3.3 ll Lines & Transversals

p. 143

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

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

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

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)

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

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

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

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

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

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

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

Ex: Find:m1=m2=m3=m4=m5=m6=x=

125o 21

3

4 6

5

x+15o

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

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.

Assignment