Question

a pipe cleaner lay across a wire shelf. the wires that make up the shelf are parallel, and the pipe cleaner is a transversal. the parallel wires are labeled a, b, and c, and the angles are labeled with numbers.

the measure of one angle is \\(130^\circ\\). which statement is true regarding the \\(130^\circ\\) angle and angle 3?

they are same-side interior angles, so angle 3 measures \\(50^\circ\\)
they are alternate interior angles, so angle 3 also measures \\(130^\circ\\)
they are corresponding angles, so angle 3 also measures \\(130^\circ\\)
they are alternate exterior angles, so angle 3 measures \\(50^\circ\\)

Explanation:

Identify the positions of the given angles

The parallel lines are \(a\), \(b\), and \(c\), cut by a transversal line.
The angle labeled \(130^\circ\) is located at the intersection with line \(c\), in the top-right position relative to the intersection (interior/exterior relative to other lines, but specifically in the upper-right quadrant of the intersection with line \(c\)).
Angle 3 is located at the intersection with line \(a\), in the bottom-left position (interior quadrant between lines \(a\) and \(b\)).

Determine the geometric relationship

The \(130^\circ\) angle is in the exterior region (above line \(c\) on the right side of the transversal, which is interior relative to the overall system of parallel lines depending on the reference, but let's look at the standard definitions).
Let's trace the positions:

• Line \(a\): Angle 3 is in the bottom-left position.
• Line \(c\): The \(130^\circ\) angle is in the top-right position.

Since they are on opposite sides of the transversal and both are in the "interior" region between the outer parallel lines \(a\) and \(c\):

• Angle 3 is below line \(a\) (interior to the region between \(a\) and \(c\)).
• The \(130^\circ\) angle is above line \(c\) (interior to the region between \(a\) and \(c\)).
• They lie on opposite sides of the transversal (left vs. right).

Therefore, they are alternate interior angles.

Calculate the measure of angle 3

Since lines \(a\) and \(c\) are parallel:

$$ \text{m}\angle 3 = 130^\circ $$

Answer:

• They are same-side interior angles, so angle 3 measures 50°
• They are alternate interior angles, so angle 3 also measures 130° (Correct answer)
• They are corresponding angles, so angle 3 also measures 130°
• They are alternate exterior angles, so angle 3 measures 50°