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Difficulty: MediumQuadrilaterals and Polygons

An architect is designing a room in the shape of a convex pentagon. The measures of the interior angles of the room are defined algebraically as (x+20)∘(x + 20)^\circ, (x+10)∘(x + 10)^\circ, (x+5)∘(x + 5)^\circ, (x−5)∘(x - 5)^\circ, and (2x)∘(2x)^\circ. What is the measure of the smallest interior angle of this room?

  1. A
    10∘10^\circ
  2. B
    50∘50^\circ
  3. 80∘80^\circAnswer
  4. D
    90∘90^\circ

Answer

The smallest interior angle measures 80∘80^\circ.
A pentagon has 5 sides, so the sum of its interior angles is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Setting the sum of the algebraic expressions equal to 540∘540^\circ yields 6x+30=5406x + 30 = 540. Subtracting 3030 leaves 6x=5106x = 510, so x=85x = 85. Substituting 8585 into the smallest expression, (x−5)∘(x - 5)^\circ, gives 85−5=80∘85 - 5 = 80^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles for a pentagon.
Using the formula (n−2)×180∘(n - 2) \times 180^\circ for a 5-sided polygon (n=5n = 5), the sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Before solving for xx, the total sum that the algebraic expressions add up to must be known.
2
Set up an equation by adding all the given angle expressions.
(x+20)+(x+10)+(x+5)+(x−5)+2x=540(x + 20) + (x + 10) + (x + 5) + (x - 5) + 2x = 540
The sum of the individual interior angles must equal the total calculated sum of the polygon's angles.
3
Combine like terms and solve for xx.
6x+30=540⇒6x=510⇒x=856x + 30 = 540 \Rightarrow 6x = 510 \Rightarrow x = 85
Simplifying the algebraic equation isolates the variable xx.
4
Substitute x=85x = 85 back into each expression to find all angle measures and identify the smallest one.
The angles are 105∘105^\circ, 95∘95^\circ, 90∘90^\circ, 80∘80^\circ, and 170∘170^\circ. The smallest of these is 80∘80^\circ.
The variable xx alone is not the answer; it must be substituted back into the expressions to find the actual angle measures.

Key Concept

The sum of the interior angles of a convex polygon is (n−2)×180∘(n-2) \times 180^\circ. This geometric principle can be combined with algebraic equations to solve for unknown angle measures.
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Topics

Quadrilaterals and Polygons

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