Question

if lines m and l are parallel and angle 4 measures 100 degrees, what is the measure of angle 7? 25 points

Explanation:

Step1: Identify angle relationships

Since lines \( m \) and \( l \) are parallel, and we have a transversal (the line with angle 4 and angle 7), we can use the concept of same - side interior angles or corresponding angles. First, angle 4 and angle 6: angle 4 and angle 6 are same - side interior angles? Wait, no. Wait, angle 4 and angle 5: Wait, let's look at the diagram. Angle 4 and angle 5: if \( m\parallel l \), then angle 4 and angle 5 are same - side interior angles? Wait, no, maybe alternate interior angles or corresponding. Wait, actually, angle 4 and angle 6: Wait, maybe first, angle 4 and angle 2: vertical angles? No, angle 4 and angle 1: adjacent supplementary? Wait, maybe a better approach. Let's see, angle 4 and angle 5: if \( m\parallel l \), then angle 4 + angle 5 = 180°? Wait, no, maybe angle 4 and angle 6: Wait, the transversal is the line that intersects \( m \) and \( l \), and the other line is \( n \). Wait, angle 4 and angle 6: are they corresponding? Wait, maybe angle 4 and angle 6 are equal? No, wait, angle 4 is 100°, let's think about angle 7. Angle 7 and angle 4: are they same - side interior angles? Wait, no, let's see the positions. Angle 4 and angle 6: if \( m\parallel l \), then angle 4 and angle 6 are same - side interior angles? Wait, no, same - side interior angles add up to 180°. Wait, angle 4 and angle 5: if \( m\parallel l \), then angle 4 and angle 5 are same - side interior angles, so angle 4 + angle 5 = 180°. Then angle 5 and angle 7: vertical angles? No, angle 5 and angle 7: are they vertical? Wait, angle 5 and angle 7: angle 5 and angle 7 are vertical angles? No, angle 5 and angle 8 are vertical, angle 6 and angle 7 are adjacent supplementary? Wait, maybe I made a mistake. Let's start over.

We know that lines \( m \) and \( l \) are parallel. The line \( n \) is a transversal? Wait, no, the two lines (the non - parallel ones) intersect, creating angles 1 - 4 and 5 - 8. Wait, angle 4 and angle 6: since \( m\parallel l \), angle 4 and angle 6 are same - side interior angles? Wait, no, same - side interior angles are on the same side of the transversal and inside the two parallel lines. Wait, angle 4 is above \( m \), angle 6 is below \( l \)? No, maybe the transversal is the line that makes angle 4 and angle 7. Wait, angle 4 and angle 7: let's see, angle 4 and angle 5: if \( m\parallel l \), then angle 4 and angle 5 are same - side interior angles, so \( \angle4+\angle5 = 180^{\circ}\). Then angle 5 and angle 7: are they vertical angles? No, angle 5 and angle 7: angle 5 and angle 7 are equal? Wait, no, angle 5 and angle 8 are vertical, angle 6 and angle 7 are adjacent supplementary. Wait, maybe angle 4 and angle 7: since \( m\parallel l \), angle 4 and angle 7 are same - side interior angles? Wait, no, let's use the fact that angle 4 and angle 6: if \( m\parallel l \), then angle 4 and angle 6 are same - side interior angles, so \( \angle4+\angle6 = 180^{\circ}\). Then angle 6 and angle 7: are they vertical angles? No, angle 6 and angle 7 are adjacent and form a linear pair? Wait, angle 6 and angle 7 are supplementary? No, angle 6 and angle 7: if they are adjacent, then \( \angle6+\angle7 = 180^{\circ}\)? No, that can't be. Wait, maybe angle 4 and angle 7 are equal? No, wait, let's look at the diagram again. The angles: angle 4 and angle 1 are adjacent, angle 4 and angle 3 are vertical? No, angle 1 and angle 3 are vertical, angle 2 and angle 4 are vertical? Wait, no, when two lines intersect, vertical angles are equal. So angle 1 = angle 3, angle 2 = angle 4. Then, since \( m\parallel l \), angle 2 and angle 6 are corresponding angles, so angle 2 = angle 6. Since angle 2 = angle 4 (vertical angles), then angle 4 = angle 6. Then angle 6 and angle 7: are they supplementary? Because they form a linear pair (adjacent angles on a straight line), so \( \angle6+\angle7 = 180^{\circ}\). Since \( \angle4 = 100^{\circ}\) and \( \angle4=\angle6 \), then \( \angle6 = 100^{\circ}\). Then \( \angle7=180^{\circ}-\angle6 = 180 - 100=80^{\circ}\).

Step1: Vertical angles and parallel lines

Angle 4 and angle 2 are vertical angles, so \( \angle4=\angle2 = 100^{\circ}\).

Step2: Corresponding angles (parallel lines)

Since \( m\parallel l \), angle 2 and angle 6 are corresponding angles, so \( \angle2=\angle6 = 100^{\circ}\).

Step3: Linear pair (supplementary angles)

Angle 6 and angle 7 form a linear pair, so \( \angle6+\angle7 = 180^{\circ}\). Substituting \( \angle6 = 100^{\circ}\), we get \( \angle7=180^{\circ}- 100^{\circ}=80^{\circ}\).

Answer:

The measure of angle 7 is \( 80^{\circ} \)