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Difficulty: MediumPolygon Angles and Properties

A convex hexagon has interior angles with measures of 2x∘2x^\circ, (3x−15)∘(3x - 15)^\circ, (2x+25)∘(2x + 25)^\circ, (3x+10)∘(3x + 10)^\circ, (4x−50)∘(4x - 50)^\circ, and (x+30)∘(x + 30)^\circ. What is the degree measure of the smallest interior angle of this hexagon?

Answer: 78 degrees

Answer

The degree measure of the smallest interior angle is 78.
The sum of the interior angles of a hexagon is calculated as (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ. Adding the algebraic expressions for the six angles yields 15x=72015x = 720, which gives x=48x = 48. Substituting x=48x = 48 back into the expressions gives the angle measures of 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest angle is 78∘78^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of the hexagon.
Sum of interior angles = 720 degrees
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For a hexagon (n=6n = 6), the sum is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ.
2
Set up an algebraic equation by summing all the given angle expressions and equating them to 720.
2x+(3x−15)+(2x+25)+(3x+10)+(4x−50)+(x+30)=7202x + (3x - 15) + (2x + 25) + (3x + 10) + (4x - 50) + (x + 30) = 720
The sum of the actual measures of the interior angles must equal the calculated sum of 720 degrees.
3
Combine like terms to simplify the equation.
15x=72015x = 720
Grouping the xx terms (2x+3x+2x+3x+4x+x=15x2x + 3x + 2x + 3x + 4x + x = 15x) and the constant terms (−15+25+10−50+30=0-15 + 25 + 10 - 50 + 30 = 0) simplifies the expression.
4
Solve for xx.
x=48x = 48
Dividing both sides of the equation 15x=72015x = 720 by 15 isolates the variable xx.
5
Substitute the value of xx back into the angle expressions to identify the smallest angle.
The angles are 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest measure is 78∘78^\circ.
Evaluating each expression at x=48x = 48 determines the actual angle measures, from which the smallest can be chosen. Evaluating (x+30)∘(x + 30)^\circ gives 48+30=78∘48 + 30 = 78^\circ, which is the minimum value.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Setting up and solving linear algebraic equations is required to determine unknown angle measures.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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