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Chapter 3
Name ________________________
Geometry Block
3.1 identify Pairs of Lines and Angles
Vocabulary:
Parallel lines – two lines that do not intersect and are coplanar.
The symbol // means “is parallel to.”
Perpendicular lines – two lines that intersect to form a right angle.
Skew lines – two lines that do not intersect and are not coplanar.
Parallel planes – two planes that do not intersect.
A transversal is a line that intersects two or more coplanar
lines at different points.
3.1 identify Pairs of Lines and Angles
Lines m and n are parallel lines . Symbol: m ǁ n
k
n
T
Lines m and k are skew lines.
m
U
Lines k and n are intersecting lines,
And there is a plane (not shown) containing them.
Planes T and U are parallel planes.
Symbol: T ǁ U
Example 1
Identify relationships in space
Think of each segment in the figure as part of a line. Which line(s) or
plane(s) in the figure appear to fit the description?
1) Line(s) parallel to 𝐶𝐷 and containing point A.
2) Lines(s) skew to 𝐶𝐷 and containing point A.
3) Line(s) perpendicular to 𝐶𝐷 and containing
point A.
4) Plane(s) parallel to plane EFG and containing
point A.
Parallel and Perpendicular Lines
Parallel Postulate
P
•
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.
l
There is exactly one line through P parallel to l
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.
There is exactly one line through P perpendicular to l.
P•
l
Example 2 Identify parallel and perpendicular lines
Photography
The given line markings show how
the roads are related to one another.
Angles and Transversals
Corresponding angles - have corresponding
positions on the lines and the transversal.
Alternate Interior angles - lie between
the two lines and opposite sides of the
transversal.
Consecutive Interior angles - lie between the
two lines and on the same side of the
transversals.
Alternate Exterior angles - lie outside the
two lines and opposite sides of the
transversal.
Example 3 Identify angle relationships
Identify all pairs of angles of the given type.
1) Corresponding
2) Alternate Interior
3) Alternate Exterior
4) Consecutive Interior
SOLUTION
3.2 Use Parallel Lines and Transversals
Corresponding Angles Postulate
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles are ≅.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are ≅.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs alternate exterior angles are ≅.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles
are complementary (equal to 180⁰).
Example 1 Identify congruent angles
The measure of three of the numbered angles is 120°.
Identify the angles. Explain your reasoning.
SOLUTION
Example 2 Use properties of parallel lines
ALGEBRA
SOLUTION
Find the value of x.
Example 3 Prove the Alternate Interior Angles Theorem
Prove that if two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are congruent.
SOLUTION
Example 4 Solve real-world problem
Science
When sunlight enters a drop of rain, different colors of light leave
the drop at different angles. This process is what makes a
0
rainbow. For violet light, m2  40 . What is m1 ?
How do you know?
3.4 Prove Lines are Parallel
How can you Prove that Lines are Parallel?
Corresponding Angles Converse:
If two lines are cut by a transversal
so the corresponding angles are
congruent (≅), then the lines are
parallel.
Examples: Find the value of x that will make line u and v parallel.
a)
b)
Example 1
Prove that if two lines are cut by a transversal so the
corresponding angles are congruent, then the lines are
parallel.
SOLUTION
GIVEN :
PROVE :
4  5
g
h
Alternate Interior Angles Converse:
If two lines are cut by a transversal so
the alternate interior angles are
congruent (≅), then the lines are
parallel.
2
6
Examples: Find the value of x that will make line u and v parallel.
a)
b)
Example 2
Prove the Alternate Interior Angles Converse
In the figure,
Prove:
p ⃦q
SOLUTION
𝒓 ⃦𝒔 and 1  3
Alternate Exterior Angles Converse:
If two lines are cut by a transversal
so the alternate exterior angles are
congruent (≅), then the lines are
parallel.
Examples: Find the value of x that will make line u and v parallel.
a)
b)
Example 3
Prove the Alternate Exterior Angles Converse
In the figure, 𝒓 ⃦𝒔 and 1  16 .
Prove:
r
s
p
1
p ⃦q
q
8
16
Consecutive Interior Angles Converse:
If two lines are cut by a transversal
so the consecutive interior angles
are supplementary (1800), then the
lines are parallel.
Examples: Find the value of x that will make line u and v parallel.
a)
b)
Example 4
Prove the Consecutive Interior Angles Converse
In the figure, 𝒓 ⃦𝒔 and 6  10 .
Prove:
r
s
10
p ⃦q
4
6
12
p
q
Transitive Property of Parallel Lines
p
If two lines are parallel to the same line,
then they are parallel to each other.
q
r
If 𝑝 ⃦ 𝑞 and 𝑞 ⃦ 𝑟 , then 𝑝 ⃦ 𝑟.
Example:
U.S. Flag
The flag of the United States has 13 alternating red and white
stripes. Each stripe is parallel to the stripe immediately below it.
Explain why the top stripe is parallel to the bottom stripe.
3.4 Find and Use Slopes of Lines
Key Concepts:
Slope  the ratio of vertical change (rise) to horizontal change (run)
between any two points on the line
‘m’ (lower case m) is the symbol used to represent the slope
Change in y or y Rise
y  y1
m

m 2
Change in x or x Run
x2  x1
Negative slope  falls from the left to the right
Positive slope  rises from left to right
Zero slope (m = 0)  horizontal (line)
Undefined slope  vertical (line)
Example 1
Find slopes of lines in coordinate plane
Find the slope of line a and line d.
Solution
Picture this!
Two parallel lines never intersect.
The slopes
(𝑟𝑖𝑠𝑒)
(𝑟𝑢𝑛)
of these
two lines are the same.
𝒎𝟏 = 𝒎𝟐
Two perpendicular lines form a right angle.
The slopes of these two lines
are exact opposite reciprocals
of each other (the product of
their slopes is -1.)
𝒎𝟏 ∗ 𝒎𝟐 = −1
Example 2
Identify parallel lines
Find the slope of each line. Which lines are parallel?
Example 3
Graphing
Line h passes through (3, 0) and (7, 6).
Graph the line perpendicular to h that
passes through the point (2, 5).
Solution
3.5 Write and Graph Equations of Lines
Slope-intercept form of a linear equation is 𝒚 = 𝒎𝒙 + 𝒃
𝒎 = slope
𝒃 = y-intercept
Rewrite the following equations in slope-intercept form.
1) 4𝑥 + 𝑦 = −8
2) 11𝑥 + 7𝑦 = −21
Example 1
Write an equation of a line from a graph
Write an equation of the line in slope-intercept form.
a)
b)
How to write an equation of a parallel line
Write an equation of the line passing through the point (4, 4) that
9
is parallel to the given line with the equation 𝑦 = 𝑥 − 1 .
2
Steps:
Determine the slope of the given line.
m = ______
Determine the slope of the parallel line.
m = ______
Plug in the given information (4, 4) into the slope-intercept form.
x = ___ and y = ____
y = mx + b______________
Solve for the y-intercept of the desired equation.
Write the equation in slope-intercept form: _________________
Example 2
Write an equation of a parallel line
3
1) Given line equation: 𝑦 = − 2 𝑥 + 4 ; passing through point (2, 1).
2) Given line equation: 𝑦 = −2 𝑥 − 2 ; passing through point (0, -3).
How to write an equation of a perpendicular line
Write the slope-intercept form of the equation of the line that is
1
perpendicular to the line 𝑦 = 𝑥 + 3 and contains the point (-2, 4).
4
Steps:
Determine the slope of the given line.
m = ________
Determine the slope of the perpendicular line.
m = _______
Plug in the information into the slope-intercept form.
x = ____ and y = ____
y = mx + b____________
Solve for the y-intercept of the desired equation.
Write the equation in slope-intercept form: ______________
Example 3
Write an equation of a perpendicular line
1
1) Given line equation: 𝑦 = − 4 𝑥 + 1 ; passing through point (1, 5).
1
2) Given line equation: 𝑦 = 3 𝑥 − 3; passing through point (-2, 5).
3.6
Distance from a Line
The distance from a point to a line is the length of the
perpendicular segment from the point to the line. This
perpendicular segment is the shortest distance between
the point and the line.
C
A
k
D
E
F
>
>
m
p
B
Distance between two parallel lines.
Distance from a point to a line.
Example 1
Find the distance between two parallel lines.
Example 2