Download 3.3 ll Lines & Transversals - Miami Killian Senior High School

3.3 ll Lines & Transversals
p. 143
Definitions
• Parallel lines (ll) – lines that are coplanar & do not
intersect. (Have the some slope. )
l ll m
l
m
• Skew lines – lines that are not coplanar & do not
intersect.
• Parallel planes – 2 planes that do not intersect.
Example: the floor & the ceiling
1. Line AB & Line DC are _______.
2. Line AB & Line EF are _______.
3. Line BC & Line AH are _______.
4. Plane ABC & plane HGF are _______.
A
B
D
C
H
1. Parallel
G
2. Parallel
3. Skew
E
F
4. Parallel
3-2
Angles Formed by Parallel Lines
and Transversals
Transversal
• A line that intersects 2 or more coplanar
lines at different points.
l
m
Holt Geometry
t
5
7
6
8
Interior <s - <3, <4, <5, <6 (inside l & m)
Exterior <s - <1, <2, <7, <8 (outside l & m)
Alternate Interior <s - <3 & <6, <4 & <5 (alternate –opposite sides of
the transversal)
Alternate Exterior <s - <1 & <8, <2 & <7
Consecutive Interior <s - <3 & <5, <4 & <6 (consecutive – same side
of transversal)
Corresponding <s - <1 & <5, <2 & <6, <3 & <7, <4 & <8 (same
location)
Post. 15 – Corresponding s Post.
• If 2  lines are cut by a transversal, then the
pairs of corresponding s are .
1
2
l
m
• i.e. If l m, then 12.
Thm 3.4 – Alt. Int. s Thm.
• If 2  lines are cut by a transversal, then the
pairs of alternate interior s are .
l
1
2
m
• i.e. If l m, then 12.
Proof of Alt. Int. s Thm.
1.
2.
3.
4.
Statements
l m
3  2
1  3
1  2
Reasons
1.
2.
3.
4.
3
l
1
2
m
Given
Corresponding s post.
Vert. s Thm
  is transitive
Thm 3.5 – Consecutive Int. s thm
• If 2  lines are cut by a transversal, then the
pairs of consecutive int. s are
supplementary.
l
1
m
2
• i.e. If l m, then 1 & 2 are supp.
Thm 3.6 – Alt. Ext. s Thm.
• If 2  lines are cut by a transversal, then the
pairs of alternate exterior s are .
l
m
1
2
• i.e. If l m, then 12.
Thm 3.7 -  Transversal Thm.
• If a transversal is  to one of 2  lines, then it
is  to the other.
t
l
m
1
2
• i.e. If l m, & t  l, then t m.
** 1 & 2 added for proof purposes.
Proof of  transversal thm
1.
2.
3.
4.
5.
6.
7.
8.
Statements
l m, t  l
12
m1=m2
1 is a rt. 
m1=90o
90o=m2
2 is a rt. 
t m
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
Given
Corresp. s post.
Def of  s
Def of  lines
Def of rt. 
Substitution prop =
Def of rt. 
Def of  lines
Ex: Find:
m1=
m2=
m3=
m4=
m5=
m6=
x=
1
125o
2
3
5
4
x+15o
6
3-2
Angles Formed by Parallel Lines
and Transversals
Check It Out! Example 1
Find mQRS.
x = 118 Corr. s Post.
mQRS + x = 180°
mQRS = 180° – x
Def. of Linear Pair
Subtract x from both sides.
= 180° – 118° Substitute 118° for x.
= 62°
Holt Geometry
3-2
Angles Formed by Parallel Lines
and Transversals
Example 2: Finding Angle Measures
Find each angle measure.
A. mEDG
mEDG = 75° Alt. Ext. s Thm.
B. mBDG
x – 30° = 75° Alt. Ext. s Thm.
x = 105 Add 30 to both sides.
mBDG = 105°
Holt Geometry
Assignment