Download Name: Geometry Unit 3: Parallel and Perpendicular Lines 3

Name:
Geometry
Unit 3: Parallel and Perpendicular Lines
3-1: Lines and Angles
In this lesson you will explore relationships of nonintersecting lines and planes.
 Warm-Up
1.
2.
1.
______________________________
2.
______________________________
3.
_______________________________
4.
_______________________________
3.
 Identifying Nonintersecting Lines and Planes.
Definition
Parallel lines are coplanar lines
that do not intersect. The
symbol || means “is parallel to”.
Parallel and Skew
Symbols
AE || BF
Diagram
AD || BC
Skew lines are noncoplanar;
they are not parallel and do
not intersect.
AB and CG are skew
Parallel planes are planes that
do not intersect.
plane ABCD || plane EFGH
 Notice the arrows in the
diagram to show parallel
lines.
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In the figure below, assume that lines and planes that appear to be parallel are parallel.
1. Which segments are parallel to AB ? _______________________________________________________________
2. Which segments are skew to CD ? __________________________________________________________________
3. What are two pairs of parallel planes? _______________________________________________________________
4. What are two segments parallel to plane BCGF?____________________________________________________
5. Name THREE line segments that are coplanar with AB. ___________________________________________
REASONING
Explain why FE and CD are not skew. ______________________________________________________________________
_________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________
Find an example in the room of:
1. Parallel lines ____________________________________________________________________________________________
2. Skew lines
____________________________________________________________________________________________
3. Parallel planes ____________________________________________________________________________________________
4. Vertical angles ____________________________________________________________________________________________
Which line(s) or plane(s) in the figure appear to fit the description?
1. Parallel to line MN and contains J ___________________________________________
2. Skew to line MN and contains J ______________________________________________
3. Perpendicular to line MN and contains J ____________________________________
4. Name the plane that contains J and appears to be parallel to plane MNO __________________________
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 When a line intersects two or more lines, the angles formed at the intersection points create
special angle pairs.
Angle Pairs Formed by Transversals
Definition
Example
Alternate interior angles:
Same-side interior angles:
Corresponding angles:
Alternate exterior angles:
1. _____________________________________
2. _____________________________________
3. _____________________________________
4. _____________________________________
5. _____________________________________
6. _____________________________________
7. _____________________________________
REASONING
Do all pairs of corresponding angles look congruent? If not, explain what you could do to make
them equal in measure. ________________________________________________________________________________
HW 3-1: p. 144-145 #11-20, 29-35, 37-42
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3-2: Properties of Parallel Lines
3-3: Proving Lines Parallel
In this lesson you will explore the relationships between special angle pairs when they are formed by
parallel lines and a transversal. You will then use the congruent and supplementary relationships of
special angle pairs to prove lines parallel.

Warm-Up
a)_________________________
b) ________________________
c) _________________________
d) ________________________
REASONING
In the diagram above…
It appears that 5  19 . Explain what is happening in the diagram to make these equal in measure.
_________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________
Therefore, if corresponding angles are congruent, then the lines are ___________________________________
(and vice-versa)! Notice something in the diagram that supports our conclusion? ____________________
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Angle Pairs Formed by Parallel Lines and a Transversal
Same-side interior angles: If a transversal intersects
two parallel lines, then the same-side interior angles
are ____________________________.
Then…
Alternate interior angles: If a transversal intersects
two parallel lines, then the alternate interior angles
are ____________________________.
Then…
Corresponding angles: If a transversal intersects two
parallel lines, then the corresponding angles are
____________________________.
Then…
Alternate exterior angles: If a transversal intersects
two parallel lines, then the alternate exterior angles
are ____________________________.
Then…
Ex.1: The measure of 3 is 55. Which angles are supplementary to 3? How do you know?
Ex. 2: What are the measures of 3 and 4? Which theorems or postulates justify your answers?
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Ex. 3: Using the diagram at the right, what is the measure of each of the other angles. Justify each answer.
Ex. 4:
a) 1 ______________
Theorem/Postulate _________________________
b) 2 ______________
Theorem/Postulate _________________________
c) 5 ______________
Theorem/Postulate _________________________
d) 6 ______________
Theorem/Postulate _________________________
e) 7 ______________
Theorem/Postulate _________________________
f) 8 ______________
Theorem/Postulate _________________________
What is the value of y?
Ex. 5:
Use the diagram on the right for questions #1-4
Ex. 1 Which lines are parallel if 1  2 ? ___________________
This is the converse of the ______________________________________ angles theorem.
Ex 2 Which lines are parallel if 4  1 ? ___________________
This is the converse of the ______________________________________ angles theorem.
Ex 3 Which lines are parallel if 3  8 ? ___________________
This is the converse of the ______________________________________ angles theorem.
Ex 4 Which lines are parallel if 1 + 5 = 180___________________
This is the converse of the ______________________________________ angles theorem.
HW 3-2: p. 153-155 #7-9, 12-20
HW 3-3: p. 160-161 #7-10, 17-28
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3-4: Parallel and Perpendicular Lines (With Visualization)
In this lesson you will relate parallel and perpendicular lines.
 Warm-Up
Using the diagram below, choose from the following words, numbers, and expressions to complete the
following sentences.
a)
___________________________
f) _____________________
b)
______________________ g) ___________________________
c)
______________________ h) find equal to 17
d)
___________________________
e)
_____________________
___________________________
i) ____________________________
You can use the relationship of
two lines to a third line
to decide whether the two
lines are parallel or perpendicular
to each other.
Notice that these lines
must be in the same plane!
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Mixed Practice
1. What is the value of x for which a||b?
2. What is the value of w for which c||d?
3. What is the value of y for which a||b? Which postulate did you use to determine this?
4. A classmate says that AB || DC based on the diagram below. Explain your classmate’s error.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
5.
6.
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For #7-8, determine the value of x for which a||b?
7.
8.
For #9-10: a, b, c and d are distinct lines in the same plane. How do a and d relate? Justify your
answer.
9. a//b, b//c ,c//d
10. a//b, b//c, c  d
11. An artist is building a mosaic. The mosaic consists of the repeating pattern shown below. What must
be true of a and b to ensure that the sides of the mosaic are parallel?
12. A student says that if AD || CF and AD  AB , then CF  AB . Explain the student’s error.
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Find examples of the following in the room:
1. Intersecting lines ______________________________________________________________________________________
What shape does the intersection make? ___________________________________________________________
2. Intersecting planes _____________________________________________________________________________________
What shape does the intersection make? ____________________________________________________________
3. A line intersecting a plane _____________________________________________________________________________
 What shape does the intersection make? ___________________________________________________________
Applying the definitions of parallel and perpendicular lines and planes, draw and visualize each
situation below:
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HW 3-4 p.167-169 #6, 11-15, 21-24, 31-32
Extra Practice: Parallel Lines and Transversals WS1 and WS2
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3-5: Parallel Lines and Triangles
In this lesson you will find measures of angles of triangles.
Practice Problems
1. What are the values of x, y and z in the diagram?
2.
4. What is the value of x?
3. What is the measure of 2?
5. What is the measure of 1?
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6. Two angles of a triangle measure 53. What is the measure of the exterior angle at each vertex of the
triangle?
7. The ratio of the angle measures of the acute angles in a right triangle is 1:2. Find the angle measures
of the triangle.
Extra Practice
HW 3-5: p. 175-176 #9-14, 17-19, 23-24
Extra Practice: Exterior Angle Theorem WS
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3-6: Constructing Parallel and Perpendicular Lines
In this lesson you will construct parallel and perpendicular lines.
Construction (Basic)
Draw a figure like the given one, then:
REASONING
What theorem is used to make these two lines be parallel?
_________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________
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Construction (Application)
Draw a figure like the given one, then:
Construction (Basic)
Draw a figure like the given one, then:
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You will use this postulate
in construction #8…
Construction (Basic)
Draw a figure like the given one, then:
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Construction (Application)
Draw a figure like the given one, then construct the line through point J that is parallel to AB
Construction (Application)
Draw two segments. Label their lengths a and b. Construct a quadrilateral with one pair of parallel
sides of lengths a and 2b.
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Construction (Application)
Construct a square with side length p.
HW 3-6: p. 186-187 #7-9, 12, 14, 16, 20
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3-7: Equations of Lines in the Coordinate Plane
In this lesson you will explore the concept of slope and how it relates to both the graph and the
equation of a line.
 The slope of a line can be positive (goes up left to right), negative (goes down left to right),
zero (horizontal) or undefined (vertical).
Graphing Lines
Ex. 1 What is the graph of y = 2/3 x + 1
Ex. 2 What is the graph of y – 2 = -1/3(x – 4)
Ex. 3 Graph y = 3x – 4
Ex. 4 Graph y – 3 = -2 (x + 3)
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Writing Equations Of Lines
Ex. 1 What is an equation of the line with slope 3 and y-intercept -5? (Using slope-intercept form)
Ex. 2 What is an equation of the line through (-1, 5) with slope 2? (Using point-slope form)
Ex. 3 What is an equation of the line with slope -1/2 and y-intercept 2?
Ex. 4 What is an equation of the line through (-1, 4) with slope -3?
Using two points to write an equation of a line
1. a) Write an equation of the line in the diagram (using point-slope form).
b) Rewrite the equation using slope-intercept form. Compare your answers.
What can you conclude?
2.
What are the equations of the vertical and horizontal lines through (2,4)?
Vertical _________________________________________
Horizontal ______________________________________
HW 3-7: p. 194-195 #8, 10, 16, 19-21, 24-25, 29-30, 34, 38-39
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3-8: Slopes of Parallel and Perpendicular Lines
In this lesson you will learn how to use slopes to determine how two lines relate graphically to each
other.
1. Are lines l1 and l2 parallel? Explain.
2.
Writing Equations of Parallel Lines
1. What is an equation of the line parallel to y = -3x -5
that contains (-1, 8)?
2. What is the equation of the line parallel to y = -x – 7
that contains (-5, 3)?
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Notice...fractions
which are negative
reciprocals of one
another have a
product of -1!
Ex. 1: Are the two lines in the diagram on the right perpendicular? Explain.
Ex. 2:
Writing Equations of Perpendicular Lines
Ex. 1: What is an equation of the line perpendicular to y = 1/5x + 2
that contains (15, -4).
Ex. 2: What is an equation of the line perpendicular to y = -3x – 5
that contains (-3, 7).
HW 3-8: p. 201-203 #10-12, 15, 17, 20, 25, 28, 30
Extra Practice: Writing Equations of Parallel Lines WS
Writing Equations of Perpendicular Lines WS
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Algebra Review: Systems of Equations
You can solve a system of equations in two variables by using substitution.
Solve the system.
y = 3x + 5
y=x+1
 The graph of a linear system with infinitely many solutions is one line, and the graph of a linear system
with no solution is two parallel lines.
Solve the system.
x+y=3
4x + 4y = 8
Extra Practice:
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Review:
1. Identify all numbered angle pairs that form the given type of angle pair. Then name the two lines
and transversal that form each pair.
a) alternate interior angles ________________________________________
b) corresponding angles___________________________________________
c) same side interior_______________________________________________
d) alternate exterior angles________________________________________
2. Find m1 and m2. Justify your answer.
3. Find the value of x for which l||m.
4. a) If b  c and b  d, then c _?_ d.
b) If c||d, then _?_  c.
5. Find the values of the variables.
6. The measures of the three angles of a triangle are
given. Find the measures of the three angles.
x + 10, x – 20, x + 25
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7. Draw a line m and point Q not on m. Construct a line perpendicular to m through Q.
8. Find the slope of the line passing through the points (6, -2), (1, 3).
9. Write an equation of the line that passes through (4,2) and (3, -2).
10. Write an equation for the line parallel to y = 8x -1 that contains (-6, 2).
11. Write an equation of the line perpendicular to y = 1/6x + 4
that contains (3, -3)
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Common Core Questions
1. Given MN shown below, with M(−6,1) and N(3,−5), what is an equation of the line that passes
through point P(6,1) and is parallel to MN?
2.
3.
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4.
5.
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