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°. which statement is true regarding the 130° angle and angle 3? they are same - side interior angles, so angle 3 measures 50°. they are alternate interior angles, so angle 3 also measures 130°. they are corresponding angles, so angle 3 also measures 130°. they are alternate exterior angles, so angle 3 measures 50°.

Explanation:

Step1: Recall angle - pair relationships

When two parallel lines are cut by a transversal:

• Same - side interior angles: If two parallel lines \(l_1\parallel l_2\) and a transversal \(t\) intersects them, same - side interior angles \(\alpha\) and \(\beta\) satisfy \(\alpha+\beta = 180^{\circ}\)
• Alternate interior angles: Alternate interior angles are equal. If \(l_1\parallel l_2\) and \(t\) is a transversal, then \(\angle x=\angle y\) (where \(\angle x\) and \(\angle y\) are alternate interior angles)
• Corresponding angles: Corresponding angles are equal. If \(l_1\parallel l_2\) and \(t\) is a transversal, then \(\angle m=\angle n\) (where \(\angle m\) and \(\angle n\) are corresponding angles)
• Alternate exterior angles: Alternate exterior angles are equal. If \(l_1\parallel l_2\) and \(t\) is a transversal, then \(\angle p=\angle q\) (where \(\angle p\) and \(\angle q\) are alternate exterior angles)

Step2: Analyze the position of \(\angle3\) and the \(130^{\circ}\) angle

The \(130^{\circ}\) angle and \(\angle3\) are on the same side of the transversal and between the parallel lines \(a\) and \(c\). So they are same - side interior angles.
Let \(\angle3=x\). Using the same - side interior angles theorem \(x + 130^{\circ}=180^{\circ}\)
Solve for \(x\): \(x=180^{\circ}- 130^{\circ}=50^{\circ}\)

Answer:

They are same - side interior angles, so angle 3 measures \(50^{\circ}\).