A civil engineer is analyzing the intersection of two straight structural beams, Beam 1 an… — Armed Services Vocational Aptitude Battery (ASVAB)

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

A
49
B
65
85Answer
D
89

Answer

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

yields

x = 35

. Substituting this back into the expression for angle Z (which is the same measure as angle Y) gives

2(35) + 15 = 85^\circ

Step-by-Step Solution

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^\circ

). Angle Y and angle Z are vertical angles, so they are equal (

Y = Z

). Therefore, angle X and angle Z must also be supplementary (

X + Z = 180^\circ

Establishing the geometric relationship allows us to set up an algebraic equation based on the given expressions.

Set up and solve the algebraic equation for x.

(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.

Substitute the value of x back into the expression for angle Z (which equals angle Y).

Angle

Y = \text{Angle } Z = 2(35) + 15 = 70 + 15 = 85^\circ

The question specifically asks for the measure of angle Y in degrees.

Key Concept

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

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