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Difficulty: MediumQuadrilaterals and Polygons

The interior angles of a quadrilateral are represented by the algebraic expressions 2x∘2x^\circ, (3x−15)∘(3x - 15)^\circ, (x+25)∘(x + 25)^\circ, and (2x+30)∘(2x + 30)^\circ. What is the measure of the largest angle in this quadrilateral?

  1. A
    135∘135^\circ
  2. B
    40∘40^\circ
  3. 110∘110^\circAnswer
  4. D
    65∘65^\circ

Answer

The largest angle measures 110∘110^\circ.
The interior angles of a quadrilateral must sum to 360∘360^\circ. By adding the four expressions together, we get 8x+40=3608x + 40 = 360. Solving for xx yields x=40x = 40. To find the largest angle, we substitute 4040 back into the original expressions: 2(40)=80∘2(40) = 80^\circ, 3(40)−15=105∘3(40) - 15 = 105^\circ, 40+25=65∘40 + 25 = 65^\circ, and 2(40)+30=110∘2(40) + 30 = 110^\circ. Comparing these four values, 110∘110^\circ is the largest.

Step-by-Step Solution

1
Set up an equation based on the sum of interior angles.
2x+(3x−15)+(x+25)+(2x+30)=3602x + (3x - 15) + (x + 25) + (2x + 30) = 360
The sum of the interior angles of any quadrilateral is always 360∘360^\circ.
2
Combine the like terms on the left side of the equation.
8x+40=3608x + 40 = 360
Grouping the xx terms (2x+3x+x+2x=8x2x + 3x + x + 2x = 8x) and constant terms (−15+25+30=40-15 + 25 + 30 = 40) simplifies the algebraic expression.
3
Solve for the variable xx.
8x=3208x = 320, which means x=40x = 40.
Subtracting 40 from both sides and then dividing by 8 isolates xx.
4
Substitute x=40x = 40 into each of the original expressions to determine the measure of each angle.
The angles are 2(40)=80∘2(40) = 80^\circ; 3(40)−15=105∘3(40) - 15 = 105^\circ; 40+25=65∘40 + 25 = 65^\circ; and 2(40)+30=110∘2(40) + 30 = 110^\circ.
The value of xx alone does not answer the question; the specific angle measures must be evaluated.
5
Identify the largest value among the calculated angles.
The largest value is 110∘110^\circ.
The question specifically asks for the maximum angle measure in the quadrilateral.

Key Concept

The sum of the interior angles of a quadrilateral is always 360 degrees. Algebraic substitution is required to convert variable expressions back into geometric measurements.
Estimated Time:1m 30s
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Topics

Quadrilaterals and Polygons

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