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Zorluk: ZorLines and Angles

A civil engineer is analyzing the intersection of two straight structural beams, Beam 1 and Beam 2. Angle XX and angle YY form a linear pair along Beam 1. Angle YY and angle ZZ are vertical angles formed by the intersection of the two beams. The measure of angle XX is given by the expression (3x−10)∘(3x - 10)^\circ and the measure of angle ZZ is represented by (2x+15)∘(2x + 15)^\circ. What is the measure of angle YY in degrees?

  1. A
    49
  2. B
    65
  3. 85Cevap
  4. D
    89

Cevap

85
The correct answer is found by recognizing that since angle X and angle Y form a linear pair, they sum to 180 degrees. Because angle Y and angle Z are vertical angles, they are equal. Substituting Z for Y means angle X and angle Z must also sum to 180 degrees. Setting up the equation (3x−10)+(2x+15)=180(3x - 10) + (2x + 15) = 180 yields x=35x = 35. Substituting this back into the expression for angle Z (which is the same measure as angle Y) gives 2(35)+15=85∘2(35) + 15 = 85^\circ.

Adım Adım Çözüm

1
Identify the relationship between angle X and angle Z.
Since angle X and angle Y form a linear pair, they are supplementary (X+Y=180∘X + Y = 180^\circ). Angle Y and angle Z are vertical angles, so they are equal (Y=ZY = Z). Therefore, angle X and angle Z must also be supplementary (X+Z=180∘X + Z = 180^\circ).
Establishing the geometric relationship allows us to set up an algebraic equation based on the given expressions.
2
Set up and solve the algebraic equation for x.
(3x−10)+(2x+15)=180→5x+5=180→5x=175→x=35(3x - 10) + (2x + 15) = 180 \rightarrow 5x + 5 = 180 \rightarrow 5x = 175 \rightarrow x = 35.
Finding the numerical value of the variable is necessary to determine the specific angle measures.
3
Substitute the value of x back into the expression for angle Z (which equals angle Y).
Angle Y=Angle Z=2(35)+15=70+15=85∘Y = \text{Angle } Z = 2(35) + 15 = 70 + 15 = 85^\circ.
The question specifically asks for the measure of angle Y in degrees.

Anahtar Kavram

Applying algebraic equations to solve for unknown values in supplementary and vertical angle relationships.
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Konular

Lines and Angles

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