Question

directions: in the diagrams below the lines are parallel as indicated. find the m∠1 and m∠2.
5.
6.
7.
8.
9.
10.

Explanation:

Problem 5:

Step1: Find \( m\angle1 \)

\( \angle1 \) and the \( 120^\circ \) angle are supplementary (linear pair), so \( m\angle1 = 180^\circ - 120^\circ = 60^\circ \)? Wait, no, wait. Wait, the two lines are parallel, and the transversal. Wait, actually, \( \angle1 \) and the \( 120^\circ \) angle: wait, no, looking at the diagram, the upper line and lower line are parallel, transversal. The \( 120^\circ \) angle and \( \angle1 \): wait, maybe \( \angle1 \) and the \( 120^\circ \) are supplementary? Wait, no, maybe I misread. Wait, the diagram for 5: upper line, transversal, angle above is \( 120^\circ \), then \( \angle1 \) is adjacent? Wait, no, let's re-examine. Wait, the two parallel lines, transversal. The angle given is \( 120^\circ \), and \( \angle1 \) is below? Wait, maybe \( \angle1 \) and \( 120^\circ \) are supplementary (linear pair), so \( m\angle1 = 180 - 120 = 60^\circ \)? No, wait, maybe \( \angle2 \) is equal to \( 120^\circ \)? Wait, no, alternate interior angles? Wait, no, the transversal: if the upper angle is \( 120^\circ \), then \( \angle1 \) is adjacent, so linear pair: \( 180 - 120 = 60^\circ \), and \( \angle2 \) is equal to \( 120^\circ \) because they are alternate interior angles? Wait, no, maybe I got it wrong. Wait, let's do problem 5 properly.

Diagram 5: Two parallel horizontal lines, transversal (diagonal line). The angle between upper line and transversal (above the line) is \( 120^\circ \). Then \( \angle1 \) is between upper line and transversal (below the line), so they are supplementary: \( m\angle1 = 180 - 120 = 60^\circ \). Then \( \angle2 \) is alternate interior angle to the \( 120^\circ \) angle? Wait, no, \( \angle2 \) is on the lower line, transversal. So \( \angle2 \) and the \( 120^\circ \) angle: are they same - side interior? No, alternate interior. Wait, no, the \( 120^\circ \) angle and \( \angle2 \): if the transversal is crossing, then \( \angle2 \) should be equal to \( 120^\circ \) because they are alternate interior angles? Wait, no, maybe \( \angle1 \) and \( \angle2 \): since lines are parallel, \( \angle1 \) and \( \angle2 \) are same - side? No, wait, let's start over.

For problem 5:

- \( \angle1 \) and the \( 120^\circ \) angle form a linear pair (adjacent, supplementary), so \( m\angle1 = 180^\circ - 120^\circ = 60^\circ \).

- \( \angle2 \) and the \( 120^\circ \) angle are alternate interior angles (since lines are parallel, transversal), so \( m\angle2 = 120^\circ \). Wait, no, that can't be. Wait, maybe the \( 120^\circ \) angle and \( \angle2 \) are same - side? No, alternate interior. Wait, maybe I mixed up. Let's check the diagram again. The upper line, transversal: angle above is \( 120^\circ \), angle below ( \( \angle1 \)) is supplementary: \( 60^\circ \). Then the lower line, transversal: \( \angle2 \) is equal to \( 120^\circ \) because it's a corresponding angle to the \( 120^\circ \) angle? Wait, no, the \( 120^\circ \) angle is above the upper line, \( \angle2 \) is below the lower line? No, maybe the \( 120^\circ \) angle and \( \angle2 \) are vertical angles? No. Wait, perhaps the correct approach:

Linear pair: \( \angle1 + 120^\circ = 180^\circ \) ⇒ \( \angle1 = 60^\circ \).

Alternate interior angles: \( \angle2 = 120^\circ \) (since \( \angle2 \) and the \( 120^\circ \) angle are alternate interior angles, lines parallel).

Problem 6:

Diagram 6: Two parallel horizontal lines, transversal (diagonal line). The angle above the upper line is \( 80^\circ \). \( \angle1 \) is below the upper line, adjacent to \( 80^\circ \), so linear pair: \( m\angle1 = 180 - 80 = 100^\circ \)? No, wait, \( \angle1 \) and \( 80^\circ \) are supplementary? Wait, no, the transversal: the angle given is \( 80^\circ \), and \( \angle1 \) is vertical to the angle adjacent? Wait, no, \( \angle1 \) and \( \angle2 \): since lines are parallel, \( \angle1 = \angle2 \) (alternate interior angles). Wait, the angle given is \( 80^\circ \), which is vertical to the angle adjacent to \( \angle1 \). Wait, the \( 80^\circ \) angle and \( \angle1 \) are supplementary? No, \( 80^\circ \) and \( \angle1 \) form a linear pair? Wait, no, the \( 80^\circ \) angle is above the upper line, \( \angle1 \) is below the upper line, so they are supplementary: \( 180 - 80 = 100^\circ \)? No, that can't be. Wait, no, the transversal: the angle of \( 80^\circ \) is an exterior angle, so \( \angle1 \) is equal to \( 180 - 80 = 100^\circ \)? No, wait, alternate interior angles: if the transversal crosses the parallel lines, then \( \angle1 \) and the angle supplementary to \( 80^\circ \)? Wait, no, let's use the fact that \( \angle1 \) and \( \angle2 \) are equal (alternate interior angles), and the \( 80^\circ \) angle and \( \angle1 \) are supplementary (linear pair). So \( m\angle1 = 180 - 80 = 100^\circ \), and \( m\angle2 = 100^\circ \)? No, that's not right. Wait, no, the \( 80^\circ \) angle and \( \angle1 \) are adjacent, forming a linear pair, so \( \angle1 = 180 - 80 = 100^\circ \), and \( \angle2 = \angle1 = 100^\circ \) (alternate interior angles).

Problem 7:

Diagram 7: Two parallel horizontal lines, transversal (diagonal line). The lower line has an angle of \( 45^\circ \). \( \angle2 \) is adjacent to \( 45^\circ \), so linear pair: \( 180 - 45 = 135^\circ \)? No, wait, \( \angle2 \) and \( 45^\circ \) are supplementary? Wait, no, \( \angle1 \) and \( \angle2 \): \( \angle1 = 45^\circ \) (corresponding angles), and \( \angle2 \) is equal to \( 180 - 45 = 135^\circ \)? No, wait, the diagram: transversal crosses two parallel lines, lower line has \( 45^\circ \), \( \angle2 \) is above, \( \angle1 \) is above the upper line. So \( \angle1 = 45^\circ \) (corresponding angles), and \( \angle2 \) is supplementary to \( 45^\circ \)? No, \( \angle2 \) and \( 45^\circ \) are alternate interior angles? Wait, no, \( \angle2 \) is on the upper line, transversal, and \( 45^\circ \) is on the lower line. So \( \angle2 = 45^\circ \) (alternate interior angles), and \( \angle1 = 180 - 45 = 135^\circ \)? Wait, no, \( \angle1 \) and \( \angle2 \) are supplementary (linear pair). So if \( \angle2 = 45^\circ \), then \( \angle1 = 180 - 45 = 135^\circ \).

Problem 8:

Diagram 8: Two parallel horizontal lines, transversal (diagonal line). Lower line has \( 40^\circ \), \( \angle2 \) is adjacent to \( 40^\circ \), so linear pair: \( 180 - 40 = 140^\circ \)? No, \( \angle2 \) and \( 40^\circ \) are supplementary? Wait, \( \angle1 \) and \( \angle2 \): since lines are parallel, \( \angle1 = \angle2 \) (alternate interior angles). Wait, the lower line has \( 40^\circ \), which is adjacent to \( \angle2 \), so \( \angle2 = 180 - 40 = 140^\circ \), and \( \angle1 = \angle2 = 140^\circ \)? No, that can't be. Wait, \( \angle1 \) is on the upper line, transversal, and \( \angle2 \) is on the lower line. So \( \angle1 \) and \( 40^\circ \) are corresponding angles, so \( \angle1 = 40^\circ \), and \( \angle2 = 180 - 40 = 140^\circ \) (supplementary to \( 40^\circ \)).

Problem 9:

Diagram 9: Two parallel horizontal lines, transversal (diagonal line). Upper line has \( 75^\circ \), \( \angle2 \) is adjacent to \( 75^\circ \), so linear pair: \( 180 - 75 = 105^\circ \)? No, \( \angle2 \) and \( 75^\circ \) are supplementary. Then \( \angle1 \) is equal to \( \angle2 \) (alternate interior angles)? Wait, \( \angle1 \) is on the lower line, transversal. The upper line has \( 75^\circ \), \( \angle2 = 180 - 75 = 105^\circ \), and \( \angle1 = \angle2 = 105^\circ \) (alternate interior angles).

Problem 10:

Diagram 10: Two parallel horizontal lines, transversal (diagonal line). Lower line has \( 110^\circ \), \( \angle1 \) is adjacent to \( 110^\circ \), so linear pair: \( 180 - 110 = 70^\circ \). \( \angle2 \) is equal to \( 110^\circ \) (alternate interior angles) or equal to \( \angle1 \)? Wait, \( \angle1 = 70^\circ \) (supplementary to \( 110^\circ \)), and \( \angle2 = 110^\circ \) (alternate interior angles) or \( \angle2 = \angle1 = 70^\circ \)? Wait, the \( 110^\circ \) angle and \( \angle2 \) are supplementary? No, \( \angle2 \) is on the upper line, transversal. So \( \angle2 = 110^\circ \) (corresponding angles), and \( \angle1 = 180 - 110 = 70^\circ \) (supplementary to \( 110^\circ \)).

Now, let's solve each problem step - by - step:

Problem 5:

Step 1: Find \( m\angle1 \)

The \( 120^\circ \) angle and \( \angle1 \) form a linear pair (they are adjacent and supplementary). So, \( m\angle1=180^\circ - 120^\circ = 60^\circ \).

Step 2: Find \( m\angle2 \)

Since the two lines are parallel and the transversal cuts them, \( \angle2 \) and the \( 120^\circ \) angle are alternate interior angles. Alternate interior angles are equal when lines are parallel. So, \( m\angle2 = 120^\circ \).

Problem 6:

Step 1: Find \( m\angle1 \)

The \( 80^\circ \) angle and \( \angle1 \) form a linear pair. So, \( m\angle1=180^\circ - 80^\circ = 100^\circ \).

Step 2: Find \( m\angle2 \)

Since the two lines are parallel, \( \angle1 \) and \( \angle2 \) are alternate interior angles. So, \( m\angle2=m\angle1 = 100^\circ \).

Problem 7:

Step 1: Find \( m\angle2 \)

The \( 45^\circ \) angle and \( \angle2 \) are alternate interior angles (parallel lines, transversal). So, \( m\angle2 = 45^\circ \).

Step 2: Find \( m\angle1 \)

\( \angle1 \) and \( \angle2 \) form a linear pair. So, \( m\angle1=180^\circ - 45^\circ = 135^\circ \).

Problem 8:

Step 1: Find \( m\angle2 \)

The \( 40^\circ \) angle and \( \angle2 \) form a linear pair. So, \( m\angle2=180^\circ - 40^\circ = 140^\circ \).

Step 2: Find \( m\angle1 \)

Since the two lines are parallel, \( \angle1 \) and \( \angle2 \) are alternate interior angles. So, \( m\angle1=m\angle2 = 140^\circ \).

Problem 9:

Step 1: Find \( m\angle2 \)

The \( 75^\circ \) angle and \( \angle2 \) form a linear pair. So, \( m\angle2=180^\circ - 75^\circ = 105^\circ \).

Step 2: Find \( m\angle1 \)

Since the two lines are parallel, \( \angle1 \) and \( \angle2 \) are alternate interior angles. So, \( m\angle1=m\angle2 = 105^\circ \).

Problem 10:

Step 1: Find \( m\angle1 \)

The \( 110^\circ \) angle and \( \angle1 \) form a linear pair. So, \( m\angle1=180^\circ - 110^\circ = 70^\circ \).

Step 2: Find \( m\angle2 \)

Since the two lines are parallel, \( \angle2 \) and the \( 110^\circ \) angle are corresponding angles (or alternate interior angles). So, \( m\angle2 = 110^\circ \).

Answers:

- Problem 5: \( m\angle1 = 60^\circ \), \( m\angle2 = 120^\circ \)
- Problem 6: \( m\angle1 = 100^\circ \), \( m\angle2 = 100^\circ \)
- Problem 7: \( m\angle1 = 135^\circ \), \( m\angle2 = 45^\circ \)
- Problem 8: \( m\angle1 = 140^\circ \), \( m\angle2 = 140^\circ \)
- Problem 9: \( m\angle1 = 105^\circ \), \( m\angle2 = 105^\circ \)
- Problem 10: \( m\angle1 = 70^\circ \), \( m\angle2 = 110^\circ \)

Answer:

s:

• Problem 5: \( m\angle1 = 60^\circ \), \( m\angle2 = 120^\circ \)
• Problem 6: \( m\angle1 = 100^\circ \), \( m\angle2 = 100^\circ \)
• Problem 7: \( m\angle1 = 135^\circ \), \( m\angle2 = 45^\circ \)
• Problem 8: \( m\angle1 = 140^\circ \), \( m\angle2 = 140^\circ \)
• Problem 9: \( m\angle1 = 105^\circ \), \( m\angle2 = 105^\circ \)
• Problem 10: \( m\angle1 = 70^\circ \), \( m\angle2 = 110^\circ \)