3-1 lines and angles
Post on 31-Dec-2015
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A
B
DC
What would you call two lines which do not intersect?
Parallel
A solid arrow placed on two lines of a diagram indicate the lines are parallel.
The symbol || is used to indicate parallel lines.
AB || CD
Skew Lines
Two lines are skew if they are not in the same plane and do not intersect.
AB does not intersect CD .
Since the lines are not in the same plane, they are skew lines.
A
BC
D
Vocabulary
Parallel lines are coplanar lines that do not intersect.
Lines in different planes that do not intersect are skew.
Parallel planes are planes that do not intersect.
Perpendicular lines – lines that intersect and form right angles
Identifying Types of Lines and Planes
Identify each of the following.
A. a pair of parallel segments
B. a pair of skew segments
C. a pair of perpendicular segments
D. a pair of parallel planes
LM ||QR
KN and PQ
NS SP
plane NMR || plane KLQ
Identify each of the following.
a. a pair of parallel segments
b. a pair of skew segments
d. a pair of parallel planes
c. a pair of perpendicular segments
BF || EJ
BF and DE are skew.
BF FJ
plane FJH || plane BCD
Identifying relationships in space
Think of each segment in the diagram. Which appear to fit the description?
a. Parallel to AB and contains D
b. Perpendicular to AB and contains D
c. Skew to AB and contains Dd. Name the plane(s) that
contains D and appear to be parallel to plane ABE
A
B
D
C
F
E H
G
Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
P
l
Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the given point perpendicular to the given line.
l
P
3-1 Lines and Angles
Today you will learn to identify angles formed by parallel lines and a transversal.
Transversals
If you have 2 coplanar lines that are intersected by a third (called a transversal), special angle pairs are formed.
Transversal
Combined Angles
The types of angles mentioned previously are only useful to us in certain combinations:– Alternate interior– Same side interior– Alternate exterior – Same side exterior
(rarely used)
Corresponding Angles
If two angles occupy the same relative position at each of the points of intersection, they are corresponding angles.
Which pairs of angles are corresponding?
1 2
3 4
5 6
7 8
Parallel Lines & Transversals
Parallel lines are lines in the same plane that do not intersect.
When a transversal intersects parallel lines, the special angles pairs take on certain properties.
b
a
ba
These arrows are how parallel lines are marked on a diagram.
Parallel Lines Cut by a Transversal
Get into pairs. On a sheet of paper, draw two parallel lines (either both horizontal or vertical).
Now use your ruler to draw a transversal that intersects both parallel lines.
13
24
5 678
Label these pairs of angles:1 & 54 & 62 & 8
Parallel Postulate
(Remember, a postulate is something we accept as true without proof.)
If parallel lines are intersected by a transversal, then the corresponding angles are congruent.
In other words, if a ll b, then 1 2.
b
a1
2
Parallel Lines & Transversal
Parallel lines and transversals form special angles:
Corresponding angles are congruent
Alternate interior angles are congruent
Same side interior angles are supplementary
Alternate exterior angles are congruent
A Special Case
If there are a pair of parallel lines, and a transversal is perpendicular to one of them, then it is perpendicular to the other.
If a ll b and a t, thenb t.
b
a
t
21
3 4
5 6
7 8
Be Able to Name Special Pairs of Angles:
1. Alternate Interior Angles
2. Corresponding Angles
3. Alternate Exterior
4. Same-Side Interior Angles
Be Able to State the Relationship Between Any Two Angles:
1. Congruent Angles
2. Supplementary Angles
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