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Difficulty: MediumQuadrilaterals and Polygons

A convex hexagon has five interior angles that measure 120∘120^\circ, 135∘135^\circ, 115∘115^\circ, 140∘140^\circ, and 110∘110^\circ. If the measure of the sixth interior angle is represented by the algebraic expression (4x+16)∘(4x + 16)^\circ, what is the value of xx?

  1. A
    55
  2. 2121Answer
  3. C
    2929
  4. D
    111111

Answer

The correct answer is 2121.
The sum of the interior angles of a hexagon (66 sides) is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ. The five given angles add up to 620∘620^\circ. This means the sixth angle must be 720∘−620∘=100∘720^\circ - 620^\circ = 100^\circ. Setting the algebraic expression 4x+164x + 16 equal to 100100 and solving for xx by subtracting 1616 and dividing by 44 gives x=21x = 21.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
The sum is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ.
The formula for the sum of interior angles is (n−2)×180∘(n - 2) \times 180^\circ, where nn is the number of sides. A hexagon has 66 sides.
2
Find the sum of the five given angles.
120∘+135∘+115∘+140∘+110∘=620∘120^\circ + 135^\circ + 115^\circ + 140^\circ + 110^\circ = 620^\circ.
Adding the known angles allows us to find the combined measure of the known portion of the polygon.
3
Calculate the measure of the sixth angle.
720∘−620∘=100∘720^\circ - 620^\circ = 100^\circ.
Subtracting the sum of the five known angles from the total sum gives the measure of the missing angle.
4
Set up and solve the equation for xx.
4x+16=100  ⟹  4x=84  ⟹  x=214x + 16 = 100 \implies 4x = 84 \implies x = 21.
The sixth angle is represented by the expression (4x+16)∘(4x + 16)^\circ, so setting it equal to 100100 allows us to isolate and solve for xx.

Key Concept

Polygon Interior Angles and Linear Equations
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Topics

Quadrilaterals and Polygons

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