CHAPTER THREE
 Parallel Lines are two lines that are coplanar and do not intersect.
 Skew Lines are lines that do not intersect and are not coplanar.
 Parallel Planes are two planes that do not intersect.
 Parallel Postulate states that 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 states that 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.
 The symbol for Parallel is . The parallel symbol with a slash through
it means not parallel.
 The symbol for Perpendicular is .
 Transversal is a line that intersects two or more coplanar lines at
different points. The two or more lines do not have to be parallel.
 If a transversal is perpendicular to one parallel line then it is
perpendicular to the other parallel line.
  does not have a transversal because there is intersection only at one
point.
 Corresponding Angles are two angles that occupy corresponding
positions or the same position on a different line.
 Alternate Exterior Angles are two angles that lie outside the two lines
on opposite sides of the transversal.
 Alternate Interior Angles are two angles that lie between the two lines
on opposite sides of the transversal.
 Consecutive Interior Angles or Same-Side Interior Angles are two
angles that lie between the two lines on the same side of the
transversal.
 If two lines are cut by a transversal 4 pairs of corresponding angles
are formed.
 Alternate Interior Angles Theorem states that if two parallel lines are
cut by a transversal, then each pair of alternate interior angles is
congruent.
 Alternate Exterior Angles Theorem states that if two parallel lines are
cut by a transversal, then each pair of alternate exterior angles is
congruent.
 Consecutive Interior Angles (Same-Side Interior) Theorem states that
if two parallel lines are cut by a transversal, then each pair of
consecutive interior angles is supplementary.
 Corresponding Angles Theorem states that if two parallel lines are cut
by a transversal, then each pair of corresponding angles is congruent.
 Flow Proof uses arrows to show the flow of the logical argument.
Each reason in a flow proof is written below the statement it justifies.
 If two lines intersect to form a linear pair of congruent angles, then the
lines are perpendicular.
 If two sides of two adjacent acute angles are perpendicular, then the
angles are complementary.
 If two lines are perpendicular, then they intersect to form 4 right
angles.
 Three types of Proofs:
o Two-Column
o Paragraph
o Flow
 If a transversal is perpendicular to one parallel line then it is
perpendicular to the other parallel line.
 Question: if a transversal is perpendicular to one of two parallel lines,
then what is the measure of all the angles formed? Answer: 90
degrees
 Corresponding Angles Converse states that if two lines are cut by a
transversal so that corresponding angles are congruent then the lines
are parallel.
 Alternate Interior Angles Converse states that if two lines are cut by a
transversal so that alternate interior angles are congruent then the lines
are parallel.
 Alternate Exterior Angles Converse states that if two lines are cut by a
transversal so that alternate exterior angles are congruent then the
lines are parallel.
 Consecutive Interior (same-side interior) Angles Converse states that
if two lines are cut by a transversal so that consecutive interior angles
are supplementary then the lines are parallel.
 Parallel Postulate states given a line and a point not on the line, then
there exists exactly one line through the point that is parallel to the
given line.
 When proving lines parallel, be sure to check for congruent
corresponding angles, alternate interior angles, alternate exterior
angles, or supplementary consecutive interior (same-side interior)
angles.
 Two lines may be perpendicular to the same line but not parallel if the
lines are not in the same plane. Lines must be coplanar to be parallel.
 If two lines are parallel to the same line, then they are parallel to each
other.
 In a plane, if two lines are perpendicular to the same line, then they
are parallel to each other.
 What can you conclude about two coplanar lines that are
perpendicular to the same line? They are parallel to each other.
 The steepness of a line is called slope.
 Horizontal axis is labeled X.
 Vertical axis is labeled Y.
 On a coordinate plane, any single point can be identified by an
ordered pair (x, y)

Slope = Change in Y / Change in X

M = (y2 – y1) / (x2 – x1) where x1  x2
 Slopes of Parallel Lines Postulate states that 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.
 The equation of a vertical line through a point at (x1, y1), is x = x1
 If a line has a slope of m and a y-intercept of b, then the SlopeIntercept Form of an equation of the line is y = mx + b.
 To graph a line using the Slope-Intercept Form, y = mx + b
o Graph the point (0, b)
o Use the slope m to find another point by moving the distance of
the change in y and then the distance of the change in x from
the
point (0, b).
 If you know two points on line p and two points on line q how could
you tell if line p was parallel to line q?
 You cannot assume that lines are parallel simply because they appear
to be.
 Slope is a measure of the change in two values. It is not the values
themselves but their relationship to one another.
 Draw graphs carefully. Graphing mistakes may lead to incorrect
answers.
 If two lines are equidistant from a third line, then the two lines are
parallel to each other.
 Parallel lines are equidistant from each other at every point on their
respective lines.
 Slopes of Perpendicular Lines Postulate states that in a coordinate
plane, two non-vertical lines are perpendicular if and only if the
product of their slopes is –1.
 Take the slope of your first line, flip it over to make the reciprocal and
then change the sign.
 Vertical and horizontal lines are perpendicular.
 The symbol for perpendicular is .
 If the slope of one of two perpendicular lines is a/b. What is the slope
of the other line? - b/a
 The converse of the Slopes of Perpendicular Lines Postulate is also
true. If the product of the slopes of two lines is –1, then the lines are
perpendicular.
 The distance from a line to a point not on the line is the length of the
segment perpendicular to the line from the point.