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Question

Difficulty: MediumLines and Angles

In a blueprint for a corner bracket, a laser cuts an exact 90∘90^\circ right angle, splitting it into two adjacent sections. The measure of one section is determined by the expression (2x+18)∘(2x + 18)^\circ, while the adjacent section is measured as (3x−8)∘(3x - 8)^\circ. Based on this design, what is the degree measure of the larger section?

  1. 50∘50^\circAnswer
  2. B
    94∘94^\circ
  3. C
    56∘56^\circ
  4. D
    30∘30^\circ

Answer

The larger section measures 50∘50^\circ.
The problem states the laser cuts a 90-degree right angle into two sections, meaning the two expressions must add up to 90 degrees. Setting up the equation (2x + 18) + (3x - 8) = 90 simplifies to 5x + 10 = 90. Solving for x gives x = 16. Substituting x back into the expressions yields 50 degrees for the first section and 40 degrees for the second section. The larger of the two is 50 degrees.

Step-by-Step Solution

1
Set up the algebraic equation based on the geometric relationship.
(2x+18)+(3x−8)=90(2x + 18) + (3x - 8) = 90
The two adjacent sections form a right angle, meaning they are complementary and must sum to 90 degrees.
2
Combine like terms on the left side of the equation.
5x+10=905x + 10 = 90
Combining the variable terms and constant terms simplifies the equation.
3
Isolate the variable x.
x=16x = 16
Subtract 10 from both sides to get 5x = 80, then divide by 5.
4
Substitute the value of x back into both angle expressions.
First section = 2(16)+18=50∘2(16) + 18 = 50^\circ; Second section = 3(16)−8=40∘3(16) - 8 = 40^\circ
The specific degree measures must be calculated to find the larger angle.
5
Compare the two angle measures.
50∘50^\circ
The question asks specifically for the measure of the larger section.

Key Concept

Complementary Angles and Algebraic Substitution
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Topics

Lines and Angles

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