Two lines are defined to be perpendicular when they intersect to form The symbol for perpendicular is " \( \perp . " \) Problem 1: Parallel Lines cut by a Perpendicular Transversal Draw two lines and mark the as parallel. Draw a perpendicular transversal. Label the angles 1 through 8. What is true about the angle measurements of each angle?

To illustrate the scenario of parallel lines cut by a perpendicular transversal, let's visualize the setup: 1. **Draw two parallel lines**: Label them as line \( l \) and line \( m \). 2. **Draw a perpendicular transversal**: Label this line as line \( t \), which intersects both lines \( l \) and \( m \) at right angles. Now, let's label the angles formed by the intersection of the transversal with the parallel lines: - The angles at the intersection of line \( t \) with line \( l \) can be labeled as: - Angle 1 (above line \( l \) to the left of line \( t \)) - Angle 2 (above line \( l \) to the right of line \( t \)) - Angle 3 (below line \( l \) to the left of line \( t \)) - Angle 4 (below line \( l \) to the right of line \( t \)) - The angles at the intersection of line \( t \) with line \( m \) can be labeled as: - Angle 5 (above line \( m \) to the left of line \( t \)) - Angle 6 (above line \( m \) to the right of line \( t \)) - Angle 7 (below line \( m \) to the left of line \( t \)) - Angle 8 (below line \( m \) to the right of line \( t \)) ### Angle Relationships 1. **Right Angles**: Since line \( t \) is perpendicular to both lines \( l \) and \( m \), angles 1, 2, 3, 4, 5, 6, 7, and 8 are all right angles. Therefore: - \( \angle 1 = \angle 2 = \angle 3 = \angle 4 = \angle 5 = \angle 6 = \angle 7 = \angle 8 = 90^\circ \) 2. **Corresponding Angles**: Angles that are in the same relative position at each intersection with the transversal are equal. Thus: - \( \angle 1 = \angle 5 \) - \( \angle 2 = \angle 6 \) - \( \angle 3 = \angle 7 \) - \( \angle 4 = \angle 8 \) 3. **Alternate Interior Angles**: The angles that lie between the two parallel lines and on opposite sides of the transversal are equal: - \( \angle 3 = \angle 5 \) - \( \angle 4 = \angle 6 \) 4. **Consecutive Interior Angles**: The angles that lie between the two parallel lines and on the same side of the transversal are supplementary: - \( \angle 3 + \angle 4 = 180^\circ \) - \( \angle 5 + \angle 6 = 180^\circ \) ### Conclusion In summary, all angles formed by the intersection of the perpendicular transversal with the parallel lines are right angles, and there are specific relationships of equality and supplementary angles among them.