Glencoe Geometry Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
Developed by FSCreations, Inc., Cincinnati, Ohio 45202
Send all inquiries to:
GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 3-1 Parallel Lines and Transversals
Lesson 3-2 Angles and Parallel Lines
Lesson 3-3 Slopes of Lines
Lesson 3-4 Equations of Lines
Lesson 3-5 Proving Lines Parallel
Lesson 3-6 Perpendiculars and Distance
Example 1 Identify Relationships
Example 2 Identify Transversals
Example 3 Identify Angle Relationships
Name all planes that are parallel to plane AEF.
Answer: plane BHG
Name all segments that intersect
Answer:
Name all segments that are parallel to
Answer:
Name all segments that are skew to
Answer:
Use the figure to name
each of the following.
a. Name all planes that are parallel to plane RST.
Answer: plane XYZ
b.
Answer:
c.
Answer:
d.
Answer:
BUS STATION Some of a bus station’s driveways
are shown. Identify the sets of lines to which line v
is a transversal.
Answer: If the lines are extended, line v intersects
lines u, w, x, and z.
BUS STATION Some of a bus station’s driveways
are shown. Identify the sets of lines to which line y is
a transversal.
Answer: lines u, w, x, z
BUS STATION Some of a bus station’s driveways
are shown. Identify the sets of lines to which line u
is a transversal.
Answer: lines v, x, y, z
BUS STATION Some of a bus station’s driveways
are shown. Identify the sets of lines to which line w
is a transversal.
Answer: lines v, x, y, z
HIKING A group of nature trails is shown. Identify the
sets of lines to which each given line is a transversal.
a. line a
Answer: lines c, d, e, f
b. line b
Answer: lines c, d, e, f
c. line c
Answer: lines a, b, d, e, f
d. line d
Answer: lines a, b, c, e, f
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: corresponding
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: alternate exterior
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: corresponding
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: alternate exterior
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: alternate interior
Identify
as alternate interior, alternate
exterior, corresponding, or consecutive interior
angles.
Answer: consecutive interior
Identify each pair of angles as alternate interior,
alternate exterior, corresponding, or consecutive
interior angles.
a.
Answer: consecutive interior
b.
Answer: corresponding
c.
Answer: alternate exterior
Identify each pair of angles as alternate interior,
alternate exterior, corresponding, or consecutive
interior angles.
d.
Answer: alternate interior
e.
Answer: corresponding
f.
Answer: alternate exterior
Example 1 Determine Angle Measures
Example 2 Use an Auxiliary Line
Example 3 Find Values of Variables
In the figure
and
Find
Corresponding Angles Postulate
Vertical Angles Theorem
Transitive Property
Definition of congruent angles
Substitution
Answer:
In the figure
Answer:
and
Find
Grid-In Test Item
What is the measure of RTV?
Read the Test Item
Be sure to identify it correctly
on the figure.
Solve the Test Item
Alternate Interior Angles Theorem
Definition of congruent angles
Substitution
Alternate Interior Angles
Theorem
Definition of congruent
angles
Substitution
Angle Addition
Postulate
Write each digit of 125 in a column of the grid.
Then shade in the corresponding bubble in each
column.
Answer:
Grid-In Test Item
What is the measure of IGE?
Answer:
ALGEBRA If
and
find x and y.
Find x.
by the Corresponding Angles
Postulate.
Definition of congruent angles
Substitution
Subtract x from each side and
add 10 to each side.
Find y.
by the Alternate Exterior Angles
Theorem.
Definition of congruent angles
Substitution
Simplify.
Add 100 to each side.
Divide each side by 4.
Answer:
and
ALGEBRA If
find x and y.
Answer:
Example 1 Find the Slope of a Line
Example 2 Use Rate of Change to Solve a Problem
Example 3 Determine Line Relationships
Example 4 Use Slope to Graph a Line
Find the slope of the line.
From (–3, 7) to (–1, –1), go
down 8 units and right 2 units.
Answer: – 4
Find the slope of the line.
Use the slope formula.
Let
be
be
and
.
Answer: undefined
Find the slope of the line.
Answer:
Find the slope of the line.
Answer: 0
a. Find the slope of the line.
Answer:
b. Find the slope of the line.
Answer: 0
c. Find the slope of the line.
Answer: 2
d. Find the slope of the line.
Answer: undefined
RECREATION For one manufacturer of camping
equipment, between 1990 and 2000, annual sales
increased by $7.4 million per year. In 2000, the total
sales were $85.9 million. If sales increase at the
same rate, what will be the total sales in 2010?
Slope formula
Simplify.
Multiply each side by 10.
Add 85.9 to each side.
The coordinates of the point representing the sales for
2010 are (2010, 159.9).
Answer: The total sales in 2010 will be about
$159.9 million.
CELLULAR TELEPHONES Between 1994 and 2000,
the number of cellular telephone subscribers
increased by an average rate of 14.2 million. In 2000,
the total subscribers were 109.5 million. If the
number of subscribers increase at the same rate,
how many subscribers will there be in 2005?
Answer: about 180.5 million
Determine whether
and
perpendicular, or neither.
are parallel,
Answer:
The slopes are not the same,
The product of the slopes is
are neither parallel nor perpendicular.
Determine whether
and
perpendicular, or neither.
are parallel,
Answer: The slopes are the same, so
are parallel.
Determine whether
and
perpendicular, or neither.
a.
Answer: perpendicular
b.
Answer: neither
are parallel,
Graph the line that contains Q(5, 1) and is parallel to
with M(–2, 4) and N(2, 1).
Slope formula
Substitution
Simplify.
The slopes of two parallel lines are the same.
The slope of the line parallel to
Graph the line.
Start at (5, 1). Move up 3
units and then move left 4
units.
Label the point R.
Answer:
Graph the line that contains R(2, –1) and is parallel to
with O(1, 6) and P(–3, 1).
Answer:
Example 1 Slope and y-Intercept
Example 2 Slope and a Point
Example 3 Two Points
Example 4 One Point and an Equation
Example 5 Write Linear Equations
Write an equation in slope-intercept form of the line
with slope of 6 and y-intercept of –3.
Slope-intercept form
Answer: The slope-intercept form of the equation of the
line is
Write an equation in slope-intercept form of the line
with slope of –1 and y-intercept of 4.
Answer:
Write an equation in point-slope form of the line whose
slope is
that contains (–10, 8).
Point-slope form
Simplify.
Answer:
Write an equation in point-slope form of the line whose
slope is
Answer:
that contains (6, –3).
Write an equation in slope-intercept form for a line
containing (4, 9) and (–2, 0).
Find the slope of the line.
Slope formula
Simplify.
Now use the point-slope form and either point to write an
equation.
Using (4, 9):
Point-slope form
Distributive Property
Add 9 to each side.
Using (–2, 0):
Point-slope form
Simplify.
Distributive Property
Answer:
Write an equation in slope-intercept form for a line
containing (3, 2) and (6, 8).
Answer:
Write an equation in slope-intercept form for a line
containing (1, 7) that is perpendicular to the line
the slope of
a line perpendicular to it is 2.
Point-slope form
Distributive Property
Add 7 to each side.
Answer:
Write an equation in slope-intercept form for a line
containing (–3, 4) that is perpendicular to the line
Answer:
RENTAL COSTS An apartment complex charges
$525 per month plus a $750 security deposit. Write
an equation to represent the total annual cost A for
r months of rent.
For each month of rent, the cost increases by $525. So
the rate of change, or slope, is 525. The y-intercept is
located where 0 months are rented, or $750.
Slope-intercept form
Answer: The total annual cost can be represented by the
equation
RENTAL COSTS An apartment complex charges $525
per month plus a $750 security deposit. Compare this
rental cost to a complex which charges a $200 security
deposit but $600 per month for rent. If a person expects
to stay in an apartment for one year, which complex
offers the better rate?
First complex:
Second complex:
Simplify.
Answer: The first complex offers the better rate: one year
costs $7050 instead of $7400.
RENTAL COSTS A car rental company charges $25 per
day plus a $100 deposit.
a. Write an equation to represent the total cost C for d days
of use.
Answer:
b. Compare this rental cost to a company which charges a
$50 deposit but $35 per day for use. If a person expects
to rent a car for 9 days, which company offers the better
rate?
Answer: The first company offers the better rate. Nine
days cost $325 instead of $365.
Example 1 Identify Parallel Lines
Example 2 Solve Problems with Parallel Lines
Example 3 Prove Lines Parallel
Example 4 Slope and Parallel Lines
Determine which lines,
if any, are parallel.
consecutive
interior angles are supplementary. So,
consecutive
interior angles are not supplementary. So, c is not parallel
to a or b.
Answer:
Determine which lines, if any, are parallel.
Answer:
ALGEBRA Find x and mZYN so that
Explore From the figure, you know that
and
You also know that
are alternate exterior angles.
Plan For line PQ to be parallel to MN, the alternate exterior
angles must be congruent.
Substitute the given angle measures into this equation
and solve for x. Once you know the value of x, use
substitution to find
Solve
Alternate exterior angles
Substitution
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Original equation
Simplify.
Examine Verify the angle measure by using the value of x to
find
Since
Answer:
ALGEBRA Find x and mGBA so that
Answer:
Given:
Prove:
Proof:
Statements
1.
2.
3.
4.
5.
6.
7.
.
.
Reasons
1. Given
2. Consecutive Interior Thm.
3. Def. of suppl. s
4. Def. of congruent s
5. Substitution
6. Def. of suppl. s
7. If cons. int. s are suppl.,
then lines are .
Given:
Prove:
Proof:
Statements
1.
2.
3.
4.
5.
6.
7.
Reasons
1. Given
2. Alternate Interior Angles
3. Substitution
. 4. Definition of suppl. s
5. Definition of suppl. s
6. Substitution
7. If cons. int. s are suppl.,
then lines are .
Answer:
Answer: Since the slopes are not equal, r is not parallel to s.
Example 1 Distance from a Point to a Line
Example 2 Construct a Perpendicular Segment
Example 3 Distance Between Lines
Draw the segment that represents the distance from
Answer:
Since 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,
Draw the segment that represents the distance from
Answer:
Construct a line perpendicular to line s through V(1, 5)
not on s. Then find the distance from V to s.
Graph line s and point V. Place the compass point at point
V. Make the setting wide enough so that when an arc is
drawn, it intersects s in two places. Label these points of
intersection A and B.
Put the compass at point A and draw an arc below line s.
(Hint: Any compass setting greater than
will work.)
Using the same compass setting, put the compass at
point B and draw an arc to intersect the one drawn in
step 2. Label the point of intersection Q.
Draw
.
and s. Use the slopes of
lines are perpendicular.
and s to verify that the
The segment constructed from point V(1, 5)
perpendicular to the line s, appears to intersect line s at
R(–2, 2). Use the Distance Formula to find the distance
between point V and line s.
Answer: The distance between V and s is about 4.24 units.
Construct a line perpendicular to line m through
Q(–4, –1) not on m. Then find the distance from
Q to m.
Answer:
Find the distance between the parallel lines a and b
whose equations are
and
respectively.
You will need to solve a system of equations to find the
endpoints of a segment that is perpendicular to both a and
b. The slope of lines a and b is 2.
First, write an equation of a line p perpendicular to a and b.
The slope of p is the opposite reciprocal of 2,
Use the y-intercept of line a, (0, 3), as one of the endpoints
of the perpendicular segment.
Point-slope form
Simplify.
Add 3 to each side.
Next, use a system of equations to determine the point
of intersection of line b and p.
Substitute 2x–3 for y in
the second equation.
Group like terms on each side.
Simplify on each side.
Substitute 2.4 for x in the equation
for p.
The point of intersection is (2.4, 1.8).
Then, use the Distance Formula to determine the distance
between (0, 3) and (2.4, 1.8).
Distance Formula
Answer: The distance between the lines is
2.7 units.
or about
Find the distance between the parallel lines a and b
whose equations are
respectively.
Answer:
and
Explore online information about the
information introduced in this chapter.
Click on the Connect button to launch your browser
and go to the Glencoe Geometry Web site. At this site,
you will find extra examples for each lesson in the
Student Edition of your textbook. When you finish
exploring, exit the browser program to return to this
presentation. If you experience difficulty connecting to
the Web site, manually launch your Web browser and
go to www.geometryonline.com/extra_examples.
Click the mouse button or press the
Space Bar to display the answers.
Click the mouse button or press the
Space Bar to display the answers.
Click the mouse button or press the
Space Bar to display the answers.
Click the mouse button or press the
Space Bar to display the answers.
Click the mouse button or press the
Space Bar to display the answers.
Click the mouse button or press the
Space Bar to display the answers.
End of Custom Shows
WARNING! Do Not Remove
This slide is intentionally blank and is set to auto-advance to end
custom shows and return to the main presentation.