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Question

Difficulty: MediumLines and Angles

A structural engineer is designing a roof truss. In the design, two parallel horizontal support beams are intersected by a diagonal brace. The consecutive interior angles (also known as same-side interior angles) formed by this intersection measure (4x−15)∘(4x - 15)^\circ and (2x+45)∘(2x + 45)^\circ. What is the measure of the larger of these two angles?

  1. A
    55∘55^\circ
  2. B
    65∘65^\circ
  3. 95∘95^\circAnswer
  4. D
    115∘115^\circ

Answer

The measure of the larger angle is 95∘95^\circ.
Consecutive interior angles formed by two parallel lines and a transversal are supplementary, which means they add up to 180∘180^\circ. Setting up the equation (4x−15)+(2x+45)=180(4x - 15) + (2x + 45) = 180 and combining like terms yields 6x+30=1806x + 30 = 180. Subtracting 3030 gives 6x=1506x = 150, so x=25x = 25. Substituting 2525 back into the expressions gives angles of 85∘85^\circ and 95∘95^\circ. Comparing the two, the larger angle is 95∘95^\circ.

Step-by-Step Solution

1
Identify the geometric relationship between the given angles.
The angles are consecutive (same-side) interior angles.
Consecutive interior angles formed by parallel lines and a transversal are supplementary, meaning their sum is 180∘180^\circ.
2
Set up an algebraic equation based on the supplementary relationship.
(4x−15)+(2x+45)=180(4x - 15) + (2x + 45) = 180
Adding the expressions for both angles and setting them equal to 180180 allows us to solve for xx.
3
Combine like terms and solve for xx.
6x+30=1806x + 30 = 180, so 6x=1506x = 150, giving x=25x = 25.
Isolating xx provides the variable value needed to find the specific angle measures.
4
Substitute x=25x = 25 back into both original expressions.
First angle: 4(25)−15=100−15=85∘4(25) - 15 = 100 - 15 = 85^\circ. Second angle: 2(25)+45=50+45=95∘2(25) + 45 = 50 + 45 = 95^\circ.
This step determines the actual degree measures of the two angles.
5
Compare the two calculated angles.
The larger angle is 95∘95^\circ.
The question asks specifically for the measure of the larger angle.

Key Concept

Same-side (consecutive) interior angles formed by parallel lines cut by a transversal are supplementary (sum to 180∘180^\circ).
Estimated Time:1m 15s
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Topics

Lines and Angles

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