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: Identify Angle Relationships

First, recall the definitions of angle relationships (same - side interior, alternate interior, corresponding, alternate exterior) when a transversal crosses parallel lines. Alternate exterior angles are equal when lines are parallel. The \(130^{\circ}\) angle and angle 3: Let's analyze their positions. The \(130^{\circ}\) angle (angle 12) and angle 3: Wait, no, let's look at the lines. The parallel lines are a, b, c. The transversal is the pipe cleaner. Angle 3 and the \(130^{\circ}\) angle (angle 12) – no, wait, let's check the other option. Wait, the first option: same - side interior angles are supplementary. If they are same - side interior, then angle 3 + \(130^{\circ}\)? No, wait, the correct approach: Let's check each option.

Option 1: Same - side interior angles. Same - side interior angles are supplementary (sum to \(180^{\circ}\)). If one is \(130^{\circ}\), the other should be \(50^{\circ}\). But are angle 3 and the \(130^{\circ}\) angle same - side interior? Wait, no, let's check the positions. Wait, the \(130^{\circ}\) angle is angle 12 (assuming). Wait, maybe I mislabeled. Let's re - examine the diagram. The parallel lines are horizontal (a, b, c). The transversal is the blue line. Angle 3 is between lines a and b, on the left of the transversal. The \(130^{\circ}\) angle is on line c, right of the transversal. Wait, no, the correct option: Let's check the fourth option. Alternate exterior angles: Alternate exterior angles are equal? No, wait, alternate exterior angles are equal when lines are parallel. Wait, no, the \(130^{\circ}\) angle and angle 3: Wait, maybe the \(130^{\circ}\) angle and angle 3 are alternate exterior? No, wait, let's think again.

Wait, the first option: "They are same - side interior angles, so angle 3 measures \(50^{\circ}\)". Same - side interior angles are supplementary. So if one angle is \(130^{\circ}\), the other (angle 3) should be \(180 - 130=50^{\circ}\). But are they same - side interior? Wait, no, maybe the correct option is the fourth? No, wait, let's check the angle positions. Wait, the \(130^{\circ}\) angle (angle 12) and angle 3: Wait, maybe I made a mistake. Let's re - check the options.

Wait, the correct answer is the first option? No, wait, let's recall: Same - side interior angles: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So if one angle is \(130^{\circ}\), the same - side interior angle is \(50^{\circ}\). Now, are angle 3 and the \(130^{\circ}\) angle same - side interior? Wait, maybe the \(130^{\circ}\) angle and angle 3 are same - side interior. Wait, no, let's look at the diagram again. The parallel lines are a, b, c (horizontal). The transversal is the blue line. Angle 3 is between a and b, left of transversal. The \(130^{\circ}\) angle is between c and the line below c, right of transversal. Wait, no, maybe the \(130^{\circ}\) angle is on line c, right of transversal, and angle 3 is on line a - b, left of transversal. Wait, maybe they are same - side interior. So if they are same - side interior, then angle 3 + \(130^{\circ}=180^{\circ}\), so angle 3 = \(50^{\circ}\). But wait, the first option says "They are same - side interior angles, so angle 3 measures \(50^{\circ}\)". But let's check the other options.

Option 2: Alternate interior angles: alternate interior angles are equal. So if they are alternate interior, angle 3 would be \(130^{\circ}\), but that's not the case here.

Option 3: Corresponding angles: corresponding angles are equal. So angle 3 would be \(130^{\circ}\), but that's not correct.

Option 4: Alternate exterior angles: alternate exterior angles are equal, but if angle 3 is \(50^{\circ}\), and alternate exterior angles should be equal, but \(130^{\circ}\) and \(50^{\circ}\) are not equal. So that's wrong.

Wait, I think I made a mistake. Let's re - define the angle relationships:

- Same - side interior angles: lie between two parallel lines, on the same side of the transversal. They are supplementary (\(m\angle1 + m\angle2 = 180^{\circ}\)).

- Alternate interior angles: lie between two parallel lines, on opposite sides of the transversal. They are equal.

- Corresponding angles: lie in the same position relative to the parallel lines and the transversal. They are equal.

- Alternate exterior angles: lie outside two parallel lines, on opposite sides of the transversal. They are equal.

Now, let's assume the \(130^{\circ}\) angle is an interior angle on the right side of the transversal, and angle 3 is an interior angle on the left side of the transversal, between the same two parallel lines (say, line a and line c? No, line a and line b). Wait, maybe the \(130^{\circ}\) angle and angle 3 are same - side interior angles. So if they are same - side interior, then \(m\angle3+130^{\circ}=180^{\circ}\), so \(m\angle3 = 50^{\circ}\). So the first option: "They are same - side interior angles, so angle 3 measures \(50^{\circ}\)" is correct? Wait, no, wait the fourth option says "They are alternate exterior angles, so angle 3 measures \(50^{\circ}\)". Wait, maybe I misidentified the angle. Let's look at the diagram again. The \(130^{\circ}\) angle is on line c, right of the transversal. Angle 3 is on line a, left of the transversal. So they are on opposite sides of the transversal and outside the region between lines a and c? No, angle 3 is between a and b, so inside. Wait, I'm confused. Let's check the answer options again.

Wait, the correct answer is the first option? No, wait, let's check the angle measures. If two angles are same - side interior, they are supplementary. So \(180 - 130 = 50\). So if angle 3 and the \(130^{\circ}\) angle are same - side interior, then angle 3 is \(50^{\circ}\). So the first option is correct? Wait, no, the fourth option says alternate exterior angles. Wait, maybe the \(130^{\circ}\) angle and angle 3 are alternate exterior. But alternate exterior angles are equal. So if they are alternate exterior, angle 3 should be \(130^{\circ}\), but the option says \(50^{\circ}\). So that's wrong. The third option: corresponding angles would be equal, so angle 3 would be \(130^{\circ}\), but that's wrong. The second option: alternate interior angles would be equal, so angle 3 would be \(130^{\circ}\), wrong. So the first option: same - side interior, supplementary, so \(50^{\circ}\). So the first option is correct.

Step2: Verify Each Option

- Option 1: Same - side interior angles are supplementary. So \(180 - 130=50^{\circ}\). If angle 3 and the \(130^{\circ}\) angle are same - side interior, this is correct.
- Option 2: Alternate interior angles are equal. So angle 3 would be \(130^{\circ}\), but this is incorrect as they are not alternate interior.
- Option 3: Corresponding angles are equal. So angle 3 would be \(130^{\circ}\), incorrect.
- Option 4: Alternate exterior angles are equal. So angle 3 would be \(130^{\circ}\), but the option says \(50^{\circ}\), incorrect.

Answer:

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