GEOMETRY
APPLICATIONS
Chapter 3: Parallel & Perpendicular Lines
Name:______________________________
Teacher:____________________________
Pd: _______
0
Table of Contents
DAY 1: (Ch. 3-1 & 3-2) SWBAT: Identify parallel, perpendicular, and skew lines.
Identify the angles formed by two lines and a transversal.
Pgs: 2-5
DAY 2: (Ch. 3-2) Calculate for missing angles when parallel lines are cut by a transversal
Pgs: 6-10
DAY 3: Full Period Quiz: Day 1 to DAY 2
DAY 4: (Ch. 3-5) SWBAT: Calculate the slope of a line using the slope formula.
Pgs: 11-15
DAY 5: SWBAT: Use slopes to identify parallel and perpendicular lines
Pgs:16-19
Take Home Quiz: Day 4 to DAY 5
DAY 6: SWBAT: Graph and Write Equations of Lines given a Slope and Point
Pgs: 20-24
DAY 7: SWBAT: Write the equation of a line given two points on the line
Pgs: 25-27
DAY 8: SWBAT: Graph Lines in Slope – Intercept and Point – Slope Form
Pgs: 28-33
DAY 9: SWBAT:
Pgs: 34-37
Graph and Write Equations of Parallel & Perpendicular Lines given a Slope and Point
DAY 10: Full Period Quiz: Day 6 to DAY 9
DAY 11: SWBAT: Graph the Solutions to Quadratic Linear Systems
Pgs: 38-43
DAY 12: SWBAT: Graph the Solutions to Quadratic Linear Systems
Pgs: 44-45
DAY 13: Chapter 3 Practice Test
DAY 14: Chapter 3 Test
1
Day 1 - 3-1 & 3-2: Lines and Angles
SWBAT: Identify parallel, perpendicular, and skew lines.
Identify the angles formed by two lines and a transversal.
Warm – Up: Matching Column
supplementary angles
points that lie in the same plane
point
two angles whose sum is 180°
coplanar points
the intersection of two distinct intersecting lines
linear pair
a pair of adjacent angles whose non-common sides are
opposite rays
Example 1: Lines
Term
Description
Example 1
Example(s)
 are coplanar
 do not intersect
 intersect at 90° angles
 are not coplanar
 are not parallel
 do not intersect
 planes that do not
intersect
2
Practice:
Identify each of the following:
a. A pair of parallel segments
b. A pair of skew segments
c. A pair of perpendicular segments
d. A pair of parallel planes
Example 2: Angles
A _____________________ is a line that intersects two coplanar lines at two different points.
Term
Description
Example 1
Example(s)
Lie on:
 the same side of the
transversal t
 on the same sides of
lines r and s
Nonadjacent angles
that lie on:
 opposite sides of the
transversal t
 between lines r and s
Lie on:
 opposite sides of the
transversal t
 outside lines r and s
Lie on:
 the same side of the
transversal t
 between lines r and s
3
Practice
Identify each of the following:
a. A pair of alternate interior angles
b. A pair of corresponding angles
c. A pair of alternate exterior angles
d. A pair of same-side interior angles
Example 3:
Line l and Line m are parallel. Find each missing angle.
Practice
Line l and Line m are parallel. Find each missing angle.
4
Homework:
In the diagram, parallel lines AB and CD are intersected by a
transversal EF at points X and Y, m FYD = 123. Find
AXY.
5
Day 2 - Chapter 3 – 2 (Parallel Lines and Related Angles)
SWBAT: Calculate for missing angles when parallel lines are cut by a transversal
Warm – Up
Classify each pair of angles as alternate interior angles, alternate exterior angles,
same-side interior angles, corresponding angles, or vertical angles.
1)
2)
1
3)
1
2
4)
1
2
2
5)
1
1
6)
2
1
2
2
State the angle relationship that justifies each statement.
7) m 3 + m 4 = 180
_______________________
8)
1
5
_______________________
9)
3
5
_______________________
10)
5
8
_______________________
1 2
4 3
5 6
11) m 4 + m 5 = 180
7 8
_______________________
Find the m 1 and explain the angle relationship.
12.
1
13.
14
165
120
55
1
1
6
Proving Lines Parallel
15.
16.
17.
Perpendicular Lines
18. Find the measure of
b.
19. Find x and measure of
b.
b
Algebra Related Questions
In the accompanying diagram, m
ABC = (4x + 22) and m
Part a: Which relationship describes
ABC and
Part b: What is the value of x and what is m
DCE = (5x) .
DCE?
DCE?
7
Homework
1) In the accompanying diagram, l ll m and m
1 = (3x + 40) and m
Part a: Which relationship describes
2?
1 and
2 = (5x – 30) .
l
1
Part b: What is the value of x and what is m
1?
2) In the accompanying diagram, l ll m and m
1 = (9x - 8) and m
Part a: Which relationship describes
2?
1 and
m
2
2 = (x + 72) .
l
1
Part b: What is the value of x and what is m
2?
2
3) In the accompanying diagram, p ll q.
m
5(x - 4)
p
(x + 12)
Part a: Which relationship describes the given angles?
q
Part b: What is the value of x?
8
4) In the accompanying diagram, p ll q. If m
1 = (4x + 1) and m
2 = (5x – 10)
p
q
1
2
Part a: Which relationship describes 1 and 2?
__________________________________________________________________________________________
__________________________________________________________________________________________
Part b: What is the value of x?
Part c: What is the m 2?
5) In the accompanying diagram, l ll m. If m
1 = (3x + 16) and m
Part a: Which relationship describes
2?
1 and
2 = (x + 12)
l
1
Part b: What is the value of x?
2
m
Part c: What is the m 1 and m 2?
9
6) Find the m
6.
7) Find the measure of
3,
4, and
5.
m 3=
m 4=
m 5=
8)
m 1=
m 2=
m 3=
m 4=
m 6=
10
Day 4 - Chapter 3-5 Slope of a Line
SWBAT: Calculate the slope of a line using the slope formula.
Warm – Up
Solve for x.
The Slope “m” of a line passing through points (x1, y1) and (x2, y2) is the ratio of the difference in the y-coordinates to
the corresponding difference in the x-coordinates.
y
Symbols: m =
run
(x1, y1)
rise
(x2, y2)
x
11
Example 1: Find the slope of (3,3) and (8,7).
Example 2: Find the slope of (2,3) and (-7,8).
Example 3: Find the slope of (-5,3) and (2,3).
12
Finding Slope From Graphs and Tables.
The graph or table shows a linear relationship. Find the slope.
4)
6)
5)
7)
Finding Slope from an Equation
8) Find the slope of the line described by 4x – 2y = 16.
9) Find the slope of the line described by 2x + 3y = 12.
13
HOMEWORK:
1) Find the slope of (2, 5) and (8, 1).
2) Find the slope of (5, –7) and (6, –4).
14
Finding Slope from an Equation
14. Find the slope of the line described by 6x – 3y = 18.
15. Find the slope of the line described by 3x + 4y = 16.
15
Day 5 - Chapter 3-6: Slopes of Parallel and Perpendicular Lines
SWBAT: Use slopes to identify parallel and perpendicular lines
Use the slope formula to determine the slope of each line.
Pairs of Lines
Parallel Lines
Y = 5x + 8
Y = 5x - 4
Same Slope
different
yintercept
Perpendicular
Lines
Y = 2x + 6
Y = -½ x - 4
Slopes are
Negative
Reciprocals
Neither
Coinciding Lines
Y = 3x – 5
Y = 2x – 4
Y = 5x + 2
Y = 2x - 4
Different Slopes
Same slope,
Same y-intercept
16
Example 1
Find the slope of a line parallel to the graph of each equation.
2
3
a) y = - x – 1
b) y = 4x - 1
slope = _____
slope = _____
c) 2x - 3y = 2
slope = _____
Independent Practice
Find the slope of a line parallel to the graph of each equation.
5
3
a) y = - x – 1
b) y = -3x - 1
c) 4x - 2y = 2
slope = _____
slope = _____
slope = _____
Example 2
Find the slope of a line perpendicular to the graph of each equation
2
a) y = 2x + 1
b) y = x - 4
c) 4x – 2y = 9
7
slope = _____
slope = _____
slope = _____
Independent Practice
Find the slope of a line perpendicular to the graph of each equation
3
a) y = -4x + 1
b) y = x - 4
c) 6x – 3y = 9
5
slope = _____
slope = _____
slope = _____
17
Example 3 Determine whether the lines are parallel, perpendicular, coincide, or neither.
3x + 5y = 2
and
3x + 6 = -5y
Determine whether the lines are parallel, perpendicular, coincide, or neither.
a) y – 5 = 2x + 6 and y – 3 = – ½x
b) 2y = 4x + 12 and 4x – 2y = 8
c) 2y – 4x = 16 and
d) y + 3 = ¾x + 16 and 3y = -4x - 9
y – 10 = 2x - 2
Regents Question
Shanaya graphed the line represented by the equation y = 2x – 6.
A. Write an equation for a line that is parallel to the given line.
B. Write an equation for a line that is perpendicular to the given line.
C. Write an equation for a line that is identical to the given line but has different coefficients.
Challenge:
Determine whether the lines are parallel, perpendicular, coincide, or neither.
y – (-3) = ¾(x + 16),
3y = -4x - 9
18
Homework:
Find the slope of a line parallel to the graph of each equation.
8
3
a) y = - x – 1
b) y = -9x - 1
c) 10x - 2y = 2
slope = _____
slope = _____
slope = _____
Find the slope of a line perpendicular to the graph of each equation
4
a) y = -6x + 1
b) y = x - 4
c) 12x – 3y = 9
5
slope = _____
slope = _____
slope = _____
Determine whether the lines are parallel, perpendicular, coincide, or neither.
19
Day 6 - Chapter 3-6: Equations of Lines Given Slope and Point
SWBAT: Graph and Write Equations of Lines given a Slope and Point
Warm – Up
1. Use the slope formula to determine the slope of the line that passes through A(3, 7) and B(-3, 1).
2. Graph the lines and use the slopes to determine whether they are parallel, perpendicular, or neither.
and
for A(-2,5) and B(-3, 1), X(0, -2) and Y(1, 2)
20
Writing Equations of Lines
Example 1
1) Write an equation of a line that passes through the given point with the given
slope:
(–1, 2) ; m = 2
Example 2
2) Write an equation of a line that passes through the given point with the given
slope:
(5, -2) ; m =
21
Practice
1) Write an equation of a line that passes through the given point with the given
slope:
(2, -5) ; m = -2
2) Write an equation of a line that passes through the given point with the given
slope:
(0, 3) ; m = 1
22
3) Write an equation of a line that passes through the given point with the given
slope:
(1, 2) ; m = -3
4) Write an equation of a line that passes through the given point with the given
slope:
(-1, 5) ; m =
23
Homework
Write an equation of a line that passes through the given point with the given slope:
1) (3, 0) ; m =
2) (2, 6) ; m =
3) (3, -1) ; m =
24
Day 7 - Chapter 3-6: Equations of Lines Given Two Points
SWBAT: Write the equation of a line given two points on the line
Warm – Up
Find the slope of the line passing through the points (6,4) and (-2,-6).
Writing Equations of Lines
Example 1
Write the equation of the line through the two points (1,1) and (2,3).
Example 2
Write the equation of the line through the two points (5,0) and (3,2)
25
Practice
1. Write the equation of the line through the two points (8,5) and (9,6)
2. Write the equation of the line through the two points (0,0) and (-3,4)
3. Write the equation of the line through the two points (-3,-4) and (-5,-6)
26
Homework
Write the equation of the line through the two points.
1. (3,1) and (6,2)
2. (-2,6) and (-4,5)
3. (1,-4) and (-2,8)
4. (-3,4) and (0,6)
27
Day 8 - Chapter 3-6: Equations of Lines in Slope Intercept Form and Point Slope Form
SWBAT: Graph Lines in Slope – Intercept and Point – Slope Form
Warm – Up
1) Write an equation of a line that passes through the point (4,-2) with slope 1.
2) Write an equation of a line that passes through the points (–1, 0) and (1, 2).
28
Linear Equations written in the form y = mx + b are called the slope-intercept form.
When an equation is written in this form, m is the ______ and b is the ____________.
Find the slope and the y-intercept, then graph.
a. y = -
2
x–4
3
b. y =
1
x+2
5
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
c. y = 4x + 1
d. y = -2x
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
29
Linear Equations written in the form y – y1 = m(x – x1) are called the point-slope form.
Find the slope and the y-intercept, then graph.
a. y + 3 = -2(x – 1)
b. y - 3 = -2(x +4)
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
c. y + 4 = 4(x +2)
d. y - 1 =
2
(x + 3)
3
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
30
Write an equation of each line below.
a.
d.
b.
e.
c.
f.
31
HOMEWORK
Find the slope and the y-intercept, then graph.
a. y = -3x + 4
b. y - 5 = 2(x +6)
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
c. x = 5
d. y + 4 =
2
(x - 6)
3
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
32
Find the slope and the y-intercept, then graph.
e. y -7 = x + 4
f. y = 2
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
g. y – x = -3
1
h. y = - x + 1
3
slope = _________
slope = _________
y - intercept = _______
y- intercept = _______
33
Day 9 –Chapter 3-6: Equations of Parallel & Perpendicular Lines
SWBAT: Graph and Write Equations of Parallel & Perpendicular Lines given a Slope and Point
Warm – Up
Writing Equations of Lines
Example 1
Practice:
A)
(-2, 2), y = 4x - 2
34
B)
(4, -2), y = -2x + 3
Example 2
(4, 2), y =
1
2
x+1
Practice:
C)
(-8, -7), y = -x - 8
35
D)
(6, -2), y = -3x - 6
Challenge Problem
(6, 4), y = 7x + 1
Wrap Up
List 3 things you learned today; 2 key terms you learned; and 1
question you have about today’s lesson.
3
2
1
36
Homework
1)
2)
3)
4)
37
Day 11 - Chapter 3 – 6: Quadratic Linear Systems
SWBAT: Graph the Solutions to Quadratic Linear Systems
Warm – Up
1. Write an equation of the line that passes through the given point and is parallel to the graph of the
equation below.
2. Write an equation of the line that passes through the given point and is perpendicular to the graph of the
equation below.
38
39
Example 2: Regents Questions
40
Practice 3:
41
Name: ______________________________________________________Date: ________ Ms. Williams
Homework
SWBAT: Solve Quadratic-Linear Systems
1.
2.
3.
4. What is the equation of a line that is perpendicular to -3y = 7x – 2 and passes through the
point (0, -8)?
42
5.
43
Day 12– Quadratic Linear Systems
Graph the lines and find the points of intersection.
1.
2.
3.
44
4.
5. y = x2 - 9
y = -5
6. y = x2 – 2x – 3
x=1
45