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3.3 ll Lines & Transversals
p. 143
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DefinitionsDefinitions• Parallel lines (ll) – lines that are coplanar & do not
intersect. (Have the some slope. ) l ll m
• Skew lines – lines that are not coplanar & do not intersect.
• Parallel planes – 2 planes that do not intersect.Example: the floor & the ceiling
l m
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H G
D C
A B
E F
1. Line AB & Line DC are _______.
2. Line AB & Line EF are _______.
3. Line BC & Line AH are _______.
4. Plane ABC & plane HGF are _______.
1. Parallel
2. Parallel
3. Skew
4. Parallel
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Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Transversal
• A line that intersects 2 or more coplanar lines at different points.
l
m
t
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1 23 4
5 6
7 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)
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Post. 15 – Corresponding s Post.
• If 2 lines are cut by a transversal, then the pairs of corresponding s are .
• i.e. If l m, then 12.
l
m
1
2
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Thm 3.4 – Alt. Int. s Thm.
• If 2 lines are cut by a transversal, then the pairs of alternate interior s are .
• i.e. If l m, then 12.
l
m
1
2
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Proof of Alt. Int. s Thm.Statements
1. l m 2. 3 23. 1 34. 1 2
Reasons1. Given2. Corresponding s post.3. Vert. s Thm4. is transitive
l
m
1
2
3
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Thm 3.5 – Consecutive Int. s thm
• If 2 lines are cut by a transversal, then the pairs of consecutive int. s are supplementary.
• i.e. If l m, then 1 & 2 are supp.
l
m 1
2
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Thm 3.6 – Alt. Ext. s Thm.
• If 2 lines are cut by a transversal, then the pairs of alternate exterior s are .
• i.e. If l m, then 12.
l m
1
2
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Thm 3.7 - Transversal Thm.• If a transversal is to one of 2 lines, then it
is to the other.
• i.e. If l m, & t l, then t m.** 1 & 2 added for proof purposes.
1
2
l
m
t
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Proof of transversal thmStatements
1. l m, t l 2. 123. m1=m24. 1 is a rt. 5. m1=90o
6. 90o=m27. 2 is a rt. 8. t m
Reasons1. Given2. Corresp. s post.3. Def of s4. Def of lines5. Def of rt. 6. Substitution prop =7. Def of rt. 8. Def of lines
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Ex: Find:m1=m2=m3=m4=m5=m6=x=
125o 21
3
4 6
5
x+15o
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Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Check It Out! Example 1
Find mQRS.
mQRS = 180° – x
x = 118
mQRS + x = 180°
Corr. s Post.
= 180° – 118° = 62°
Subtract x from both sides.Substitute 118° for x.
Def. of Linear Pair
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Holt Geometry
3-2 Angles Formed by Parallel Lines and Transversals
Find each angle measure.
Example 2: Finding Angle Measures
A. mEDG
B. mBDG
mEDG = 75° Alt. Ext. s Thm.
mBDG = 105°
x – 30° = 75° Alt. Ext. s Thm.x = 105 Add 30 to both sides.
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Assignment