3.1: lines and angles 1. todays objectives to identify relationships between figures in space to...
DESCRIPTION
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 ||. 3TRANSCRIPT
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3.1: Lines and Angles
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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
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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
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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
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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