A civil engineer is designing a railroad crossing where a straight diagonal track intersec… — Armed Services Vocational Aptitude Battery (ASVAB)

A civil engineer is designing a railroad crossing where a straight diagonal track intersects two parallel roads. The two same-side interior angles between the parallel roads and the track are represented by the algebraic expressions $(4x - 20)^\circ$ and $(3x + 25)^\circ$ . Based on the geometric properties of parallel lines intersected by a transversal, match each description on the left with its correct calculated numerical value on the right.

The algebraic value of $x$ $25$
The measure of the acute angle formed at the intersections $80^\circ$
The measure of the obtuse angle formed at the intersections $100^\circ$
The difference in measure between the obtuse angle and the acute angle $20^\circ$

Answer

The correct matches are: the value of x is 25, the acute angle is 80°, the obtuse angle is 100°, and the difference between them is 20°.

Same-side interior angles formed by parallel lines and a transversal are supplementary, meaning they add up to 180 degrees. Setting up the equation (4x - 20) + (3x + 25) = 180 allows us to solve for x, which equals 25. Substituting x = 25 back into the original expressions provides the individual angle measures: 80 degrees (acute) and 100 degrees (obtuse). Finally, subtracting the acute angle from the obtuse angle yields a difference of 20 degrees.

Step-by-Step Solution

Set up an equation based on the geometric relationship of the angles.

(4x - 20) + (3x + 25) = 180

When a transversal intersects parallel lines, same-side interior angles are supplementary, meaning their sum is exactly 180 degrees.

Solve the algebraic equation to find the value of x.

7x + 5 = 180 \rightarrow 7x = 175 \rightarrow x = 25

Combining the like terms and isolating x provides the necessary variable value to determine the actual angle measures.

Substitute the value of x back into the original expressions.

4(25) - 20 = 80^\circ

and

3(25) + 25 = 100^\circ

This calculation determines the specific measures of the acute and obtuse angles formed at the intersection.

Calculate the difference between the obtuse and acute angle measures.

100^\circ - 80^\circ = 20^\circ

This satisfies the final required numerical value for the matching pairs.

Key Concept

Properties of parallel lines intersected by a transversal, specifically that same-side interior angles are supplementary.

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