Download Unit 3 Parallel and Perpendicular Lines

Geometry
Unit 3
Parallel and Perpendicular Lines
Name:________________________________________________
1
Geometry
Chapter 3 – Parallel and Perpendicular Lines
***In order to get full credit for your assignments they must me done on time and you must SHOW
ALL WORK. ***
1.____ (3-1) Parallel Lines and Transversals –Day 1 Page 129 – 130 # 22 – 46 even
2. ____ (3-2) Angles and Parallel Lines – Day 1 Page 136 – 137 # 14 – 25, 32 – 36
3. ____ Wrap-Up (3-1) and (3-2) 3-1/3-2 Practice Worksheet and 3-2 Application WS
4.____ (3-3) Slopes of Lines – Day 1 Page 142 # 16 – 38 even
5. ____ (3-3) Slopes of Lines – Day 2 3-3 Practice Worksheet and 3-3 Application WS
6. ____ (3-4) Equations of Lines – Day 1 Page 148 # 15 – 43 odd, 46 – 49
7. ____ (3-4) Equations of Lines – Day 2 3-4 Practice Worksheet and 3-4 Application WS
8. ____ (3-5) Proving Lines Parallel – Day 1 Page 155 # 13 – 24, 26 – 31, 38, 39
9. ____ (3-5) Proving Lines Parallel – Day 2 Section 3-5 Practice Worksheet
10. ____ Chapter 3 Review WS
2
3
Date: _____________________________
Section 3 – 1: Parallel Lines and Transversals
Notes
Parallel Lines: coplanar lines that do not ____________________
 Symbol:
 Ex:
Parallel Planes: ______________ that do not ___________________
 Ex:
Skew Lines: lines that do not ___________________ and that are not ________________
Example #1: Refer to the figure on the right.
a.) Name all planes that are parallel to plane AEF.
b.) Name all segments that intersect AF .
c.) Name all segments that are parallel to DC .
d.) Name all segments that are skew to AD .
4
Transversal: a line that intersects
Ex:
_________ or more lines in a plane
at different _______________
Example #2: Identify the pairs of lines to which each given line is a transversal.
a.) p
b.) r
c.) q
d.) t
Key Concept (Transversals and Angles):
Transversal p intersects lines q and r.
a.) Exterior Angles
b.) Interior Angles
c.) Consecutive Interior Angles
d.) Alternate Exterior Angles
e.) Alternate Interior Angles
f.) Corresponding Angles
5
CRITICAL THINKING
1.) Find the Error: Kenisha and Eric are naming alternate interior angles in the figure
at the right. One of the angles must be 4 . Who is correct? Explain your reasoning.
2.) A good way to remember new mathematical terms is to relate them to words that
you use in everyday life. Many words start with the prefix trans-, which is a Latin root
meaning across. List four English words that start with trans-. How can the meaning of
this prefix help you remember the meaning of transversal?
6
7
Discovering Properties of Parallel Lines
Directions:
1. Using the lines on a piece of paper as a guide, draw a pair of parallel lines.
2. Draw a transversal intersecting the parallel lines and number the angles formed from 1 to 8.
3. Use a protractor to measure each of the numbered angles and fill in the chart below.
Angle
Measure
Angle
1
5
2
6
3
7
4
8
Measure
Using the table and your picture, make as many different observations as possible. (Make sure to include
all eight angles)
8
Using the table, your picture, and your observations, identify every pair of each of the following types of
angles and make a conjecture about their measures.
1. Vertical angles:
Conjecture:
2. Linear pairs:
Conjecture:
3. Corresponding angles:
Conjecture:
4. Alternate Interior Angles:
Conjecture:
5. Alternate Exterior Angles
Conjecture:
6. Consecutive Interior Angles:
Conjecture:
9
10
Date: _____________________________
Section 3 – 2: Angles and Parallel Lines
Notes
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN:
 By the Corresponding Angles Postulate each pair of corresponding angles is
_____________________.
 By the Alternate Interior Angles Theorem each pair of alternate interior angles is
_____________________.
 By the Alternate Exterior Angles Theorem each pair of alternate exterior angles is
_____________________.
 By the Consecutive Interior Angles Theorem each pair of consecutive interior
angles is __________________________.
Example #1: In the figure, m3  110 and m12  55 . Find the measure of each
angle.
a.) 1
b.) 6
c.) 2
d.) 10
e.) 13
f.) 15
11
Example #2: Find x and y in the following figure.
Example #3: Find x and y in the following figure.
Example #4: Find 1 .
12
CRITICAL THINKING
For questions 1 and 2, refer to the picture at the right.
1.) Determine whether 1 is always, sometimes, or
never congruent to 2 . Explain.
2.) Determine the minimum number of angle measures you would have to know to
find the measures of all of the angles in the figure. Explain your reasoning.
13
14
Date:
_____________________________
Section 3 – 3: Slopes of Lines
Notes
Slope – the ratio of a line’s __________________ ________ to its ___________________ _________
slope =
Example #1: Find the slope.
a.)
b.)
c.)
d.)
Example #2: 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.
15
Example #3: Find the slope of l, m, and n. Compare the slopes of each.
Two nonvertical lines have the _________ slope if and only if they are ___________.
Two nonvertical lines are _________________ if and only if the product of their slopes is _________.
Example #4: Determine whether ⃖ ⃗ and ⃖ ⃗ are parallel, perpendicular, or neither.
a.
F(1, -3), G(-2,-1), H(5,0), J(6,3)
b. F(4,2), G(6, -3), H(-1, 5), J(-3, 10)
16
CRITICAL THINKING
1.) Find the Error: Curtis and Laura calculated the slope of a line containing A(15, 4)
and B(-6, -13). Who is correct? Explain your reasoning.
2.) The maximum recommended slope of a wheelchair ramp is 1/12. A business is
installing a wheelchair ramp that rises 22 inches over a horizontal length of 228 inches.
Is the ramp steeper then recommended? Be sure to show all of your work, and explain
your reasoning.
17
18
Date:
_____________________________
Section 3 – 4: Equations of Lines
Notes
Slope-Intercept Form –
Example #1: Write an equation in slope-intercept form of the line with slope of –4 and y-intercept
of 1.
Example #2:
at (0, -8).
Write an equation in slope-intercept form of the line with slope 5 and a point
Example #3:
at (-3, 6).
Write an equation in slope-intercept form of the line with slope –2 and a point
19
Point-Slope Form –
Example #4: Write an equation in point-slope form of the line whose slope is 
(-10, 8).
3
that contains
5
Example #5: Write an equation in slope intercept form for a line containing (4, 9) and (-2, 0).
Write an equation in slope-intercept form for a line containing (1, 7) that is
1
perpendicular to the line y   x  1 .
2
Example #6:
20
CRITICAL THINKING
1.) The price of a pair of “Jordan’s” can be represented by the linear equation
P = 3.52t + 65, where t = 0 corresponds to 1985. Explain the meaning of the slope and
y-intercept. Use this equation to determine how much a pair of “Jordan’s” will cost in
2020.
21
22
Date: _____________________________
Section 3 – 5: Proving Lines Parallel
Notes
IF TWO LINES IN A PLANE ARE CUT BY A TRANSVERSAL SO THAT:
 Corresponding
angles
are
__________________,
then
the
lines
are
__________________.
 A pair of alternate exterior angles is __________________, then the two lines are
___________________.
 A pair of alternate interior angles is ___________________, then the lines are
__________________.
 A pair of consecutive interior angles is ________________________, then the lines
are ___________________.
Ask yourself, “What if they were parallel…”
Example #1: Determine which lines, if any, are parallel.
23
Example #2: Given the following information, determine which lines, if any, are
parallel. State the postulate or theorem that justifies your answer.
a.) 2  8
b.) 9  16
c.) 2  10
d.) 6  15
Example #3: Find x so that a b.
]
Example #4: Find y so that l  m.
24
CRITICAL THINKING
1.) Describe two situations in your own life in which you encounter parallel lines.
How could you verify that the lines are parallel?
2.) When striping the practice football field, Coach Novak first painted the sidelines.
Next he marked off 10-yard increments on one sideline. He then constructed lines
perpendicular to the sidelines at each 10-yard mark. Why does this guarantee that the
10-yard lines will be parallel?
25