Chapter 3 Midterm
Review
Parallel and Perpendicular Lines
By: James, Jeremy, Eric, and Karthik
Introduction
 Chapter 3 is about parallel and perpendicular lines. Parallel and
perpendicular line theorems are very important.
 Section 1-3: Identifying angle relations given parallel lines with transversals.
Incorporate angle relationships into proofs. Use algebra to find angle
measures.
 Section 3-4: Using slope equations to identify parallel and perpendicular
lines.
 Section 5: Using angle relationships to prove lines parallel.
 Section 6: Finding the distance between a point and a line as well a
between two parallel lines.
Section 1: Key Terms
 Parallel Lines: Lines or segments that do not intersect with one another.
Parallel lines are usually mark with corresponding arrows.
 Parallel Planes: Like parallel lines, planes can be parallel to one another
 Skew Lines: Lines that do not intersect and are not coplanar
 Transversal: A line that intersects two or more lines in a plane at different
points
g
e
f
Lines e and f are parallel. Line g is the
transversal.
Section 1: Angle Pair
Relationships
 Corresponding angles are angles in the same place on two sides of a
transversal.
 Alternate interior angles are on alternate sides of the inside of the
transversal.
 Alternate exterior angles are on alternate sides of the outside of the
transversal.
 Consecutive interior angles are consecutive angles on the inside of the
transversal.
Section 1 Example
1
 Angle 3 and Angle 5 are alternate interior.
2
4
3
 Angles 2 and 7 are alternate exterior.
5
6
7
 Angles 4 and 5 are consecutive interior.
8
Section 2: Angles and Parallel Lines
 Corresponding Angles Postulate (CAP): When two parallel lines are cut by a
transversal, the corresponding angles are congruent.
c
Line a parallel to line b
a
1 2
<1 ≡ <5, <2 ≡ <6
3 4
<7 ≡ <3 , <8 ≡ <4
b
5 6
7 8
Section 2 Continued
 Alternate Interior Angles Theorem: If two parallel lines are cut by a
transversal, each pair of alternate interior angles is congruent
 Consecutive Interior Angles Theorem: If two parallel lines are cut by a
transversal, each pair of consecutive interior angles are supplementary
 Alternate Exterior Angles: If two parallel lines are cut by a transversal, the
alternate exterior angles are congruent
Section 3: Slopes of Lines
 Words to Know:
 Slope: the rise and run of a line.
 Formula: Delta y over delta x
 Commonly referred to with Cartesian Coordinates.
 Rate of Change: how a quantity changes over time.
 Special Slopes:
 Parallel Lines: will have the same slope.
 Perpendicular Lines: will have opposite reciprocals as their slopes.
Section 3 Example
 Find the slope of this line
Math History
 Rene Descartes – French Mathematician who invented the Cartesian
Coordinate system and is widely recognized as the father of analytical
geometry
 Cartesian Coordinates: Specifies a point in a plane with horizontal and
vertical coordinates (x,y). This can be expanded into higher dimensions with
simply adding more variables (x,y,z, etc.)
 Analytical Geometry: Also known as coordinate geometry, deals with
geometry on the coordinate plane. It uses algebraic principles to solve
geometric problems.
How Math History relates to this
Chapter
 Analytic Geometry, which Descartes developed is very closely tied to this
chapter
 Used to find slopes of lines in cartesian coordinates, along with their
parallels and perpendiculars.
 Used to find distances and midpoints between points
 Used for finding the distance from a point to a line (perpendicular distance)
as well as the distance between 2 parallel lines
Section 4: Equations of Lines
 Equations of lines:
 Point-slope form: y-y1=m(x-x1)
 Slope-intercept form: y=mx+b
 Uses for equations:
 One slope, one point: Point-slope form
 Two points: Slope -intercept form
 One point, one equation: slope-intercept form
Section 4 Example
 Write the equation of a line with point A(1,6) and the slope of -6 point slope
form
 What is the equation of a line containing the points (1,9) and (-8, 7)?
Section 5: Proving Lines Parallel
Postulates:
 CAP Converse: If two lines are cut by a transversal so that the corresponding
angles are congruent, then the lines are parallel.
 Parallel Lines Postulate: If given a line and a point not on the line, there is exactly
one line through the point that is parallel to the given line.
Section 5: Continued
 Theorems:
 AEA Converse: If two lines are cut by a transversal so that the alternate exterior
angles are congruent, then the two lines are parallel.
 AIA Converse: If two lines are cut by a transversal so that the alternate interior
angles are congruent, then the two lines are parallel.
 CIA Converse: If two lines are cut by a transversal so that the consecutive interior
angles are supplementary, then the lines are parallel.
 Parallel Perpendicular Theorem: If two lines are perpendicular to the same line,
then they are parallel.
Section 5 Example
2
9
1
3 4
11 12
8

7
5
If angles 1 and 7 are 6
congruent, then the lines are
If angles 1 and 6 are congruent,
14 13
15 16
then the lines are parallel.
congruent.
If angles 3 and 7 are congruent, then the
lines are parallel.
10
If the two lines are both perpendicular to
the transversal, they are parallel.
Section 6: Perpendicular Lines and
Distance
 Distance between a point and a line is the length of the segment
perpendicular to the line from the point.
 The distance between parallel lines is the distance from one point on one
line to another point on the other line.
 Theorem 3.9: In a plane, if two lines are equidistant from a third line, then
the two lines are parallel.
Section 6 Example 1
 Find the distance between two lines with the equations of y = 6x – 9 and
y = 6x + 3
Section 6 Example 2
 Find the distance between point A(-3, -4) and a line with the equation of
y = -4x + 9
Questions?