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Difficulty: MediumLines and Angles

A city planner is reviewing a design where two straight, parallel avenues are intersected by a diagonal rail line. The measure of one obtuse angle at the first intersection is (12x−20)∘(12x - 20)^\circ, and the measure of its corresponding angle at the second intersection is (8x+40)∘(8x + 40)^\circ.

What is the measure of the acute angle formed by the intersection of the rail line and the avenues?

  1. A
    76∘76^\circ
  2. B
    80∘80^\circ
  3. 20∘20^\circAnswer
  4. D
    64∘64^\circ

Answer

The measure of the acute angle is 20∘20^\circ.
Because the avenues are parallel, their corresponding angles are equal. Setting the expressions equal gives 12x−20=8x+4012x - 20 = 8x + 40. Subtracting 8x8x and adding 2020 to both sides results in 4x=604x = 60, which simplifies to x=15x = 15. Substituting 1515 back into either expression yields the obtuse angle measure: 12(15)−20=160∘12(15) - 20 = 160^\circ. The question asks for the adjacent acute angle, which forms a linear pair with the obtuse angle. Subtracting from 180∘180^\circ gives 180∘−160∘=20∘180^\circ - 160^\circ = 20^\circ.

Step-by-Step Solution

1
Set the expressions for the two corresponding angles equal to each other.
12x−20=8x+4012x - 20 = 8x + 40
When two parallel lines are intersected by a transversal, corresponding angles are equal in measure.
2
Solve the algebraic equation for xx.
4x=60⇒x=154x = 60 \Rightarrow x = 15
Subtracting 8x8x from both sides and adding 2020 to both sides isolates the variable xx.
3
Substitute x=15x = 15 back into either expression to find the measure of the obtuse angle.
12(15)−20=180−20=160∘12(15) - 20 = 180 - 20 = 160^\circ
The expressions represent the obtuse corresponding angles, so plugging in the value of xx gives the actual angle measure.
4
Subtract the obtuse angle from 180∘180^\circ to find the measure of the adjacent acute angle.
180∘−160∘=20∘180^\circ - 160^\circ = 20^\circ
Angles on a straight line form a linear pair and are supplementary, meaning their sum must be 180∘180^\circ.

Key Concept

Corresponding angles on parallel lines are equal, and adjacent angles on a straight line are supplementary (sum to 180∘180^\circ).
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Topics

Lines and Angles

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