Geometry Chapter 3: Lines and Planes

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Questions and Answers

What are skew lines?

Lines that do not intersect and are coplanar

Lines that do not intersect and are not coplanar (correct)

Lines that intersect

Lines that are parallel

What are parallel lines?

Lines in the same plane that never intersect.

What are parallel planes?

Planes that do not intersect.

State Postulate 3.1.

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Signup and view all the answers

State Postulate 3.2.

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Signup and view all the answers

What is a transversal?

A line that intersects two or more coplanar lines at different points. Signup and view all the answers

What are corresponding angles?

Angles formed by a transversal cutting through 2 or more lines that are in the same relative position. Signup and view all the answers

What are alternate interior angles?

Angles that lie within a pair of lines and on opposite sides of a transversal. Signup and view all the answers

What are alternate exterior angles?

Angles that lie outside a pair of lines and on opposite sides of a transversal. Signup and view all the answers

What are consecutive interior angles?

Two angles that lie between the two lines on the same side of the transversal. Signup and view all the answers

State Theorem 3.1.

If two parallel lines are cut by a transversal, then the pair of corresponding angles are congruent. Signup and view all the answers

State Theorem 3.2.

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Signup and view all the answers

State Theorem 3.3.

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Signup and view all the answers

State Theorem 3.4.

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Signup and view all the answers

State Theorem 3.5.

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Signup and view all the answers

State Theorem 3.6.

If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel. Signup and view all the answers

State Theorem 3.7.

If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel. Signup and view all the answers

State Theorem 3.8.

If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel. Signup and view all the answers

State Theorem 3.9.

If two lines are parallel to the same line, then they are parallel to each other. Signup and view all the answers

What is the distance from a point to a line?

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

What is a perpendicular bisector?

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Signup and view all the answers

State Theorem 3.10.

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Signup and view all the answers

State Theorem 3.11.

In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. Signup and view all the answers

State Theorem 3.12.

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Signup and view all the answers

What is a directed line segment?

A segment, AB, that represents moving from point A to point B. Signup and view all the answers

State Theorem 3.13.

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Signup and view all the answers

State Theorem 3.14.

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Signup and view all the answers

Study Notes

Skew and Parallel Lines

Skew Lines: Lines that do not intersect and are not in the same plane.
Parallel Lines: Lines in the same plane that never meet or intersect.
Parallel Planes: Planes that do not intersect each other.

Postulates

Parallel Postulate: Given a line and a point not on it, there is exactly one line through the point that is parallel to the original line.
Perpendicular Postulate: Given a line and a point not on it, there is exactly one line through the point that is perpendicular to the original line.

Transversal and Angles

Transversal: A line that intersects two or more coplanar lines at different points.
Corresponding Angles: Angles formed by a transversal that are located in the same relative position with respect to the intersected lines.
Alternate Interior Angles: Angles that lie between two lines and are on opposite sides of a transversal.
Alternate Exterior Angles: Angles located outside two lines and on opposite sides of a transversal.
Consecutive Interior Angles: Two angles located between two intersected lines and on the same side of a transversal.

Angle Theorems

Corresponding Angles Theorem: If two parallel lines are cut by a transversal, corresponding angles are congruent.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, alternate exterior angles are congruent.
Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, pairs of consecutive interior angles are supplementary.

Converse Theorems

Corresponding Angles Converse: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
Alternate Interior Angles Converse: If two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.
Alternate Exterior Angles Converse: If two lines are cut by a transversal and alternate exterior angles are congruent, the lines are parallel.
Consecutive Interior Angles Converse: If two lines are cut by a transversal and consecutive interior angles are supplementary, the lines are parallel.

Additional Theorems

Transitive Property of Parallel Lines: If two lines are both parallel to the same line, then they are parallel to each other.
Distance from a Point to a Line: Defined as the length of the perpendicular segment from a point to a line.
Perpendicular Bisector: A line or segment that divides another segment into two equal parts at a right angle.

Perpendicularity Theorems

Linear Pair Perpendicular Theorem: If two lines intersect to form a linear pair of congruent angles, the lines are perpendicular.
Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line.
Lines Perpendicular to a Transversal Theorem: If two lines are perpendicular to the same line, those two lines are parallel.

Slopes of Lines

Slopes of Parallel Lines: Two nonvertical lines in a coordinate plane are parallel if and only if their slopes are equal.
Slopes of Perpendicular Lines: Two nonvertical lines in a coordinate plane are perpendicular if and only if the product of their slopes equals -1.

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