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Geometry Chapter 3: Parallel and Perpendicular Lines
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Geometry Chapter 3: Parallel and Perpendicular Lines

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Questions and Answers

What describes parallel lines?

  • Lines that always intersect
  • Lines that intersect at right angles
  • Lines in the same plane that never intersect (correct)
  • Lines that are not in the same plane
  • What are skew lines?

    Lines that do not intersect and are not in the same plane.

    What are parallel planes?

    Planes that never intersect.

    Define a transversal.

    <p>A line that intersects two or more lines in a plane at different points.</p> Signup and view all the answers

    What are interior angles?

    <p>Angles that lie inside parallel lines.</p> Signup and view all the answers

    What are exterior angles?

    <p>Angles that lie outside parallel lines.</p> Signup and view all the answers

    Define consecutive interior angles.

    <p>Angles that are on the same side of the transversal and inside the parallel lines.</p> Signup and view all the answers

    What are alternate interior angles?

    <p>Angles that are on opposite sides of the transversal and inside the parallel lines.</p> Signup and view all the answers

    What are alternate exterior angles?

    <p>Angles that are on opposite sides of the transversal and outside the parallel lines.</p> Signup and view all the answers

    Define corresponding angles.

    <p>Angles that are in the same position on the parallel lines with respect to the transversal.</p> Signup and view all the answers

    The Corresponding Angles Postulate states that if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

    <p>True</p> Signup and view all the answers

    The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

    <p>True</p> Signup and view all the answers

    Consecutive Interior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.

    <p>True</p> Signup and view all the answers

    The Alternate Exterior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

    <p>True</p> Signup and view all the answers

    What is the slope of a line?

    <p>The ratio of the change along the y-axis to the change along the x-axis.</p> Signup and view all the answers

    The slope of a vertical line is defined and equal to zero.

    <p>False</p> Signup and view all the answers

    The slope of a horizontal line is equal to zero.

    <p>True</p> Signup and view all the answers

    A line with positive slope rises from left to right.

    <p>True</p> Signup and view all the answers

    A line with negative slope falls from left to right.

    <p>True</p> Signup and view all the answers

    What is the rate of change?

    <p>A description of how one quantity changes with respect to another quantity.</p> Signup and view all the answers

    Two non-vertical lines are parallel if and only if they have different slopes.

    <p>False</p> Signup and view all the answers

    Two non-vertical lines are perpendicular if and only if the product of their slopes is equal to -1.

    <p>True</p> Signup and view all the answers

    What is the y-intercept?

    <p>The point where a line intersects the y-axis.</p> Signup and view all the answers

    What is the equation of a horizontal line?

    <p>y = b</p> Signup and view all the answers

    What is the equation of a vertical line?

    <p>x = a</p> Signup and view all the answers

    If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

    <p>True</p> Signup and view all the answers

    Define the distance between a point and a line.

    <p>The length of the perpendicular line segment from the line to the point.</p> Signup and view all the answers

    What is the Perpendicular Postulate?

    <p>There exists exactly one line through a point not on a line that is perpendicular to the given line.</p> Signup and view all the answers

    What is the distance formula for the distance between two points?

    <p>This formula can be derived from the Pythagorean Theorem.</p> Signup and view all the answers

    What does it mean for two lines to be equidistant?

    <p>The same distance apart at every point.</p> Signup and view all the answers

    What is the distance between parallel lines?

    <p>The perpendicular distance between one of the lines and any point on the other line.</p> Signup and view all the answers

    If two lines are each equidistant from a third line, then the two lines are parallel to each other.

    <p>True</p> Signup and view all the answers

    Study Notes

    Parallel and Perpendicular Lines

    • Parallel Lines: Never intersect and exist in the same plane.
    • Skew Lines: Do not intersect and are not in the same plane.
    • Parallel Planes: Planes that never meet.

    Angles Formed by Transversals

    • Transversal: A line that crosses two or more other lines at varying points.
    • Interior Angles: Found between two parallel lines.
    • Exterior Angles: Located outside two parallel lines.
    • Consecutive Interior Angles: Same-side angles inside parallel lines; supplementary (sum to 180 degrees).
    • Alternate Interior Angles: Opposite-side angles inside parallel lines; congruent if lines are parallel.
    • Alternate Exterior Angles: Opposite-side angles outside parallel lines; congruent if lines are parallel.
    • Corresponding Angles: Matching position angles across parallel lines concerning the transversal; congruent if lines are parallel.

    Angle Theorems and Postulates

    • Corresponding Angles Postulate: If parallel lines are cut by a transversal, each pair of corresponding angles is congruent.
    • Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
    • Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, consecutive interior angles sum to 180 degrees.
    • Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, alternate exterior angles are congruent.

    Slope and Line Characteristics

    • Slope Definition: Represents the ratio of vertical change (rise) to horizontal change (run).
    • Vertical Line Slope: Undefined slope due to a zero denominator.
    • Horizontal Line Slope: Slope equals zero since there's no vertical change.
    • Positive Slope: Line rises from left to right.
    • Negative Slope: Line falls from left to right.
    • Rate of Change: Measures how one variable changes concerning another.

    Slopes of Parallel and Perpendicular Lines

    • Postulate: Slopes of Parallel Lines: Non-vertical lines are parallel if they share the same slope (m1 = m2).
    • Postulate: Slopes of Perpendicular Lines: Non-vertical lines are perpendicular if the product of their slopes equals -1 (m1 * m2 = -1).

    Line Equations

    • Equation of a Horizontal Line: Represented as y = b, where y is constant.
    • Equation of a Vertical Line: Represented as x = a, where x is constant.

    Relationships Between Angles and Lines

    • Converse of Corresponding Angles: If corresponding angles are congruent when two lines are cut by a transversal, the lines are parallel.
    • Distance Between a Point and a Line: Measured by the length of the perpendicular segment from the point to the line.
    • Perpendicular Postulate: For a given line, there exists exactly one line perpendicular to it through any point not on that line.
    • Distance Between Parallel Lines: Calculated as the perpendicular distance from one line to any point on the other line.

    Equidistance and Parallelism

    • Equidistant Lines: Two lines are equidistant if they maintain the same distance from each other at all points.
    • Theorem: Two Lines Equidistant from a Third Line: If two lines are equidistant from a third line in a plane, they are parallel.

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