Chapter 3 VPTF

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parallel lines

Two lines that do not intersect and are coplanar

skew lines

Lines that do not intersect are not coplanar.

parallel planes

Two planes that do not intersect

transversal

a line that intersects two or more coplanar lines at different points.

corresponding angles

Two angles that are formed by two lines and a transversal and occupy corresponding positions.

alternate interior angles

Two angles that are formed by two lines and a transversal and lie between the two lines and on opposite sides of the transversal.

alternate exterior angles

Two angles that are formed by two lines and a transversal and lie outside the two lines and on opposite sides of the transversal.

consecutive interior angles

Two angles that are formed by two lines and a transversal and lie between the two lines and on the same side of the transversal. Also called same-side interior angles.

paragraph proof

A type of proof written in paragraph form.

slope

The slope m of a non-vertical line is the ratio of the vertical change (the rise) to horizontal change (the run) between any two points on the line.

slope-intercept form

A linear equation written in the form y = mx + b where m is the slope and b is the y-intercept of the equation's graph.

standard form of a linear equation

A linear equation written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero.

distance from a point to a line

The length of the perpendicular segment from the point to the line.

Parallell Postulate

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.

Perpendicular Postulate

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.

Corresponding Angles Postulate

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

Corresponding Angles Converse

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

Slopes of Parallell Lines

In a coordinate plane, two non vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

Slopes of Perpendicular Lines

In a coordinate plane, two non vertical lines are perpendicular if and only if the product of their slopes is -1. Horizontal lines are perpendicular to vertical lines.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

Alternative Interior Angles Converse

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

Alternate Exterior Angles Converse

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

Consecutive Interior Angles Converse

If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

Transitive Property of Parallel Lines

If two lines are parallel to the same line, then they are parallel to each other.

If two lines intersect to form a linear pair of congruent angles

then the lines are perpendicular.

It two lines are perpendicular

then they intersect to form four right angles.

If two sides of two adjacent acute angles are perpendicular

then the angles are complementary.

Perpendicular Transversal Theorem

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Lines Perpendicular to a Transversal Theorem

In a plan,e if two lines are perpendicular to the same line, then they are parallel to each other.

Example: