Geometry
Unit 3
Parallel Lines
DAY
Monday
Day 1
3.1
ACTIVITY/OBJECTIVE
ASSIGNMENT
CC.9-12.G.CO.1
Identify parallel, perpendicular, and skew lines.
Identify the angles formed by two lines and a transversal.
Packet Page 1-4
Day 2
3.2
CC.9-12.G.CO.9
Prove and use theorems about the angles formed by parallel lines
and a transversal.
Constructions for Parallel and Perpendicular Lines
Packet Page 5-8
Day 3
3.3
CC.9-12.G.CO.9
Use angles formed by a transversal to prove two lines are parallel.
Packet Page 9-11
Day 4
3.4
CC.9-12.G.CO.9
Prove and apply theorems about perpendicular lines.
Quiz 3.1-3.3
Packet Pages 12-14
Day 5
3.5 and 3.6
CC.9-12.G.GPE.5
Fine the slope of a line.
Use slopes to identify parallel and perpendicular lines.
Packet Page 15-18
Day 6
3.1-3.6
7
Review
Quiz 2 3.1-3.6
Unit 3 Test
1
Define Transversal:
Name the obvious transversal(s):
c
1.
a
a
2.
b
3.
a
c
b
c
b
When 2 coplanar lines are cut by a transversal, 8 angles are formed:
INTERIOR ’s:
EXTERIOR ’s:
Some of these angles have a relationship that we have previously studied.
LINEAR PAIRS:
VERTICAL ’s:
c
b
6
7
a
5
8
1 2
3 4
TYPES OF ANGLES
Alternate Interior ’s:
______________ interior ’s on ______________ sides of the transversal.
Name the Alt. Int. ’s:
Same–Side Interior ’s (consecutive int. ’s): Two ____________ ’s on the same side of the
transversal.
Name the S.S.int. ’s:
Corresponding ’s: Two angles in ______________ ______________ relative to the two lines.
Name the Corr. ’s:
Alternate Exterior ’s: ______________ exterior ’s on ______________ sides of the transversal.
2
Name the Alt. Ext. ’s:
2
Use the given line as a transversal:
1. Name alt. int ’s using line x:
1
3
4
y
2. Name s.s. int. ’s using line y:
3. Name corr. ’s using line z:
4. Name alt. ext. ’s using line y:
6
8 7
5
10
9
11
12
5. Name alt. int. ’s using line z:
6. Name s.s. int ’s using line z:
z
x
Tell whether the statement is true or false. If false, sketch a counterexample. Do not assume points are coplanar unless
specified.
_______1. If a line intersects one of two parallel lines, then it must intersect the other.
_______2. If two lines are coplanar, then they must be parallel.
_______3. Two coplanar line segments, which have no point in common, must be parallel.
_______4. Two lines, which are parallel to the same line, must be parallel to each other.
_______5. If a plane contains one of two parallel lines, then it must contain the other.
_______6. If a line is parallel to a plane, then it is parallel to every line in the plane.
_______7. If two lines are  to the same line, then they must be parallel to each other.
_______8. If two planes are  to the same line, then they must be parallel to each other.
_______9. If two lines are skew to a third line, then they must be skew to each other.
_______10. Two planes, which are parallel to the same plane, must be parallel to each other.
3
a
y
x
5
8
1
4
2
7
6
b
3
9
12
14
10
15
13 11
16
17
18
19
Assume a ⁄⁄ b. Complete the chart.
ANGLES
1.
 1 and  14
2.
 2 and  15
3.
 7 and  9
4.
 9 and  16
5.
 10 and  17
6.
 16 and  14
7.
 9 and  14
8.
 18 and  19
9.
 1 and  16
10.
 3 and  8
11.
 6 and  9
12.
 12 and  13
13.
 7 and  11
14.
 6 and  8
15.
 4 and  13
16.
 9 and  12
TRANSVERSAL
TYPE
 , SUPPL., OR NONE
(relationship between angles)
4
Day 2 - If 2 lines are parallel and they are intersected by a transversal, then the following is true about
each pair of angles:
:
Alternate Interior Angles:
3  6
4  5
Examples – Find the measures of the angles (or value of the variables(s)).
3.
2.
1.
6.
5.
9.
4.
7.
10.
8.
11.
5
6
The Converse of each theorem also works:
Converse of the Alternate Interior Angles Theorem:
Converse of the Alternate Exterior Angles Theorem:
Converse of the Same-Side Interior Angles Theorem:
Converse of Corresponding Angles Postulate:
Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or theorem
you would use. If not, state cannot be determined.
1.
2.
3.
92°
88°
l
k
7
4.
5.
6.
l
k
7. A
105°
E
8.
B
I
C
D
9.
k
122°
75°
H
55°
l
F
58°
55°
m
G
Find the value of x so that n || m. State the theorem or postulate that justifies your solution.
5x
10.
n
11.
m
5x+23
n
7x+13
m
5x-18
12.
n
m
8x-5
3x+48
x=
x=
x=
_____________________
_____________________
_____________________
Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you
would use.
p
13.
p
q
14.
p
15.
q
_____________________
Name the type for each pair of angles
16. 1   8
17. 4  6
18. 10   7
19. m3 + m4 = 180
20. 5   3
21. 6  7
q
_____________________
_____________________
k
j
3
1
4 5
2
10
l
6
8
9 7
n
p
8
Day 3: Parallel Proofs
t
l
1
2
m
3
4
5
1. Given: l // m; 1  4
Prove: s // t
1. l // m ; 1  4
1. _______________________________________
2. 3  1
2. _______________________________________
3. 3  4
3. _______________________________________
4. s // t
4. _______________________________________
2. Given: l // m ; 2  5
Prove: s // t
1. l // m ; 2  5
1. ________________________________________
2. 2  3
2. ________________________________________
3. 3  5
3. ________________________________________
4. s // t
4. ________________________________________
3. Given: l // m; s // t
Prove: 2  4
1. l // m ; s // t
1. ________________________________________
2. 2   3
3  4
3. 2  4
2. ________________________________________
3. ________________________________________
4. Given: l // m; s // t
Prove: 1  5
1. l // m; s // t
1. _________________________________________
2. 1  3
3  5
3. 1  5
2. _________________________________________
3. _________________________________________
9
5. Given: 3 is supplementary to 5.
Prove:
BD // FE
1. 3 is supplementary to 5
1. _________________________________________
2. m3 + m5 = 180
2. _________________________________________
3. 3  4
3. _________________________________________
4. m3 = m4
4. _________________________________________
5. m4 + m5 = 180
5. _________________________________________
6. 4 is supplementary to 5
6. _________________________________________
7.
7. _________________________________________
BD // FE
B
A
6. Given: 2  5;
BE bisects  CBD.
Prove: AC // DE
1
D
1. 2  5;
BE bisects  CBD.
2.
3. 3  5
3.
4.
4.
AC // DE
8. Given: l // m ; s // t
Prove: 1  5
9. Given: l // m; 1  4
Prove: s // t
C
3
4
5
E
1.
2. 3  2
7. Given: l // m ; s // t
Prove: 2  4
2
s
Diagram for # 7 - 10
l
t
1
2
m
3
5
4
10. Given: l // m; 2  5
Prove: s // t
10
11. Given: BC // EF ; BA // ED
Prove: B  E
A
D
C
P
B
E
12. Given: AB //DE
Prove: mACD = mBAC + mCDE
F
A
B

C

E
D
13. Given: g // h; g // j
Prove: 2  3
g
1
2
h
3
14. Given: AB //CD ; BC //DE
Prove: B  D
j
C
A
E
B
D
15. Given: a // c; 1  2
Prove: b // c
2
1
a
16. Given: C is a supplement of D
Prove: A is a supplement of B
b
B
A
c
C
D
11
Day 4: Perpendicular Lines
Perpendicular Bisector is a line perpendicular to a segment at the segment's midpoint.
Distance from a point to a line the length of the perpendicular segment from the point to the line.
1- 6 Use the given diagram on the right, in which AM = MB.
 C
1. Name a pair of  rays.
2. ____ is the  bisector of ____.
3. Name a linear pair of angles which are .
4. If t in X is  to
5. If

A
M

B
AB at M, what can you say about t and CM ? Why?
MR in X is a  bisector of AB , then R is on CM . Why?
6. If p contains M and is  to the plane determined by
CM and AB , then p ___ CM and p ___ AB . Why?
7. In a plane , how many lines can be  to a given line at a given point?
8. Would your answer be different if the words “in a plane” were omitted from the question?
Homework on Perpendicular Lines.
True or False. If false, give a counterexample.
_____1. If
PQ  PR , then QPR is a right angle.
______2. If
AB  CD , then ABC is a right angle.
_____3. If 2 lines intersect to form a right angle,
then the lines are .
______4. There is exactly one line  to a given at a
given point on the line.
_____5. If 2 angles are a linear pair, then each
is a right .
______6. A given segment has exactly one 
bisector.
_____7. If M is the midpoint of AB and if AB is  to
plane X at M, there is exactly one line in X which
______8. If 2 adjacent angles are , then each is a
right angle.
is a  bisector of
AB .
12
In 9 – 13 refer to the diagram below and the given info. : mCAB = 90; CDA  BDA;
EA  AB ; mECB = 90
C
**Mark the diagram with the given information**
X
9. What pairs of lines are ?
D
10. ____ is a  bisector of ____. Why?
E
A
11. If
FC in X is a  bisector of EB , then F is on AC . Why?
12. If
t is a line in the plane of the diagram, and t  BC at D, how are t and AD related? Why?
13. If G is on
B
CE and EGA  CGA, how are AG and EC related? Why?
14. If
l  m and m  n, is l  n? Explain.
15. If
m  n, is n  m ? Explain.
13
1.The perpendicular bisector of a segment is a line ______________________ to a segment at the segment’s
______________________.
2. The shortest segment from a point to a line is ______________________ to the line.
For Exercises 3 and 4, name the shortest segment from the point to the line and
write an inequality for x.
3.
4.
________________________________________
________________________________________
Fill in the blanks to complete these theorems about parallel and
perpendicular lines.
5. If two coplanar lines are perpendicular to the same line, then the two lines are ______________________
to each other.
6. If two intersecting lines form a linear pair of ______________________ angles,
then the lines are perpendicular.
7. In a plane, if a transversal is perpendicular to one of two parallel lines, then it is
______________________ to the other line.
Use the drawing of a basketball goal for Exercises 8–10.
In each exercise, justify Esperanza’s conclusion with one
of the completed theorems from Exercises 5–7. Write the
number 5, 6, or 7 in each blank to tell which theorem you used.
8. Esperanza knows that the basketball pole intersects the
court to form a linear pair of angles that are congruent.
She concludes that the pole and the court are perpendicular.
______________________
9. Esperanza knows that the hoop and the court are both
perpendicular to the pole. She concludes that the hoop
and the court are parallel to each other.
______________________
10. Esperanza knows that the hoop and the court are parallel to
each other. She also knows that the hoop is perpendicular to
the pole. Esperanza concludes that the pole and the court
are perpendicular.
______________________
14
Day 5 Slopes of Lines
15
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or
neither.
2. IJ and KL for I(1, 0), J(5, 3), K(6, 1),
and L(0, 2) _______________________
3. PQ and RS for P(5, 1), Q(1, 1), R(2, 1),
and S(3, 2) _______________________
whether each pair of lines is parallel, perpendicular, or neither.
4. EF with slope3 and GH with slope1
5. PQ with slope 
_____________________
2
3
and RS with slope  
3
2
_____________________
Match the letter of each example to the correct form of a line.
6. point-slope form
_________
A. x3
7. slope-intercept form
_________
B. yx1
8. horizontal line
_________
C. y  7
9. vertical line
_________
D. y  2  ( x  6)
Write the equation of each line in the given form.
10. the horizontal line through (3, 7) in
point-slope form
________________________________________
 1 7
12. the line through   ,   and (2, 14) in
 2 2
slope-intercept form
1
2
11. the line with slope 
8
through (1, 5) in
5
point-slope form
_________________________________________
13. the line with x-intercept 2 and y-intercept
1 in slope-intercept form
16
Write the equation of each line in the given form. Graph each line.
14.the line with slope 2 and y-intercept 1
in slope-intercept form
________________________________________
16. the line through (0, 0) and (2, 2) in
slope-intercept form
15. the line with slope
2
through (4, 4) in
3
point-slope form
_________________________________________
17. the line through (1, 1) and (0, 2) in point-slope form
Graph each line.
18. y  3 
3
( x  1)
4
19. y  
4
x2
3
Determine whether the lines are parallel, intersect, or coincide.
20. x  5y  0, y  1 
1
( x  5)
5
1
x   1 y
2
3
1
22. y  4( x  3),  4y   x
4
4
21. 2y  2  x,
____________________
____________________
____________________
17
Write the equation of each line in the given form.
1. the horizontal line through (3, 7) in
point-slope form
2. the line with slope 
8
through (1, 5) in
5
point-slope form
________________________________________
 1 7
3. the line through   ,   and (2, 14) in
 2 2
slope-intercept form
________________________________________
_________________________________________
4. the line with x-intercept 2 and y-intercept
1 in slope-intercept form
_________________________________________
Graph each line.
5. y  3 
3
( x  1)
4
6. y  
4
x2
3
Determine whether the lines are parallel, intersect, or coincide.
7. x  5y  0, y  1 
1
( x  5)
5
1
x   1 y
2
3
1
9. y  4( x  3),  4y   x
4
4
8. 2y  2  x,
____________________
____________________
____________________
18