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3.1: Lines and Angles

Today’s Objectives

To identify relationships between figures in space

To identify angles formed by two lines and a transversal

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Parallel Lines

Parallel lines are coplanar lines that do not intersect. Arrows are used to indicate lines are parallel. The symbol used for parallel lines is ||.

In the above figure, the arrows show that line AB is parallel to line CD.

With symbols we denote, .

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Skew Lines and Parallel Planes

Two lines are skew if they do not intersect and are not in the same plane (not coplanar).

Ex: are skew.

All planes are either parallel or intersecting.

Parallel planes are two planes that do not intersect.

CG and EF# # # # # # # # # # # # # # #

H

E

G

DC

BA

F

Ex: Plane ABC and Plane EFG

are parallel planes

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Examples:

1. Name all segments that are parallel to2. Name all segments that intersect 3. Name all segments that are skew to4. Name all planes that are parallel to plane BCG.

Answers:5. Segments BC, FG, & EH.6. Segments DH, DC, AE & AB.7. Segments CG, BF, FE, & GH.8. Plane ADH. H

E

G

DC

BA

F

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Example:

1) What are the lines parallel to CD? 

   2) What lines are skew to CD?  

3) Name the parallel plane to CDA.  

𝐴𝐵 𝐸𝐹

𝐻𝐴 𝐺𝐵 ´𝐸𝐻 𝐹𝐺

𝐸𝐹𝐺

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Transversal

Definition: A line that intersects two or more lines in a plane at different points is called a transversal.

When a transversal t intersects line n and m, EIGHT angles are formed

Exterior angles: Outside the lines

Interior angles : Between the lines

tm

n

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Corresponding Angles

Corresponding Angles: Two angles, on the same side of the transversal, that occupy corresponding positions, one interior and one exterior.

2 and 6, 1 and 5, 3 and 7, 4 and 8

1 2

3 4

5 6

7 8

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Alternate Angles

Alternate Interior Angles: Two angles that lie between the lines on opposite sides of the transversal (but not a linear pair).

Alternate Exterior Angles: Two angles that lie outside the lines on opposite sides of the transversal.

3 and 6, 4 and 5

2 and 7, 1 and 81 2

3 4

5 6

7 8

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Consecutive Angles (aka Same-Side)

Consecutive Interior Angles: Two angles that lie between the lines, both on the same side of the transversal.

Consecutive Exterior Angles: Two angles that lie outside the lines, both on the same side of the transversal.

3 and 5 , 4 and 6

1 and 7 , 2 and 8 1 2

3 4

5 6

7 8

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Vertical Angles & Linear Pair

Vertical Angles:

Linear Pair:

1 4, 2 3, 5 8, 6 7

Two angles that are opposite angles. Vertical angles are congruent.

1 & 2 , 2 & 4 , 4 &3, 3 & 1, 5 & 6, 6 & 8, 8 & 7, 7 & 5

Supplementary angles that form a straight line (sum = 180)

1 2

3 4

5 67 8

ExampleList all pairs that fit the description

a. Corresponding

b. Alternate Exterior

c. Alternate Interior

d. Consecutive Interior

< 4 and < 2 < 3 and < 1 < 5 and < 7 < 6 and < 8

< 2 and < 6 < 3 and < 7

< 4 and < 8 < 1 and < 5

< 3 and < 2 < 6 and < 7

ExampleComplete the statement with corresponding, alternate exterior, alternate interior, or consecutive interior.

1. < 4 and < 8 are

2. < 2 and < 6 are

3. < 1 and < 8 are

4. < 7 and < 2 are

5. < 4 and < 6 are

6. < 5 and < 7 are

Alternate interior

Consecutive exterior

Corresponding

Vertical

Alternate exterior

Consecutive interior

Take Home Message

Parallel and skew lines

When a transversal intersects two other lines, 8 different angles are formed Alternate vs Consecutive (same-side) Interior vs. Exterior