Transcript Ch 3 Perpendicular and Parallel Lines

Lesson 3.1
Lines and Angles
You will learn to …
* identify relationships between lines
* identify angles formed by transversals
If two lines are coplanar and
do not intersect, then they are
_______________
parallel lines
Parallel Lines
B
AB || CD
A
D
C
r || p
r
p
If two lines are NONcoplanar
and do not intersect, then they
skew lines
are ____________
If two lines intersect to form
one right angle, then they are
perpendicular lines
___________________.
AB | CD
r | p
Determine whether the lines are
parallel, perpendicular, or neither.
1. t and u
neither
2. t and s
parallel
3. r and u
perpendicular
s
t
r
u
Determine whether the lines are
intersecting or skew.
4. t and u
skew
t
u
Postulate 13
Parallel Postulate
Given a line and a point not on the line,
exactly one line
then there is _________________
through the point parallel to the line.
Postulate 14
Perpendicular Postulate
Given one line and one point,
exactly one line through
there is ________________
the point perpendicular to the line.
transversal – a line that
intersects two or more coplanar
lines at different points
no
transversal
a
b
c
Identify
transversal(s):
A) Line a
B) Line b
C) Line c
D) Lines a and b
E) All 3 lines
corresponding angles
1 2
4
3
5 6
7 8
alternate interior angles
1 2
4
3
5 6
7 8
alternate exterior angles
1 2
4
3
5 6
7 8
consecutive interior angles
1 2
4
3
5 6
7 8
same-side interior angles
same-side exterior angles
1 2
4
3
5 6
7 8
Describe the relationship between
the given angles.
5.
1
and
2
6. 3 and 4
7. 5 and 6
corresponding
1
alternate exterior
consecutive
interior 5
4
3
2
6
Lesson 3.2
Proof &
Perpendicular Lines
You will learn to …
* write different types of proofs
* prove results about perpendicular lines
1.
2. Which angles are congruent?
adjacent?
1  2
3  4
1
1 4 4
3 3 22
1 and 4
2 and 4
1 and 3
2 and 3
3. How would the diagram change if
adjacent angles were congruent?
Theorem 3.1
Perpendicular Lines
Theorem
If 2 lines intersect to form a
linear pair of congruent
angles, then the lines are
perpendicular
_______________.
4. Is RP  ST ? How do you know?
R
Yes, RP  ST.
2
1
T
P
m1 = m2 and
m1 + m2 = 180
m1 = 90 
m2 = 90 
Perpendicular
Lines Theorem
S
Draw two adjacent acute
angles such that their
uncommon sides are
perpendicular.
What do you know about the
two acute angles?
Theorem 3.2
Adjacent Complements
Theorem
If 2 sides of two adjacent acute
angles are perpendicular,
then the angles are
complementary
________________.
5. AC  BD Find x.
B
(6x + 4) + 20 = 90
x = 11
C
A
E
20 P
D
Theorem 3.3
4 Right Angles Theorem
If 2 intersecting lines are
perpendicular then they
_____________,
form 4 right angles.
Determine whether enough
information is given to conclude that
the statement is true.
d
6. 1  2
yes
7. 2  3
8. 3  4
4
yes
no
3
b
Paragraph Proofs
9. Given: AB = BC
Prove: ½ AC = BC
C
A
B
AC = AB + BC by the Segment Addition
Postulate. Since AB=BC, AC = BC + BC
by substitution. By the Distributive Prop,
AC = 2BC. ½ AC = BC by the Division
Prop.
Look at your Ch 2 Celebration.
Paragraph Proofs
10. Given: 1 and 3 are a linear pair
2 and 3 are a
3
linear pair
1
2
Prove: m1 = m2
Since 1 & 3 and 2 & 3 are linear
pairs, 1 & 3 are supplementary and
2 & 3 are supplementary by the
Linear Pair Postulate. So, 1  2 by
the Congruent Supplements Theorem.
Look at your
By definition
of ,Chm2Celebration.
1 = m 2
Flow Proofs
11. Given: 1 and 2 are a linear pair
2 and 3 are a linear pair
Prove: m1 = m3
1
2
3
Flow Proofs
12. Given: 5  6
5 and 6 are a linear pair
Prove: j  k
j
5 6
k
Practice!
The best way for
you to get better
at writing proofs
is to practice.
Don’t give up!
1. Write a two-column proof of
Theorem 3.1
Perpendicular Lines Theorem
Given:  1   2,  1 and  2 are
a linear pair
g
Prove: g  h
12
h
2. Write a two-column proof of
Theorem 3.2
Adjacent Complements Theorem
Given: BA  BC
Prove: 1 and 2 are
complementary
A
1
B
2
C
3. Write a two-column proof of
Theorem 3.3
4 Right Angles Theorem
Given: j  k ,
Prove: 2 is a right angle
j
1
2
k
4. Given: j  k ,
3 and 4 are complementary
Prove:  5   6
j
4
5
3
6
k
Lesson 3.3
Parallel Lines and
Transversals
Students need a protractor and straight edge
You will learn to …
* prove and use results about parallel
lines and transversals
* use properties of parallel lines
Postulate 15 &
Theorems 3.4 – 3.6
If 2 parallel lines are cut by
a transversal, then…
Use a straight edge to create 2 parallel lines cut by a transversal.
…corresponding angles are
congruent
______________.
Corresponding Angles Postulate
corresponding angles
1 2
4
3
1  5
2  6
5 6
7 8
3  7
4  8
…alternate interior angles
congruent
are ______________.
Alternate Interior Angles Theorem
alternate interior angles
4
3
5 6
3  6
4  5
… alternate exterior angles are
congruent
_______________.
Alternate Exterior Angles Theorem
alternate exterior angles
1 2
1  8
2  7
7 8
…consecutive interior angles
supplementary
are _______________.
Consecutive Interior Angles Theorem
consecutive interior angles
3 4
5 6
m3 + m5 = 180
m4 + m6 = 180
…same-side exterior angles
supplementary
are _______________.
Same-Side Exterior Angles Theorem
same-side exterior angles
1 2
7 8
m1 + m7 = 180
m2 + m8 = 180
Find the measure of the
numbered angle.
1. m1 =
110
110
1
Corresponding s Postulate
Find the measure of the
numbered angle.
2. m2 =
100
100
2
Alt. Ext. s Theorem
Find the measure of the
numbered angle.
3. m3 =
112
112
3
Alt. Int. s Theorem
Find the measure of the
numbered angle.
4. m4 =
60
120
4
Cons. Int. s Theorem
Find the measure of the
numbered angle.
5. m5 =
70
110
5
Same-side Ext. s Theorem
6. Find x.
125
(12x – 5)
12x – 5 + 125 = 180
12x + 120 = 180
12x = 60
x=5
7. Find x.
100
(5x + 40)
5x + 40 = 100
5x = 60
x = 12
Theorem 3.7
 Transversal Theorem
If a transversal is
perpendicular to one of two
parallel lines, then …
________________________.
it
is perpendicular to the other
m
t
n
If t  n and n || m, then t  m
A# 3.3
#28
Statements
1) j || k
2)  1   3
3)  2   3
4)  1   2
#29
Statements
1) p  q ; q || r
2)  1 is a right angle
3)  1   2
4)  2 is a right angle
5) p  r
Lesson 3.4 & 3.5
Parallel Lines
You will learn to …
* prove that two lines are parallel
* use properties of parallel lines
What is the converse
of a conditional
if-then statement?
Write the converse of the statement.
1. If two parallel lines are cut by a
transversal, then corresponding
angles are congruent.,
If corresponding angles are
congruent, then the two lines cut
by the transversal are parallel.
Postulate 16
Corresponding Angles Converse

If corresponding angles are___,
then…
1
2
3 4
5 6
7 8
Theorem 3.8 –
Alternate Interior s Converse
If alternate interior angles
 then…
are___,
3 4
5 6
Theorem 3.9
Consecutive Interior s Converse
If same-side interior angles
are supplementary
, then…
3 4
5 6
Theorem 3.10 –
Alternate Exterior Angles Converse
If alternate exterior angles
 then…
are___,
1
2
7
8
Same-side Exterior s Converse
If same-side exterior angles
are supplementary
, then…
1
2
7
8
…the
2 lines cut by the
transversal are parallel
Postulate and Theorems
IF
|| lines THEN angles
Converse
IF
angles
THEN
|| lines
2. Can you prove that n || m ?
Explain.
112
n
112
m
Yes, Corresponding s Converse
3. Can you prove that n || m ?
Explain.
78
n
78
m
Yes, Alternate Ext. s Converse
4. Can you prove that n || m ?
Explain.
72
108
n
m
Yes, Consecutive Interior s Converse
5. Can you prove that n || m ?
Explain.
n
m
102
102
Yes, Alternate Interior s Converse
6. Can you prove that n || m ?
Explain.
123
n
47
m
NO 123 + 47  180
7. Can you prove that n || m ?
Explain.
100
100
n
m
NO
p || n
If p || k and n || k, then…?
p
k
n
Theorem 3.11
3 Parallel Lines Theorem
If 2 lines are parallel to the
same line, then they are
parallel
____________
to each other.
r || t.
If r || s and s || t, then ____
p || n
If p  k and n  k, then…?
n
Theorem 3.12
  ||
Theorem
If 2 lines are perpendicular to
the same line, then they are
parallel to each other.
_________
r || t
If r  s and t  s, then ____
A# 3.4
#30
Statements
1)  4   5
2)  4 &  6 are vertical angles
3)  4   6
4)  5   6
5) g || h
A# 3.4
#32
Statements
1)  B   BEA
2)  BEA   CED
3)  CED   C
4)  B   C
5) AB || CD
A# 3.4
#34
Statements
1) m  7 = 125°; m  8 = 55°
2) m  7 + m  8 = 125° + m  8
3) m  7 + m  8 = 125° + 55°
4) m  7 + m  8 = 180°
5)  7 &  8 are supplementary
6) j || k
Do Practice Proofs…
Students need scissors, glue, and 2 sheets of paper.
Lesson 3.6 & 3.7
Parallel and
Perpendicular Lines
You will learn to …
* find slopes of lines and use slope to
identify parallel lines and
perpendicular lines
* write equation of parallel lines
* write equations of perpendicular lines
The slope of a line is the
ratio of the
vertical change (rise) to the
horizontal change (run).
rise y2 – y1
Slope =
= x –x
run
2
1
Find the slope of the line that
passes through the given points.
1. (-5,7) (-2,4)
m = -1
2. (3, -2) (-5, -2)
m=0
3. (-6, 2) (-6, -2) m = undefined
Postulate 17
Slopes of Parallel Lines
Lines are parallel
if and only if they have the
same slope.
All vertical lines are parallel.
All horizontal lines are parallel.
Vertical
v
Lines
v
Horizontal
Lines
h
x=#
y=#
Slope is 0
Slope is undefined
slope-intercept form
y = mx + b
slope
y–intercept
(0,b)
Identify the slope of the line.
4. y = -5x + 14
m=-5
5. 2x – 4y = -3
– 4y = – 2x – 3
-4
-4
-4
y=½x+¾ m=½
Write the equation of the line with
the given slope and y-intercept.
6. slope = - 2
y-int = 3
y = -2x + 3
7. slope = 0
y-int = -5
y=-5
Write the equation of the line that
has a y-intercept of -7 and is
parallel to the given line.
8. y = - ½ x + 10
m=-
y=-½x–7
½
point-slope form
y – y1 = m (x – x1)
9. Use the point-slope form to write
the equation of the line through the
point (2, 3) that has a slope of 5.
y – y1 = m (x – x1)
y - 3 = 5 (x - 2)
y - 3 = 5x - 10
y = 5x - 7
10. Write the equation of the line
through the point (-2, -4) that is
parallel to y = - ½ x + 5.
y – y1 = m (x – x1)
y - 4 = - ½ (x - 2)
y + 4 = - ½ (x + 2)
y+4=-½x-1
y=-½x-5
Postulate 18
Slopes of Perpendicular Lines
Lines are perpendicular
if and only if
the product of their slopes is -1.
Opposite & Reciprocals
Identify the slope of the line that is
perpendicular to the given line.
1
11. y = -5x + 14 m =
5
12. y = ½ x - 7
13. y = - 8
m=-2
m = undefined
Find the equation of the line that is
perpendicular to the given line and
passes through the given point.
14. y = 3x – 2 (9,4) m
1
y – 4 = - /3 (x – 9)
y–4=-
y=-
1/ x
3
1/
+3
x
+
7
3
1
= -? /3
Find the equation of the line that is
perpendicular to the given line and
passes through the given point.
15. y =
m = -?7
– 11 (5, -10)
y + 10 = - 7 (x – 5)
1/ x
7
y + 10 = - 7x + 35
y = - 7x + 25
Workbook
Page 55 (2, 6, 8, 9, 11)
Workbook
Page 59 (1-3)