Download Parallel Lines

Lesson 3-1
Parallel Lines and
Transversals
1
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 ||.
A
C
B
D
In the above figure, the arrows show that line AB is parallel to
line CD.
With symbols we denote, AB CD .
2
PERPENDICULAR LINES



Perpendicular lines are lines that intersect to form a
right angle.
The symbol used for perpendicular lines is  .
4 right angles are formed.
m
In this figure line m is perpendicular to line n.
n
With symbols we denote, m  n
3
OBLIQUE LINES


Oblique lines are lines that intersect, but do NOT form
a right angle.
m  n
4
Skew Lines and Parallel Planes


Two lines are skew if they do not intersect and are not in the
same plane (not coplanar).
Ex: CG and EF
All planes are either parallel or intersecting. Parallel planes are
two planes that do not intersect.
A
Ex: Plane ABC and Plane EFG
D
B
C
E
F
H
5
G
Examples:
1.
2.
3.
4.
Name all segments that are parallel to AD
Name all segments that intersect AD
Name all segments that are skew to AD
Name all planes that are parallel to plane ABC.
A
Answers:
1. Segments BC, FG, & EH.
2. Segments DH, DC, AE & AB.
3. Segments CG, BF, FE, & GH.
4. Plane FGH.
D
B
C
E
F
H
G
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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
of the following types are formed:
m
Exterior angles
Interior angles
Consecutive interior angles
Alternative exterior angles
Alternative interior angles
Corresponding angles
t
n
7
Vertical Angles & Linear Pair
Vertical Angles: Two angles that are opposite angles. Vertical
angles are congruent.
 1   4,  2   3,  5   8,  6   7
Linear Pair: Supplementary angles that form a line (sum = 180)
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5
1
2
3 4
5
7
6
8
8
Angles and Parallel Lines

1.
2.
3.

1.
2.
If two parallel lines are cut by a transversal, then the
following pairs of angles are congruent.
Corresponding angles
Alternate interior angles
Alternate exterior angles
If two parallel lines are cut by a transversal, then the
following pairs of angles are supplementary.
Consecutive interior angles
Consecutive exterior angles
Continued…..
9
Corresponding Angles & Consecutive Angles
Corresponding Angles: Two angles that occupy
corresponding positions.
 2   6,  1   5,  3   7,  4   8
1
3
2
4
5 6
7 8
10
Consecutive Angles
Consecutive Interior Angles: Two angles that lie between parallel
lines on the same sides of the transversal.
m3 +m5 = 180º, m4 +m6 = 180º
Consecutive Exterior Angles: Two angles that lie outside parallel
lines on the same sides of the transversal.
1
m1 +m7 = 180º, m2 +m8 = 180º
3
4
5
7
2
6
8
11
Alternate Angles

Alternate Interior Angles: Two angles that lie between parallel
lines on opposite sides of the transversal (but not a linear pair).
 3   6,  4   5

Alternate Exterior Angles: Two angles that lie outside parallel
lines on opposite sides of the transversal.  2   7,  1   8
1
3
4
5
7
2
6
8
12
Example: If line AB is parallel to line CD and s is parallel to t, find
the measure of all the angles when m< 1 = 100°. Justify your answers.
A
1
4
C
5
8
m<2=80° m<3=100° m<4=80°
2
12
3
6
10
11
B
D
13 14
16 15
7
s
9
t
m<5=100° m<6=80° m<7=100° m<8=80°
m<9=100° m<10=80° m<11=100° m<12=80°
m<13=100° m<14=80° m<15=100° m<16=80°
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Example: If line AB is parallel to line CD and s is parallel to t, find:
1. the value of x, if m<3 = 4x + 6 and the m<11 = 126.
2. the value of x, if m<1 = 100 and m<8 = 2x + 10.
3. the value of y, if m<11 = 3y – 5 and m<16 = 2y + 20.
ANSWERS:
1. 30
4
C
5
8
2. 35
s
3. 33
1
A
2
9
12
3
6
B
10
11
D
13 14
16 15
7
t
14