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Difficulty: MediumPolygon Angles and Properties

A convex hexagon has four interior angles that measure 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ. The remaining two interior angles are in the ratio 3:53:5. What is the degree measure of the largest interior angle of this hexagon?

  1. A
    90∘90^\circ
  2. B
    140∘140^\circ
  3. C
    144∘144^\circ
  4. 150∘150^\circAnswer
  5. E
    160∘160^\circ

Answer

The degree measure of the largest interior angle is 150∘150^\circ.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the four given angles (100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ) from 720∘720^\circ leaves a remaining sum of 240∘240^\circ. Since the two remaining angles are in the ratio 3:53:5, they can be written as 3x3x and 5x5x, where 3x+5x=240∘3x + 5x = 240^\circ. Solving for xx gives 8x=240∘  ⟹  x=30∘8x = 240^\circ \implies x = 30^\circ. The two remaining angles are 90∘90^\circ and 150∘150^\circ. Comparing all six interior angles of the hexagon, the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a hexagon.
The sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ.
The sum of the interior angles of any convex nn-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Find the sum of the four given interior angles.
100∘+115∘+125∘+140∘=480∘100^\circ + 115^\circ + 125^\circ + 140^\circ = 480^\circ.
This determines the total measure of the known angles.
3
Determine the remaining sum of the two unknown angles.
720∘−480∘=240∘720^\circ - 480^\circ = 240^\circ.
Subtracting the sum of the known angles from the total sum yields the combined measure of the remaining two angles.
4
Set up an equation using the ratio 3:53:5 to find the measures of the two remaining angles.
3x+5x=240∘  ⟹  8x=240∘  ⟹  x=30∘3x + 5x = 240^\circ \implies 8x = 240^\circ \implies x = 30^\circ. The two angles are 3(30∘)=90∘3(30^\circ) = 90^\circ and 5(30∘)=150∘5(30^\circ) = 150^\circ.
Representing the two angles in terms of a common variable xx allows us to solve for their individual measures.
5
Identify the largest interior angle among all six angles of the hexagon.
The largest angle is 150∘150^\circ.
Comparing all six angles (90∘90^\circ, 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, 140∘140^\circ, and 150∘150^\circ), the maximum value is 150∘150^\circ.

Key Concept

Calculating the interior angle sum of a polygon and solving for unknown angles using ratios.
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Topics

Polygon Angles and Properties

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