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Question

Difficulty: MediumLines and Angles

Two adjacent angles, ∠A\angle A and ∠B\angle B, form a straight line. If the measure of ∠A\angle A is (12x−10)∘(12x - 10)^\circ and the measure of ∠B\angle B is (6x+28)∘(6x + 28)^\circ, what is the measure of the larger angle?

  1. A
    52∘52^\circ
  2. B
    58∘58^\circ
  3. 98∘98^\circAnswer
  4. D
    118∘118^\circ

Answer

The larger angle measures 98 degrees.
The correct answer is 98∘98^\circ because the angles are supplementary, meaning their sum is 180∘180^\circ. Setting up the equation (12x−10)+(6x+28)=180(12x - 10) + (6x + 28) = 180 simplifies to 18x+18=18018x + 18 = 180. Solving this gives x=9x = 9. Substituting x=9x = 9 into both expressions reveals the angles are 98∘98^\circ and 82∘82^\circ. Comparing the two, 98∘98^\circ is the larger measure.

Step-by-Step Solution

1
Recognize the geometric relationship between the two angles.
Since the angles form a straight line, they are a linear pair and are supplementary. Their sum must equal 180∘180^\circ.
Adjacent angles on a straight line add up to 180 degrees.
2
Set up the algebraic equation based on the supplementary relationship.
(12x−10)+(6x+28)=180(12x - 10) + (6x + 28) = 180
This translates the geometric concept into a solvable algebraic equation.
3
Combine like terms to simplify the equation.
18x+18=18018x + 18 = 180
Grouping the xx variables (12x+6x12x + 6x) and constants (−10+28-10 + 28) simplifies the problem.
4
Solve for xx.
18x=162⇒x=918x = 162 \Rightarrow x = 9
Subtracting 18 from both sides and then dividing by 18 isolates the variable.
5
Substitute x=9x = 9 into both original expressions to find the angle measures.
∠A=12(9)−10=108−10=98∘\angle A = 12(9) - 10 = 108 - 10 = 98^\circ and ∠B=6(9)+28=54+28=82∘\angle B = 6(9) + 28 = 54 + 28 = 82^\circ.
Finding the value of xx is not the final step; the exact angle measurements must be calculated.
6
Identify the larger angle.
Since 98∘>82∘98^\circ > 82^\circ, the larger angle is 98∘98^\circ.
The question specifically asks for the measure of the larger angle.

Key Concept

Adjacent angles forming a straight line are supplementary and sum to 180 degrees.
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Topics

Lines and Angles

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