

Polygon Angles and Properties

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