3.1: lines and angles 1. todays objectives to identify relationships between figures in space to...
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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
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