Parallel Lines: Coplanar lines that never intersect (have same slope)
Skew Lines: Non-coplanar lines that never intersect
Transversal Lines: Lines that intersect 2 or more coplanar lines
Interior Angles: Angles that are on the interior of a line
Exterior Angles: Angles that are on the exterior of a line
Consecutive Interior Angles: Interior angles that lie on the same side of the transversal
Alternative Interior Angles: Nonadjacent interior angles that lie on the opposite sides of the transversal
Alternative Exterior Angles: Nonadjacent exterior angles that lie on the opposite sides of the transversal
Corresponding Angles: Angles that lie on the same side of the transversal

3.2

Thanks to Ishwari Veerkar & Lekhana Chennuru

Corresponding Angles Postulate: If 2 parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
Alternate Interior Angles Theorem: If 2 parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.
Consecutive Interior Angles Theorem: If 2 parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary (add up to 180 degrees).
Alternate Exterior Angles Theorem: If 2 parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
Perpendicular Transversal Theorem: In a plane, if a line is perpendicular to 1 of the 2 parallel lines then it is perpendicular to the other.
Since postulates are accepted without proof, you can use the corresponding angles postulate to prove each of the theorems above.

3.3

Thanks to Lekhana Chennuru

In a coordinate plane, the slope of a line is the ratio of the change along the y-axis to the change along the x-axis between any 2 points on the line. [m = rise/run = y2 - y1 / x2 - x1]
Different types of slope: Positive Slope = / Negative Slope = \ Zero Slope = <----------> Undefined Slope = |
Slope can be interpreted as a rate of change, describing how a quantity, y, changes in relation to quantity, x. The slope of a line can also be used to identify the coordinates of any point on the line.
Postulate 3.2; Slopes of Parallel Lines = 2 distinct non vertical lines have the same slope if and only if they are parallel. All vertical lines are parallel.
Postulate 3.3; Slopes of Perpendicular Lines = 2 non vertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.
Slopes of Perpendiculars; If a line L has a slope of a/b, then the slope of a line perpendicular to line L is the opposite reciprocal, -b/a, since a/b (-b/a) = -1.

3.4

Thanks to Lekhana Chennuru

Slope Intercept form is "y = mx + b" and the point-slope form is "y - y1 = m (x - x1)".
When substituting negative coordinates, use parentheses to avoid making errors with the signs. When the slope of a line is not given, use 2 points on the line to calculate the slope. Then, use the point slope form to write an equation of the line.
Equation of a horizontal line = "y = b" --> b is the y-intercept of the line. Equation of a vertical line = "x = a" --> a is the x-intercept of the line.
Parallel lines that are not vertical have equal slope. 2 non vertical lines are perpendicular if the product of their slope is -1. Vertical and horizontal lines are ALWAYS perpendicular to one another. The word "linear" indicates a line.
The equations of a horizontal and vertical lines involve only one variable.

3.5

Thanks to Lekhana Chennuru

Converse of Corresponding Angles Postulate: If 2 lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
Parallel Postulate: If given a line and a point not on the line, then there exists exactly 1 line through the point that is parallel to the given point.
Parallel lines that are cut by a transversal create several pairs of congruent angles. These special angle pairs can also be used to prove that a pair of lines are parallel.
Theorem 3.5 -- Alternate Exterior Angles Converse -- If 2 lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the 2 lines are parallel.
Theorem 3.6 -- Consecutive Interior Angles Converse -- If 2 lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.
Theorem 3.7 -- Alternate Interior Angles Converse -- If 2 lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.
Theorem 3.8 -- Perpendicular Transversal Converse -- In a plane, if 2 lines are perpendicular to the same line, then they are parallel
Angle Relationships can be used to solve problems involving unknown values. The angle pair relationships formed by a transversal can be used to prove that 2 lines are parallel.
When 2 parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. When a pair of lines form angles that do not meet this criterion, the lies cannot possibly be parallel.

3.6

Thanks to Lekhana Chennuru and Ishwari Veerkar

Distance between a point and a line --> The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.
Postulate 3.6 -- Perpendicular Postulate: If given a line and a point not on the line, then there exists exactly 1 line through the point that is perpendicular to the given line.
An alternate definition of parallel lines is that 2 lines in a plane are parallel if they are everywhere equidistant. Equidistant means that the distance between 2 lines measured along a perpendicular line to the lines is always the same.
The distance between 2 parallel lines is the perpendicular distance between 1 of the lines and any point on the other line.
Theorem 3.9 --> Two lines equidistant from a third: In a plane, if 2 lines are each equidistant from a third line, then the 2 lines are parallel to each other.
Conversely, the locus of points in a plane that are equidistant from 2 parallel lines is a third line that is parallel to and centered between the 2 parallel lines.

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