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Difficulty: MediumPolygon Angles and Properties

A convex pentagon has one interior angle that measures 108∘108^\circ. The remaining four interior angles have measures in the ratio 3:4:5:63:4:5:6. What is the measure of the largest interior angle of this pentagon?

  1. A
    84∘84^\circ
  2. B
    120∘120^\circ
  3. 144∘144^\circAnswer
  4. D
    162∘162^\circ
  5. E
    180∘180^\circ

Answer

The measure of the largest interior angle of the pentagon is 144∘144^\circ.
The correct answer is the option stating 144∘144^\circ. To find this, we calculate the sum of all interior angles of a pentagon using (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtracting the known angle of 108∘108^\circ leaves 432∘432^\circ for the remaining four angles. The ratio 3:4:5:63:4:5:6 implies these angles can be represented as 3x3x, 4x4x, 5x5x, and 6x6x, summing to 18x18x. Solving 18x=43218x = 432 gives x=24x = 24. The largest angle is 6x=6×24∘=144∘6x = 6 \times 24^\circ = 144^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a pentagon.
The sum of the interior angles of any convex pentagon (n=5n = 5) is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Before finding individual angles, we must determine the total sum of all interior angles in the polygon.
2
Determine the sum of the remaining four angles.
The remaining four angles sum to 540∘−108∘=432∘540^\circ - 108^\circ = 432^\circ.
We subtract the measure of the known angle from the total sum to find the portion allocated to the remaining angles.
3
Set up and solve an algebraic equation using the given ratio.
Let the four remaining angles be represented as 3x3x, 4x4x, 5x5x, and 6x6x. Their sum is 3x+4x+5x+6x=18x3x + 4x + 5x + 6x = 18x. Setting this equal to the remaining sum gives 18x=43218x = 432, which simplifies to x=24x = 24.
Using the ratio allows us to define the relative sizes of the remaining angles in terms of a single variable, which we can solve for using their sum.
4
Find the measure of the largest interior angle.
The largest of these four angles is represented by 6x6x, which is 6×24∘=144∘6 \times 24^\circ = 144^\circ. Since 144∘144^\circ is also larger than the given angle of 108∘108^\circ, it is the largest interior angle of the pentagon.
We multiply the value of xx by the largest coefficient in the ratio and compare it to the other given angle to identify the maximum measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For an irregular polygon, when some angles are known and others are in a ratio, we subtract the known angles from the total sum and distribute the remaining sum proportionally according to the ratio.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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